TWO-CARDINAL DERIVED TOPOLOGIES, INDESCRIBABILITY AND RAMSEYNESS

Abstract

We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

1 Introduction

The derived set of a subset A of a topological space $(X,\tau )$ is the collection $d(A)$ of all limit points of A in the space. We refer to the function d as the Cantor derivative of the space $(X,\tau )$ . Recently, Bagaria showed [2] that the derived topologies on ordinals, whose definition we review now, are closely related to certain widely studied stationary reflection properties and large cardinal notions. Suppose $\delta $ is an ordinal and $\tau _0$ is the order topology on $\delta $ . That is, $\tau _0$ is the topology on $\delta $ generated by $B_0=\{\{0\}\}\cup \{(\alpha ,\beta )\mid \alpha <\beta \leq \delta \}$ . For a set $A\subseteq \delta $ , it easily follows that the collection $d_0(A)$ of all limit points of A in the space $(\delta ,\tau _0)$ , is equal to $\{\alpha <\delta \mid \ A\ \text {is unbounded in}\ \alpha \}$ . Beginning with the interval topology on $\delta $ and declaring more and more derived sets to be open, Bagaria [2] introduced the sequence of derived topologies $\langle \tau _\xi \mid \xi <\delta \rangle $ on $\delta $ . For example, $\tau _1$ is the topology on $\delta $ generated by $B_1=B_0\cup \{d_0(A)\mid A\subseteq \delta \}$ , and $\tau _2$ is the topology on $\delta $ generated by $B_2=B_1\cup \{d_1(A)\mid A\subseteq \delta \}$ where $d_1$ is the Cantor derivative of the space $(\delta ,\tau _1)$ . Bagaria showed that limit points of sets in the spaces $(\delta ,\tau _\xi )$ , for $\xi \in \{1,2\}$ , can be characterized as follows. For $A\subseteq \delta $ and $\alpha <\delta $ : $\alpha $ is a limit point of A in $(\delta ,\tau _1)$ if and only if A is stationary in $\alpha $ , and $\alpha $ is a limit point of A in $(\delta ,\tau _2)$ if and only if whenever S and T are stationary subsets of $\alpha $ there is a $\beta \in A$ such that $S\cap \beta $ and $T\cap \beta $ are stationary subsets of $\beta $ . Furthermore, Bagaria proved that limit points of sets in the spaces $(\delta ,\tau _\xi )$ for $\xi>2$ can be characterized in terms of an iterated form of pairwise simultaneous stationary reflection called $\xi $ -s-stationarity.

In this article we address the following natural question: is there some analogue of the sequence of derived topologies on an ordinal in the two-cardinal setting? Specifically, suppose $\kappa $ is an uncountable cardinal and X is a set of ordinals with $\kappa \subseteq X$ . Is there a topology $\tau $ on $P_\kappa X$ such that, for all $A \subseteq P_\kappa X$ , the limit points of A in the space $(P_\kappa X,\tau )$ are precisely the points $x\in P_\kappa X$ such that the set A satisfies:

• • Some unboundedness condition at x?

• • Some stationarity condition at x?

• • Some pairwise simultaneous stationary reflection-like condition at x?

Recall that, for an infinite cardinal $\kappa $ and a set X, $P_\kappa X = \{x \subseteq X \mid |x| < \kappa \}$ . Given $x \in P_\kappa X$ , we denote $|x \cap \kappa |$ by $\kappa _x$ . For $x,y\in P_\kappa X$ we say that x is a strong subset of y and write $x\prec y$ if $x\subseteq y$ and $|x|<\kappa _y$ . Let us note that the ordering $\prec $ , and its variants, are used in the context of supercompact Prikry forcings [19]. In Section 3.1, we show that the ordering $\prec $ induces a natural topology $\tau _0$ on $P_\kappa X$ analogous to the order topology on an ordinal $\delta $ . Furthermore, beginning with $\tau _0$ and following the constructions of [2], in Section 3.2 we define a sequence of derived topologies $\langle \tau _\xi \mid \xi <\kappa \rangle $ on $P_\kappa X$ . Let us note that after submitting the current article, the authors learned that Torres, working under the supervision of Bagaria, simultaneously and independently defined a sequence of two-cardinal derived topologies and obtained results similar to those in Sections 3.2–3.6 involving the relationship between various two-cardinal notions of $\xi $ -s-stationarity and two-cardinal derived topologies.

We show (see Propositions 3.10 and 3.14) that in the space $(P_\kappa X,\tau _1)$ , for $x\in P_\kappa X$ with $\kappa _x=x\cap \kappa $ an inaccessible cardinal, x is a limit point of a set $A\subseteq P_\kappa X$ if and only if A is strongly stationary in $P_{x\cap \kappa }x$ (see Section 2 for the definition of strongly stationary set). Let us note that although the notion of strong stationarity is distinct from the widely popular notion of two-cardinal stationarity introduced by Jech [23] (see [11, Lemma 2.2]), it has previously been studied by several authors [10, 11, 25, 26, 31]. The analogy with the case of derived topologies on ordinals continues: in the space $(P_\kappa X,\tau _2)$ , when $x\in P_\kappa X$ is such that $\kappa _x=x\cap \kappa <\kappa $ and $P_{x\cap \kappa }x$ satisfies a two-cardinal version of $\Pi ^1_1$ -indescribability, x is a limit point of a set $A\subseteq P_\kappa X$ if and only if for every pair $S,T$ of strongly stationary subsets of $P_{\kappa \cap x}x$ there is a $y\prec x$ in A with $y\cap \kappa <x\cap \kappa $ such that S and T are both strongly stationary in $P_{y\cap \kappa }y$ (see Proposition 3.29). Additionally, using a different method, we show (see Corollary 3.35) that if $\kappa $ is weakly inaccessible and X is a set of ordinals with $\kappa \subseteq X$ , then there is a topology on $P_\kappa X$ such that for $A\subseteq P_\kappa X$ , $x\in P_\kappa X$ is a limit point of A if and only if $\kappa _x$ is weakly inaccessible and A is stationary in $P_{\kappa _x}x$ in the sense of Jech [23].

In order to prove the characterizations of limit points of sets in the spaces $(P_\kappa X,\tau _\xi )$ (Theorem 3.16(1)), we introduce new iterated forms of two-cardinal stationarity and two-cardinal pairwise simultaneous stationary reflection, which we refer to as $\xi $ -strong stationarity and $\xi $ -s-strong stationarity (see Definition 3.7). Let us note that the notions of $\xi $ -strong stationarity and $\xi $ -s-strong stationarity introduced here are natural generalizations of notions previously studied in the cardinal context by Bagaria, Magidor, and Sakai [4], Bagaria [2], and by Brickhill and Welch [8], as well as those previously studied in the two-cardinal context by Sakai [28], by Torres [30], as well as by Benhamou and the third author [7].

We establish some basic properties of the ideals associated with $\xi $ -strong stationarity and $\xi $ -s-strong stationarity and introduce notions of $\xi $ -weak club and $\xi $ -s-weak club which provide natural filter bases for the corresponding filters (see Corollary 3.19). The consistency of the non-discreteness of the derived topologies $\tau _\xi $ on $P_\kappa X$ is obtained using various two-cardinal indescribability hypotheses, all of which follow from appropriate local instances of supercompactness (see Section 3.5). We also show that by restricting our attention to a certain natural club subset of $P_\kappa X$ , some questions about the resulting spaces, such as questions regarding when particular subbases are in fact bases, become more tractable (see Section 3.6).

Additionally, in Section 4, we answer several questions asked by the first author and Holy [15] and the first author and White [16] concerning the relationship between Ramseyness and indescribability. Suppose $\kappa $ is an uncountable cardinal. For example, answering [15, Question 10.9] in the affirmative, we show that the existence of an uncountable cardinal $\kappa $ such that for every regressive function $f:[\kappa ]^{<\omega }\to \kappa $ there is a set $H\subseteq \kappa $ which is positive for the Ramsey ideal and homogeneous for f, is strictly stronger in consistency strength than the existence of a cardinal $\kappa $ such that for every regressive function $f:[\kappa ]^{<\omega }\to \kappa $ there is a set $H\subseteq \kappa $ that is positive for the $\Pi ^1_1$ -indescribability ideal and homogeneous for f.

2 Strong stationarity and weak clubs

An ideal I on $P_\kappa X$ is strongly normal if whenever $S\in I^+$ and $f:S\to P_\kappa X$ is such that $f(x)\prec x$ for all $x\in S$ , then there is some $T\in P(S)\cap I^+$ such that $f\upharpoonright T$ is constant. It is easy to see that an ideal I is strongly normal if and only if the dual filter $I^*$ is closed under $\prec $ -diagonal intersections in the following sense: whenever $A_x\in I^*$ for all $x\in P_\kappa X$ , the $\prec $ -diagonal intersection

$$\begin{align*}\bigtriangleup_\prec\{A_x\mid x\in P_\kappa X\}=\{y\in P_\kappa X\mid y\in\bigcap_{x\prec y}A_x\}\end{align*}$$

is in $I^*$ . Carr, Levinski, and Pelletier [10] showed that there is a strongly normal ideal on $P_\kappa X$ if and only if $\kappa $ is a Mahlo cardinal or $\kappa =\mu ^+$ for some cardinal $\mu $ with $\mu ^{< \mu }=\mu $ . Furthermore, they proved that when a strongly normal ideal exists on $P_\kappa X$ , the minimal such ideal is that consisting of the non-strongly stationary subsets of $P_\kappa X$ , which are defined as follows. Given a function $f:P_\kappa X\to P_\kappa X$ we let

$$\begin{align*}B_f=\{x\in P_\kappa X\mid x\cap\kappa\neq\emptyset\land f[P_{\kappa_x}x]\subseteq P(x)\}.\end{align*}$$

A set $S\subseteq P_\kappa X$ is strongly stationary in $P_\kappa X$ if for all $f:P_\kappa X\to P_\kappa X$ we have $S\cap B_f\neq \emptyset $ . The non-strongly stationary ideal on $P_\kappa X$ is the collection

$$\begin{align*}{\mathop{\mathrm{NSS}}}_{\kappa,X}=\{X\subseteq P_\kappa X\mid\ X\ \text{is not strongly stationary}\}.\end{align*}$$

Thus, when $\kappa $ is Mahlo or $\kappa =\mu ^+$ where $\mu ^{< \mu }=\mu $ , the ideal ${\mathop{\mathrm{NSS}}}_{\kappa ,X}$ is the minimal strongly normal ideal on $P_\kappa X$ .

When $\kappa $ is Mahlo, we can identify a filter base for the filter dual to ${\mathop{\mathrm{NSS}}}_{\kappa ,X}$ consisting of sets which are, in a sense, cofinal in $P_\kappa X$ and satisfy a certain natural closure property. We say that a set $C\subseteq P_\kappa X$ is $\prec $ -cofinal in $P_\kappa X$ if for all $x\in P_\kappa X$ there is a $y\in C$ such that $x\prec y$ . A set $C\subseteq P_\kappa X$ is said to be a weak club in $P_\kappa X$ if C is $\prec $ -cofinal in $P_\kappa X$ and $\prec $ -closed in $P_\kappa X$ , meaning that for all $x\in P_\kappa X$ , if C is $\prec $ -cofinal in $P_{\kappa _x}x$ then $x\in C$ . Given a function $f:P_\kappa X\to P_\kappa X$ let

$$\begin{align*}C_f=\{x\in P_\kappa X\mid x\cap\kappa\neq\emptyset\land f[P_{\kappa_x}x]\subseteq P_{\kappa_x}x\}.\end{align*}$$

Fact 2.1. If $\kappa $ is a Mahlo cardinal, the sets

$$\begin{align*}\mathcal{C}_0&=\{B_f\mid f:P_\kappa X\to P_\kappa X\},\\\mathcal{C}_1&=\{C_f\mid f:P_\kappa X\to P_\kappa X\}\end{align*}$$

and

$$\begin{align*}\mathcal{C}_2=\{C\subseteq P_\kappa X\mid\ C\ \text{is a weak club in}\ P_\kappa X\}\end{align*}$$

generate the same filter on $P_\kappa X$ , namely, the filter ${\mathop{\mathrm{NSS}}}_{\kappa ,X}^*$ dual to the ideal ${\mathop{\mathrm{NSS}}}_{\kappa ,X}$ . Hence, when $\kappa $ is a Mahlo cardinal, a set $S\subseteq P_\kappa X$ is strongly stationary in $P_\kappa X$ if and only if $S\cap C\neq \emptyset $ for every C which is a weak club in $P_\kappa X$ .

Proof By definition, the filter on $P_\kappa X$ generated by ${\mathcal C}_0$ is ${\mathop{\mathrm{NSS}}}_{\kappa ,X}^*$ .

Let us show that the filter generated by ${\mathcal C}_1$ equals that generated by ${\mathcal C}_2$ . Suppose $C\in {\mathcal C}_2$ is a weak club in $P_\kappa X$ . Define $f:P_\kappa X\to P_\kappa X$ by letting $f(x)$ be some member of C with $x\prec f(x)$ . Then $C_f\subseteq C$ because if $x\in C_f$ then C is $\prec $ -cofinal in $P_{\kappa _x}x$ and hence $x\in C$ .

For the other direction, we fix a function $g:P_\kappa X\to P_\kappa X$ and show that $C_g$ is a weak club in $P_\kappa X$ . First let us check that $C_g$ is $\prec $ -cofinal in $P_\kappa X$ . Fix $x\in P_\kappa X$ . We define an increasing chain $\langle x_\eta \mid \eta <\kappa \rangle $ in $P_\kappa X$ as follows. Let $x_0=x$ . Given $x_\eta $ we choose $x_{\eta +1}\in P_\kappa X$ with $\kappa _{x_{\eta +1}}=x_{\eta +1}\cap \kappa $ and $\bigcup g[P_{\kappa _{x_\eta }}x_\eta ]\prec x_{\eta +1}$ . When $\eta <\kappa $ is a limit ordinal we let $x_\eta =\bigcup _{\alpha <\eta }x_\alpha $ . Then $\langle \kappa _{x_\eta }\mid \eta <\kappa \rangle $ is a strictly increasing sequence in $\kappa $ and the set

$$\begin{align*}C=\{\eta<\kappa\mid(\forall\zeta<\eta)\kappa_{x_\zeta}<\eta\}=\{\eta<\kappa\mid\kappa_{x_\eta}=\eta\}\end{align*}$$

is a club in $\kappa $ . Since $\kappa $ is Mahlo, we can fix some regular $\kappa _{x_\eta }=\eta \in C$ . Clearly $x\prec x_\eta $ . Let us show that $x_\eta \in C_g$ . Suppose $a\in P_{\kappa _{x_\eta }}x_\eta $ . Since $\kappa _{x_\eta }=\eta $ is regular, $|a|<\kappa _{x_\eta }$ implies that $a\in P_{\kappa _{x_\zeta }}x_\zeta $ for some $\zeta <\eta $ , and therefore

$$\begin{align*}g(a)\subseteq \bigcup g[P_{\kappa_{x_\zeta}}x_\zeta]\prec x_{\zeta+1},\end{align*}$$

which implies $g(a)\in P_{\kappa _{x_\eta }}x_\eta $ and hence $x_\eta \in C_g$ . Since $x\prec x_\eta $ , it follows that $C_g$ is $\prec $ -cofinal.

Now we verify that $C_g$ is $\prec $ -closed in $P_\kappa X$ . Suppose $C_g\cap P_{\kappa _x}x$ is $\prec $ -cofinal in $P_{\kappa _x}x$ . We must show that $x\in C_g$ . Suppose $y\in P_{\kappa _x}x$ . Then there is some $z\in C_g$ with $y\prec z\prec x$ . Thus $g(y)\prec z\prec x$ and hence $x\in C_g$ .

Now let us verify that the filter generated by ${\mathcal C}_0$ equals that generated by ${\mathcal C}_1$ . For any function $f:P_\kappa X\to P_\kappa X$ we have $C_f\subseteq B_f$ , so the filter generated by ${\mathcal C}_0$ is contained in the filter generated by ${\mathcal C}_1$ . Let us fix a function $g:P_\kappa X\to P_\kappa X$ . We must show that there is a function $h:P_\kappa X\to P_\kappa X$ such that $B_h\subseteq C_g$ . Define $h:P_\kappa X\to P_\kappa X$ by letting $h(x)$ be some member of $C_g$ with $g(x)\prec h(x)$ , for all $x\in P_\kappa X$ . Suppose $x\in B_h$ . To show $x\in C_g$ , suppose $y\prec x$ . Then it follows that $g(y)\prec h(y)\subseteq x$ , which implies $g(y)\prec x$ and thus $x\in C_g$ . Therefore $B_h\subseteq C_g$ and hence the filter generated by ${\mathcal C}_0$ equals the filter generated by ${\mathcal C}_1$ .

We end this section by discussing the more common variants of “club” and “stationary” subsets of $P_\kappa X$ , introduced by Jech in [23]. Recall that, for a regular cardinal $\kappa $ and a set $X \supseteq \kappa $ , a set $C \subseteq P_\kappa X$ is said to be a club in $P_\kappa X$ if it is $\subseteq $ -cofinal in $P_\kappa X$ and, whenever $D \subseteq C$ is a $\subseteq $ -linearly ordered set of cardinality less than $\kappa $ , we have $\bigcup D \in C$ . This latter requirement is equivalent to the following formal strengthening: whenever $D \subseteq C$ is $\subseteq $ -directed and $|D| < \kappa $ , we have $\bigcup D \in C$ . We then say that a set $S \subseteq P_\kappa X$ is stationary if, for every club C in $P_\kappa X$ , we have $S \cap C \neq \emptyset $ . The following basic observation justifies the use of the name “weak club” for the notion thusly designated above.

Proposition 2.2. If $\kappa $ is weakly inaccessible, $X \supseteq \kappa $ is a set of ordinals, and C is a club in $P_\kappa X$ , then C is a weak club in $P_\kappa X$ .

Proof Suppose that C is a club in $P_\kappa X$ . Since $\kappa $ is a limit cardinal, the fact that C is $\subseteq $ -cofinal implies that it is also $\prec $ -cofinal. To verify closure, fix $x \in P_\kappa X$ such that C is $\prec $ -cofinal in $P_{\kappa _x} x$ . Then it is straightforward to construct a set $D \subseteq C \cap P_{\kappa _x} x$ such that:

• • D is $\subseteq $ -directed;

• • $|D| \leq |x| < \kappa $ ;

• • $\bigcup D = x$ .

Since C is a club, it follows that $\bigcup D = x \in C$ . Thus, C is a weak club.

3 Two-cardinal derived topologies and $\xi $ -strong stationarity

Fix for this section an arbitrary regular uncountable cardinal $\kappa $ and a set of ordinals $X \supseteq \kappa $ . We will investigate a sequence of derived topologies $\langle \tau _\xi \mid \xi < \kappa \rangle $ on $P_\kappa X$ , simultaneously isolating a hierarchy of stationary reflection principles that characterize the existence of limit points with respect to these topologies. We emphasize that all definitions and arguments in this section are in the context of the ambient space of $P_\kappa X$ . We begin by describing $\tau _0$ , a generalization of the order topology.

3.1 A generalization of the order topology to $P_\kappa X$

Given $x,y\in P_\kappa X$ with $x\prec y$ , let

$$\begin{align*}(x,y]=\{z\in P_\kappa X\mid x\prec z\prec y \lor z=y\}\end{align*}$$

and

$$\begin{align*}(x,y)=\{z\in P_\kappa X\mid x\prec z\prec y\}.\end{align*}$$

Let $\tau _0$ be the topology on $P_\kappa X$ generated by

$$\begin{align*}{\mathcal B}_0=\{(x,y]\mid x,y\in P_\kappa X\land x\prec y\} \cup \{\{y\} \mid y \in P_\kappa X\land \kappa_y = 0\}.\end{align*}$$

It is easy to see that ${\mathcal B}_0$ is a base for $\tau _0$ : If $y \in P_\kappa X$ and $\kappa _y = 0$ , then $\{y\} \in {\mathcal B}_0$ . If $0 < \kappa _y < \omega $ , then, letting x be any subset of y of cardinality $\kappa _y - 1$ , we have $\{y\} = (x,y] \in {\mathcal B}_0$ . Finally, if $\kappa _y$ is infinite and $y\in (a_0,b_0]\cap \cdots \cap (a_{n-1},b_{n-1}]$ , then $y\in (a,y]\subseteq (a_0,b_0]\cap \cdots \cap (a_{n-1},b_{n-1}]$ where $a=\bigcup _{i < n}a_i$ .

For $A \subseteq P_\kappa X$ , let

$$\begin{align*}d_0(A)=\{x\in P_\kappa X\mid\ x\ \text{is a limit point of}\ A\ \text{in}\ (P_\kappa X,\tau_0)\}.\end{align*}$$

Proposition 3.1. For every $A \subseteq P_\kappa X$ ,

$$\begin{align*}d_0(A)=\{x\in P_\kappa X\mid\ A\ \text{is}\ \prec \!{\text{-}} \text{cofinal in}\ P_{\kappa_x}x\}.\end{align*}$$

Proof Fix $A \subseteq P_\kappa X$ and $x\in d_0(A)$ . By the above argument that ${\mathcal B}_0$ is a base for $\tau _0$ , it follows that $\kappa _x\ge \omega $ . Suppose that $y\in P_{\kappa _x}x$ . Since $(y,x]$ is an open neighborhood of x, we can choose a $z\in (y,x]\cap A$ with $z\neq x$ . This implies $z\in (y,x)\cap A$ , and hence A is $\prec $ -cofinal in $P_{\kappa _x}x$ . Conversely, suppose A is $\prec $ -cofinal in $P_{\kappa _x}x$ and let $(a,b]$ be a basic open neighborhood of x. Then $a\in P_{\kappa _x}x$ and we may choose some $y\in A$ with $a\prec y\in P_{\kappa _x}x$ . Hence $y\in (a,b]\cap A\setminus \{x\}$ .

Corollary 3.2. A point $x\in P_\kappa X$ is not isolated in $\tau _0$ if and only if $\kappa _x=|x\cap \kappa |$ is a limit cardinal.

The following proposition connects the order topology $\tau _0$ on $P_\kappa X$ to the notion of weak club discussed in Section 2, in the case where $\kappa $ is a weakly Mahlo cardinal.

Proposition 3.3. If $\kappa $ is weakly Mahlo and A is $\prec $ -cofinal in $P_\kappa X$ then $d_0(A)$ is a weak club in $P_\kappa X$ .

Proof First let us show that $d_0(A)$ is $\prec $ -closed. Suppose $d_0(A)$ is $\prec $ -cofinal in $P_{\kappa _x}x$ . Fix $a\in P_{\kappa _x}x$ and let $b\in d_0(A)\cap (a,x)$ . Then A is $\prec $ -cofinal in $P_{\kappa _b}b$ , so we may choose $c\in A\cap (a,b)\subseteq A\cap (a,x)$ and hence $x\in d_0(A)$ .

Let us show that $d_0(A)$ is $\prec $ -cofinal in $P_\kappa X$ . Fix $x\in P_\kappa X$ . We define an increasing chain $\langle x_\eta \mid \eta <\kappa \rangle $ in $P_\kappa X$ as follows. Let $x_0=x$ . Given $x_\eta $ choose $x_{\eta +1}\in A$ with $x_\eta \prec x_{\eta +1}$ . If $\eta <\kappa $ is a limit let $x_\eta =\bigcup _{\zeta <\eta }x_\zeta $ . Then $\langle \kappa _{x_\eta }\mid \eta <\kappa \rangle $ is a strictly increasing sequence in $\kappa $ and the set

$$\begin{align*}C=\{\eta<\kappa\mid (\forall\zeta<\eta)\ \kappa_{x_\zeta}<\eta\}=\{\eta<\kappa\mid\kappa_{x_\eta}=\eta\}\end{align*}$$

is a club in $\kappa $ . Thus, since $\kappa $ is weakly Mahlo, we can fix some regular $\kappa _{x_\eta }=\eta \in C$ . Let us show that $x_\eta \in d_0(A)$ . Suppose $a\in P_{\kappa _{x_\eta }}x_\eta $ . Since $\kappa _{x_\eta }=\eta $ is regular, $|a|<\kappa _{x_\eta }$ implies $a\in P_{\kappa _{x_\zeta }}x_\zeta $ for some $\zeta <\eta $ , and therefore $a\in P_{\kappa _{x_{\zeta +1}}}x_{\zeta +1}$ which entails that $a\prec x_{\zeta +1}\in A$ .

Recall that an ordinal $\delta $ has uncountable cofinality if and only if for every $A\subseteq \delta $ which is unbounded in $\delta $ , there is an $\alpha <\delta $ such that A is unbounded in $\alpha $ . The following proposition is the analogous result for the notion of $\prec $ -cofinality in $P_\kappa X$ when $\kappa $ is weakly inaccessible.

Proposition 3.4. If $\kappa $ is weakly inaccessible, then the following are equivalent.

1. (1) $\kappa $ is a weakly Mahlo cardinal.

2. (2) For all $A\subseteq P_\kappa X$ if A is $\prec $ -cofinal in $P_\kappa X$ then there is an $x\in P_\kappa X$ such that A is $\prec $ -cofinal in $P_{\kappa _x}x$ .

Proof The fact that (1) implies (2) follows from Proposition 3.3. Let us show that (2) implies (1). We assume (2) holds, and that $\kappa $ is weakly inaccessible but not weakly Mahlo. Let $C \subseteq \kappa $ be a club consisting of singular cardinals, and let ${D=\{x\in P_\kappa X\mid x\cap \kappa \in C\}}$ . Then D is $\prec $ -cofinal in $P_\kappa X$ . By (2), there is a ${y\in P_\kappa X}$ such that D is $\prec $ -cofinal in $P_{\kappa _y}y$ . Then $\kappa _y$ is a limit cardinal and C is cofinal in $\kappa _y$ , so $\kappa _y\in C$ is a singular cardinal.

Let us argue that $y\cap \kappa $ is an ordinal less than $\kappa $ . Suppose $\alpha \in y\cap \kappa $ . Since D is $\prec $ -cofinal in $P_{\kappa _y}y$ there is some $z\in D\cap P_{\kappa _y}y$ such that $\{\alpha \}\prec z$ . Thus $\alpha \in z$ and $z\cap \kappa $ is an ordinal, which implies $\alpha \subseteq z\cap \kappa \subseteq y\cap \kappa $ . Hence $y\cap \kappa $ is transitive.

Let $a\subseteq \kappa _y$ be cofinal in $\kappa _y$ with $|a|=\mathop{\mathrm{cf}}(\kappa _y)<\kappa _y$ . Since $y\cap \kappa $ is an ordinal we have $a\subseteq \kappa _y=|y\cap \kappa |\subseteq y\cap \kappa $ and thus $a\in P_{\kappa _y}y$ . However, there is no $x\in D\cap P_{\kappa _y}y$ with $a\prec x$ because for such an x, $\kappa \cap x\in C$ would be an ordinal containing the set a which is cofinal in $\kappa _y$ , and hence $\kappa _x\geq \kappa _y$ .

We note that the assumption that $\kappa $ is weakly inaccessible is necessary in Proposition 3.4, but only for the somewhat trivial reason that, if $\kappa $ is a successor cardinal, then there are no $\prec $ -cofinal subsets of $P_\kappa X$ .

3.2 Definitions of derived topologies and iterated stationarity in $P_\kappa X$

With the topology $\tau _0$ on $P_\kappa X$ , the base ${\mathcal B}_0$ for $\tau _0$ and the Cantor derivative $d_0$ in hand, we can now define the derived topologies on $P_\kappa X$ as follows. Given $\tau _\xi $ , ${\mathcal B}_\xi $ and $d_\xi $ , we let

$$\begin{align*}{\mathcal B}_{\xi+1}={\mathcal B}_\xi\cup\{d_\xi(A)\mid A\subseteq P_\kappa X\},\end{align*}$$

we let $\tau _{\xi +1}$ be the topology generated by ${\mathcal B}_{\xi +1}$ and we let $d_{\xi +1}$ be defined by

$$\begin{align*}d_{\xi+1}(A)=\{x\in P_\kappa X\mid\ x\ \text{is a limit point of}\ A\ \text{in}\ (P_\kappa X,\tau_{\xi+1})\}\end{align*}$$

for $A\subseteq P_\kappa X$ . When $\xi $ is a limit ordinal we let $\tau _\xi $ be the topology generated by ${\mathcal B}_\xi :=\bigcup _{\zeta <\xi }{\mathcal B}_\zeta $ and we let $d_\xi $ be the Cantor derivative of the space $(P_\kappa X,\tau _\xi )$ .

Since ${\mathcal B}_0$ is a base for $\tau _0$ , it easily follows that the sets of the form

$$\begin{align*}I\cap d_{\xi_0}(A_0)\cap\cdots\cap d_{\xi_{n-1}}(A_{n-1}),\end{align*}$$

where $I\in {\mathcal B}_0$ , $n<\omega $ , $\xi _i<\xi $ and $A_i\subseteq P_\kappa X$ for $i<n$ , form a base for $\tau _\xi $ whenever $\xi <\kappa $ . We return to the question of whether or not ${\mathcal B}_\xi $ forms a base for $\tau _\xi $ in Theorem 3.21, as well as in Section 3.6.

Let us note here that the next two lemmas can easily be established using arguments similar to those for [2, Proposition 2.1 and Corollary 2.2].

Lemma 3.5. For all $\zeta <\xi $ and all $A_0,\ldots ,A_n\subseteq P_\kappa X$ ,

$$\begin{align*}d_\zeta(A_0)\cap\cdots\cap d_\zeta(A_{n-1})\cap d_\xi(A_n)=d_\xi(d_\zeta(A_0)\cap\cdots\cap d_\zeta(A_{n-1})\cap A_n).\end{align*}$$

Lemma 3.6. For every ordinal $\xi $ , the sets of the form

$$\begin{align*}I\cap d_\zeta(A_0)\cap\cdots\cap d_\zeta(A_{n-1}),\end{align*}$$

where $I\in {\mathcal B}_0$ , $n<\omega $ , $\zeta <\xi $ and $A_i\subseteq P_\kappa X$ for $i<n$ , form a base for $\tau _\xi $ .

In the next few sections, we will characterize the non-isolated points of the spaces $(P_\kappa X,\tau _\xi )$ in terms of the following two-cardinal notions of $\xi $ -s-strong stationarity.

Definition 3.7.

1. (1) For $A\subseteq P_\kappa X$ and $x\in P_\kappa X$ , we say that A is $0$ -strongly stationary in $P_{\kappa _x}x$ if and only if A is $\prec $ -cofinal in $P_{\kappa _x}x$ . For an ordinal $\xi>0$ , we say that A is $\xi $ -strongly stationary in $P_{\kappa _x}x$ if and only if $\kappa _x$ is a limit cardinal ¹ and, whenever $\zeta < \xi $ and $S\subseteq P_{\kappa _x}x$ is $\zeta $ -strongly stationary in $P_{\kappa _x}x$ , there is some $y\in A\cap P_{\kappa _x}x$ such that S is $\zeta $ -strongly stationary in $P_{\kappa _y}y$ .

2. (2) A set $C\subseteq P_\kappa X$ is a $0$ -weak club in $P_{\kappa _x}x$ if and only if it is $\prec $ -cofinal and $\prec $ -closed in $P_{\kappa _x}x$ . For an ordinal $\xi>0$ , we say that C is a $\xi $ -weak club in $P_{\kappa _x}x$ if and only if it is $\xi $ -strongly stationary in $P_{\kappa _x}x$ and it is $\xi $ -strongly stationary closed in $P_{\kappa _x}x$ , meaning that whenever $y \prec x$ and C is $\xi $ -strongly stationary in $P_{\kappa _y}y$ we have $y\in C$ .

3. (3) We say that A is $0$ -simultaneously strongly stationary in $P_{\kappa _x}x$ or ( $0$ -s-strongly stationary in $P_{\kappa _x}x$ for short) if and only if A is $\prec $ -cofinal in $P_{\kappa _x}x$ . For an ordinal $\xi>0$ , we say that A is $\xi $ -simultaneously strongly stationary in $P_{\kappa _x}x$ (or $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ for short) if and only if $\kappa _x$ is a limit cardinal and, whenever $\zeta <\xi $ and $S,T\subseteq P_\kappa X$ are $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ there is some $y\in A\cap P_{\kappa _x}x$ such that S and T are both $\zeta $ -s-strongly stationary in $P_{\kappa _y}y$ .

4. (4) A set $C\subseteq P_\kappa X$ is a $0$ -s-weak club in $P_{\kappa _x}x$ if and only if it is $\prec $ -cofinal and $\prec $ -closed in $P_{\kappa _x}x$ . For an ordinal $\xi>0$ , we say that C is a $\xi $ -s-weak club in $P_{\kappa _x}x$ if and only if it is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ and it is $\xi $ -s-closed in $P_{\kappa _x}x$ , meaning that whenever $y<x$ and C is $\xi $ -s-strongly stationary in $P_{\kappa _y}y$ we have $y\in C$ .

In what follows, given $x \in P_\kappa X$ and $\xi < \kappa $ , we will simply say that, e.g., $P_{\kappa _x}x$ is $\xi $ -s-strongly stationary to mean that it is $\xi $ -s-strongly stationary in $P_{\kappa _x} x$ . Let us first note the following simple proposition, which justifies the restriction of our attention to values of $\xi $ less than $\kappa $ . By the results of Section 3.5, the proposition is sharp, at least assuming the consistency of certain large cardinals.

Proposition 3.8. For all $x \in P_\kappa X$ , $P_{\kappa _x} x$ is not $(\kappa _x+1)$ -strongly stationary.

Proof Suppose otherwise, and let $x \in P_\kappa X$ be a counterexample such that $\kappa _x$ is minimal among all counterexamples. Since $P_{\kappa _x} x$ is $(\kappa _x+1)$ -strongly stationary, it is a fortiori $\kappa _x$ -strongly stationary. Therefore, by the definition of $(\kappa _x+1)$ -strong stationarity, we can find $y \in P_{\kappa _x} x$ such that $P_{\kappa _x} x$ is $\kappa _x$ -strongly stationary in $P_{\kappa _y} y$ . Since $\kappa _x> \kappa _y$ , this implies that $P_{\kappa _y} y$ is $(\kappa _y + 1)$ -strongly stationary, contradicting the minimality of $\kappa _x$ .

Considering the previous proposition, it is natural to wonder whether the definitions of $\xi $ -strong stationarity and $\xi $ -s-strong stationarity can be modified using canonical functions to allow for settings in which some $x\in P_\kappa X$ can be $\xi $ -strongly stationary for $\kappa _x<\xi <|x|^+$ ; this was done in the cardinal setting by the first author in [13]. See the discussion before Questions 5.7 and 5.8 for more information.

Definition 3.7 leads naturally to the definition of the following ideals, which can be strongly normal under a certain large cardinal hypothesis by Proposition 3.30 below.

Definition 3.9. Suppose that $x \in P_\kappa X$ . We define

$$\begin{align*}\overline{{\mathop{\mathrm{NS}}}}_{\kappa_x,x}^\xi=\{A\subseteq P_\kappa X\mid\ A\ \textrm{is not}\ \xi\text{-}\textrm{strongly stationary in}\ P_{\kappa_x} x\}\end{align*}$$

and

$$\begin{align*}{\mathop{\mathrm{NS}}}_{\kappa_x,x}^\xi=\{A\subseteq P_\kappa X\mid\ A\ \textrm{is not}\ \xi\text{-}\textrm{s-strongly stationary in}\ P_{\kappa_x} x\}.\end{align*}$$

Let us show that for the x’s in $P_\kappa X$ that we will care most about, namely those for which $\kappa _x$ is regular, $1$ -strong stationarity and $1$ -s-strong stationarity are equivalent in $P_{\kappa _x}x$ ; moreover, if $\kappa _x$ is inaccessible, then these notions are equivalent to strong stationarity in $P_{\kappa _x}x$ plus the Mahloness of $\kappa _x$ .

Proposition 3.10. Suppose $A \subseteq P_\kappa X$ and $x\in P_\kappa X$ with $\kappa _x$ regular. Then the following are equivalent, and both imply that $\kappa _x$ is weakly Mahlo.

1. (1) A is $1$ -strongly stationary in $P_{\kappa _x}x$ .

2. (2) A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ .

If, moreover, $\kappa _x$ is strongly inaccessible, then these two statements are also equivalent to the following:

1. (3) $\kappa _x$ is Mahlo and A is strongly stationary in $P_{\kappa _x}x$ .

Proof Note that, if A is $1$ -strongly stationary in $P_{\kappa _x}x$ , then $\kappa _x$ is a limit cardinal and hence weakly inaccessible. We can thus assume that this is the case. (2) $\implies $ (1) is trivial. Let us now assume that A is $1$ -strongly stationary in $P_{\kappa _x} x$ . By Proposition 3.4, it follows that $\kappa _x$ is weakly Mahlo. To see that A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ , fix sets $S_0,S_1 \subseteq P_\kappa X$ that are both $\prec $ -cofinal in $P_{\kappa _x} x$ . Let T be the set of $y \in P_{\kappa _x} x$ such that $S_0$ and $S_1$ are both $\prec $ -cofinal in $P_{\kappa _y} y$ . We claim that T is $\prec $ -cofinal in $P_{\kappa _x} x$ . To see this, fix an arbitrary $y_0 \in P_{\kappa _x} x$ . Define a continuous, $\prec $ -increasing sequence $\langle y_\eta \mid \eta < \kappa _x \rangle $ in $P_{\kappa _x} x$ as follows. The set $y_0$ is already fixed. Given $y_\eta $ , find $z^0_\eta \in S_0$ and $z^1_\eta \in S_1$ such that, for all $i < 2$ , we have $y_\eta \prec z^i_\eta \prec x$ . Then let $y_{\eta + 1} = z^0_\eta \cup z^1_\eta $ . If $\xi < \kappa _x$ is a limit ordinal, let $y_\xi = \bigcup \{y_\eta \mid \eta < \xi \}$ . The set of $\eta < \kappa _x$ for which $\kappa _{y_\eta } = \eta $ is a club in $\kappa _x$ , so, since $\kappa _x$ is weakly Mahlo, we can fix some regular cardinal $\eta < \kappa _x$ such that $\kappa _{y_\eta } = \eta $ . A now-familiar argument then shows that $S_0$ and $S_1$ are both $\prec $ -cofinal in $y_\eta $ , and hence $y_\eta \in T$ .

Since A is $1$ -strongly stationary in $P_{\kappa _x}x$ , we can find $w \in A$ such that T is $\prec $ -cofinal in $P_{\kappa _w}w$ . It follows immediately that $S_0$ and $S_1$ are both $\prec $ -cofinal in $P_{\kappa _w} w$ ; therefore, A is $1$ -s-strongly stationary in $P_{\kappa _x} x$ .

For the “moreover” clause, assume that $\kappa _x$ is strongly inaccessible and A is $1$ -strongly stationary in $P_{\kappa _x} x$ . The fact that $\kappa _x$ is Mahlo follows from the previous paragraphs. To show that A is strongly stationary in $P_{\kappa _x} x$ , suppose C is a weak club subset of $P_{\kappa _x}x$ . Since A is $1$ -strongly stationary there is some $y\in A$ such that C is $\prec $ -cofinal in $P_{\kappa _y}y$ . Since C is weakly closed we have $y\in A\cap C$ .

Finally, suppose $\kappa _x$ is Mahlo and A is strongly stationary in $P_{\kappa _x}x$ . To show that A is $1$ -strongly stationary in $P_{\kappa _x}x$ , fix a set S which is $\prec $ -cofinal in $P_{\kappa _x}x$ . Since $\kappa _x$ is Mahlo, it follows from Proposition 3.3 that $d_0(S)$ is a weak club in $P_{\kappa _x}x$ . Thus, by Fact 2.1, $d_0(S)\cap A$ is strongly stationary in $P_{\kappa _x}x$ and hence there is a $y\in A\cap P_{\kappa _x}x$ such that S is $\prec $ -cofinal in $P_{\kappa _y}y$ .

3.3 The $\tau _1$ topology on $P_\kappa X$

We now discuss the first derived topology $\tau _1$ on $P_\kappa X$ . Recall that this is the topology generated by

$$\begin{align*}{\mathcal B}_1={\mathcal B}_0\cup\{d_0(A)\mid A\subseteq P_\kappa X\}.\end{align*}$$

Remark 3.11. Recall that the subbase for the first derived topology on an ordinal $\delta $ is always a base for that topology (see [2]). By definition, ${\mathcal B}_1$ is a subbase for the first derived topology $\tau _1$ on $P_\kappa X$ , but it is not clear whether it is a base for $\tau _1$ for the following reason. Suppose $x\in I\cap d_0(A_0)\cap \cdots \cap d_0(A_{n-1})$ where $n\geq 1$ . Then each $A_i$ , for $i<n$ , is $\prec $ -cofinal in $P_{\kappa _x}x$ and hence $\kappa _x$ is a limit cardinal. If $\kappa _x$ is a Mahlo cardinal, then it follows by Proposition 3.3 that each $d_0(A_i)$ , for $i<n$ , is a weak club in $P_{\kappa _x}x$ , and hence $I\cap d_0(A_0)\cap \cdots \cap d_0(A_{n-1})$ is a weak club in $P_{\kappa _x}x$ . Thus, $x\in d_0(I\cap d_0(A_0)\cap \cdots \cap d_0(A_{n-1}))\subseteq I\cap d_0(A_0)\cap \cdots \cap d_0(A_{n-1})$ . However, if $\kappa _x$ is not Mahlo, then it is not clear whether $d_0(A_i)$ is a weak club in $P_{\kappa _x}x$ for $i<n$ , and furthermore, it is not clear how to proceed. This difference seems not to create too much difficulty so we proceed with our definition as is, but in Section 3.6 we show that, if we pass to a certain club subset C of $P_\kappa X$ , then the natural restriction of ${\mathcal B}_1$ to C is a base for the subspace topology on C induced by $\tau _1$ .

We will need the following lemma.

Lemma 3.12. Fix $x \in P_\kappa X$ , and suppose that A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ and $A_0,\ldots ,A_{n-1}$ are all $0$ -s-strongly stationary (i.e., $\prec $ -cofinal) in $P_{\kappa _x}x$ , where $n\geq 2$ . Then $d_0(A_0)\cap \cdots \cap d_0(A_{n-1})\cap A$ is $1$ -s-strongly stationary in $P_{\kappa _x}x$ .

Proof First let us use a straightforward inductive argument on $n\geq 2$ to show that whenever $A_0,\ldots ,A_{n-1}$ are $0$ -strongly stationary in $P_{\kappa _x}x$ , the set $d_0(A_0)\cap \cdots \cap d_0(A_{n-1})\cap A$ is $0$ -strongly stationary in $P_{\kappa _x}x$ . Suppose $A_0$ and $A_1$ are $\prec $ -cofinal in $P_{\kappa _x}x$ and note that $\kappa _x$ must be a limit cardinal. To show that $d_0(A_0)\cap d_0(A_1)\cap A$ is $\prec $ -cofinal in $P_{\kappa _x}x$ , fix $y\in P_{\kappa _x}x$ . Then $A_0\cap (y,x)$ and $A_1\cap (y,x)$ are $\prec $ -cofinal in $P_{\kappa _x}x$ . Since A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ there is an $a\in A\cap P_{\kappa _x}x$ such that $A_0\cap (y,x)$ and $A_1\cap (y,x)$ are both $\prec $ -cofinal in $P_{\kappa _a}a$ , and hence $y<a$ . Therefore $a\in d_0(A_0)\cap d_0(A_1)\cap A\cap (y,x)$ . Now suppose the result holds for n, and suppose $A_0,\ldots ,A_{n-1},A_n$ are all $\prec $ -cofinal in $P_{\kappa _x}x$ . By our inductive hypothesis, $d_0(A_0)\cap \cdots \cap d_0(A_{n-1})$ is $\prec $ -cofinal in $P_{\kappa _x}x$ , and by the base case the set $d_0(d_0(A_0)\cap \cdots \cap d_0(A_{n-1}))\cap d_0(A_n)\cap A$ is $\prec $ -cofinal in $P_{\kappa _x}x$ . But

$$ \begin{align*} d_0(d_0(A_0)\cap\cdots\cap d_0(A_{n-1}))\cap d_0(A_n) &\subseteq d_0(d_0(A_0))\cap \cdots\cap d_0(d_0(A_{n-1}))\cap d_0(A_n) \\ &\subseteq d_0(A_0)\cap \cdots\cap d_0(A_{n-1})\cap d_0(A_n). \end{align*} $$

Now we prove the statement of the lemma. Fix sets $A_0,\ldots ,A_{n-1}\subseteq P_\kappa X$ that are $\prec $ -cofinal in $P_{\kappa _x}x$ . To show that $d_0(A_0)\cap \cdots \cap d_0(A_{n-1})\cap A$ is $1$ -s-strongly stationary in $P_{\kappa _x}x$ , fix sets S and T that are $\prec $ -cofinal in $P_{\kappa _x}x$ . By the previous paragraph, it follows that the set

$$\begin{align*}d_0(S)\cap d_0(T)\cap d_0(A_0)\cap\cdots\cap d_0(A_{n-1})\cap A\end{align*}$$

is $\prec $ -cofinal in $P_{\kappa _x}x$ and hence there is some

$$\begin{align*}y\in d_0(S)\cap d_0(T)\cap d_0(A_0)\cap\cdots\cap d_0(A_{n-1})\cap A,\end{align*}$$

which establishes that $d_0(A_0)\cap \cdots \cap d_0(A_{n-1})\cap A$ is $1$ -s-strongly stationary.

Corollary 3.13. Suppose $P_{\kappa _x}x$ is $1$ -s-strongly stationary. Then a set A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ if and only if for every set C which is a $0$ -s-weak club in $P_{\kappa _x}x$ we have $A\cap C\cap P_{\kappa _x}x\neq \emptyset $ .

Proof Suppose A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ and C is a $0$ -s-weak club in $P_{\kappa _x}x$ . Then $d_0(C)\cap P_{\kappa _x}x\subseteq C\cap P_{\kappa _x}x$ and by Lemma 3.12, $d_0(C)\cap A$ is $1$ -s-strongly stationary in $P_{\kappa _x}x$ . Thus $A\cap C\cap P_{\kappa _x}x\neq \emptyset $ . Conversely, assume that $A \cap C\cap P_{\kappa _x}x\neq \emptyset $ whenever C is a $0$ -s-weak club in $P_{\kappa _x}x$ . Fix sets S and T that are $0$ -s-strongly stationary in $P_{\kappa _x}x$ . Then $d_0(S)\cap d_0(T)$ is a $0$ -s-weak club in $P_{\kappa _x}x$ because $d_0(S)\cap d_0(T)\cap P_{\kappa _x}x$ is $1$ -s-strongly stationary and hence $0$ -s-strongly stationary in $P_{\kappa _x}x$ by Lemma 3.12, and $d_0(S)\cap d_0(T)$ is $0$ -s-closed in $P_{\kappa _x}x$ since

$$\begin{align*}d_0(d_0(S)\cap d_0(T))\subseteq d_0(S)\cap d_0(T)\end{align*}$$

as a consequence of the fact that $d_0$ is the limit point operator of the space $(P_\kappa X,\tau _0)$ .

Proposition 3.14. If $A\subseteq P_\kappa X$ then

$$\begin{align*}d_1(A)=\{x\in P_\kappa X\mid\ A\ \text{is}\ 1\text{-}\text{s-strongly stationary in}\ P_{\kappa_x}x\}.\end{align*}$$

Proof Suppose A is not $1$ -s-strongly stationary in $P_{\kappa _x}x$ . If $\kappa _x$ is a successor cardinal then x is isolated in $(P_\kappa X,\tau _1)$ by Corollary 3.2 and hence $x\notin d_1(A)$ . Suppose $\kappa _x$ is a limit cardinal. Then there are sets S and T which are $0$ -strongly stationary in $P_{\kappa _x}x$ such that $d_0(S)\cap d_0(T)\cap A\cap P_{\kappa _x}x=\emptyset $ . Then it follows that $d_0(S)\cap d_0(T)\cap (0,x]$ is an open neighborhood of x in the $\tau _1$ topology that does not intersect A in some point other than x. Hence $x\notin d_1(A)$ .

Conversely, suppose A is $1$ -s-strongly stationary in $P_{\kappa _x}x$ . We must show that x is a limit point of A in the $\tau _1$ topology. Suppose $x\in I\cap d_0(A_0)\cap \cdots \cap d_0(A_{n-1})$ , where $I\in {\mathcal B}_0$ and $A_0,\ldots ,A_{n-1}\subseteq P_\kappa X$ . Then the sets $A_0,\ldots ,A_{n-1}$ are all $\prec $ -cofinal in $P_{\kappa _x}x$ , and by Lemma 3.12, the set $I\cap d_0(A_0)\cap \cdots d_0(A_{n-1})\cap A$ is $\prec $ -cofinal in $P_{\kappa _x}x$ , which implies that the open neighborhood $I\cap d_0(A_0)\cap \cdots \cap d_0(A_{n-1})$ of x intersects A in some point other than x.

Corollary 3.15. A point $x\in P_\kappa X$ is not isolated in $(P_\kappa X,\tau _1)$ if and only if $P_{\kappa _x}x$ is $1$ -s-strongly stationary in $P_{\kappa _x}x$ .

3.4 The $\tau _\xi $ topology on $P_\kappa X$ for $\xi \geq 2$

We now move to the general setting. Let us first characterize limit points of sets in the spaces $(P_\kappa X,\tau _\xi )$ in terms of $\xi $ -s-strong stationarity.

Theorem 3.16. For all $\xi <\kappa $ the following hold.

1. (1)_ξ We have

   $$\begin{align*}d_\xi(A)=\{x\in P_\kappa X\mid\ A\ \textrm{is}\ \xi\text{-}\textrm{s-strongly stationary in}\ P_{\kappa_x}x\}.\end{align*}$$

2. (2)_ξ For all $x\in P_\kappa X$ , a set A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ if and only if for all $\zeta \leq \xi $ and every pair $S,T$ of subsets of $P_{\kappa _x}x$ that are $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ , we have $A\cap d_\zeta (S)\cap d_\zeta (T)\neq \emptyset $ (equivalently $A\cap d_\zeta (S)\cap d_\zeta (T)$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ ).

3. (3)_ξ For all $x\in P_\kappa X$ , if A is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ and $A_i$ is $\zeta _i$ -s-strongly stationary in $P_{\kappa _x}x$ for some $\zeta _i<\xi $ and all $i<n$ , then $A\cap d_{\zeta _0}(A_0)\cap \cdots \cap d_{\zeta _{n-1}}(A_{n-1})$ is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ .

Proof We have already established that $(1)_\xi $ and $(3)_\xi $ hold for $\xi \leq 1$ and $(2)_0$ holds. Given these base cases, the fact that (1), (2), and (3) hold for all $\xi <\kappa $ can be established by simultaneous induction using an argument which is essentially identical to that of [2, Proposition 2.10]. For the reader’s convenience, we include the argument here.

First, suppose $(1)_\zeta $ , $(2)_\zeta $ , and $(3)_\zeta $ hold for all $\zeta $ less than some limit ordinal $\xi <\kappa $ . It is clear that $(1)_\xi $ and $(3)_\xi $ also must hold. Let us prove that $(2)_\xi $ holds. Notice that the backward direction of $(2)_\xi $ easily follows from the definition of $\xi +1$ -s-strong stationarity and the fact that $(1)_\zeta $ holds for $\zeta \leq \xi $ . For the forward direction of $(2)_\xi $ , suppose A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ . Fix $\zeta \leq \xi $ and a pair $S,T$ of $\zeta $ -s-strongly stationary subsets of $P_{\kappa _x}x$ . To show that $A\cap d_\zeta (S)\cap d_\zeta (T)$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ , fix sets $A,B$ that are $\eta $ -s-strongly stationary in $P_{\kappa _x}x$ where $\eta <\zeta $ . Using the fact that (3) holds for $\zeta $ , we see that $S\cap d_\eta (A)\cap d_\eta (B)$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ . Since A is $\xi +1$ -s-strongly stationary, and applying the fact that $(1)_\zeta $ holds, we have

$$\begin{align*}\emptyset\neq d_\zeta(d_\eta(A)\cap d_\eta(B)\cap S)\cap d_\zeta(T)\cap A.\end{align*}$$

But, by Lemma 3.5,

$$\begin{align*}d_\zeta(d_\eta(A)\cap d_\eta(B)\cap S)\cap d_\zeta(T)\cap A = d_\eta(A)\cap d_\eta(B)\cap d_\zeta(S)\cap d_\zeta(T)\cap A.\end{align*}$$

Thus, $d_\zeta (S)\cap d_\zeta (T)\cap A$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ .

One can show that if $(1)_{\leq \xi }$ , $(2)_{\leq \xi }$ , and $(3)_{\leq \xi }$ hold then, by induction on n, $(3)_{\xi +1}$ must also hold. For the reader’s convenience we provide a proof that $(3)_{\xi +1}$ holds for $n=1$ , the remaining argument is the same as [2, Proposition 2.10]. Suppose $n=1$ . To prove that $A\cap d_{\zeta _0}(A_0)$ is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ , fix sets S and T that are $\eta $ -s-strongly stationary in $P_{\kappa _x}x$ for some $\eta \leq \xi $ . By $(1)_{\leq \xi }$ , it will suffice to show that $A\cap d_{\zeta _0}(A_0)\cap d_\eta (S)\cap d_\eta (T)\neq \emptyset $ . If ${\zeta _0}=\eta $ , then by $(2)_{\leq \xi }$ , it follows that the set $A\cap d_{\zeta _0}(A_0)\cap d_\eta (d_\eta (S)\cap d_\eta (T))$ , which is contained in ${A\cap d_{\zeta _0}(A_0)\cap d_\eta (S)\cap d_\eta (T)}$ , is ${\zeta _0}$ -s-strongly stationary in $P_{\kappa _x}x$ , and thus $A\cap d_{\zeta _0}(A_0)\cap d_\eta (S)\cap d_\eta (T)\neq \emptyset $ . If ${\zeta _0}<\eta $ , then by $(3)_\eta $ , if follows that $d_{\zeta _0}(A_0)$ is $\eta $ -s-strongly stationary in $P_{\kappa _x}x$ , and by $(2)_{\xi }$ , the set $A\cap d_\eta (d_{\zeta _0}(A_0))\cap d_\eta (d_\eta (S)\cap d_\eta (T))$ , which is contained in $A\cap d_{\zeta _0}(A_0)\cap d_\eta (S)\cap d_\eta (T)$ , is $\eta $ -s-strongly stationary in $P_{\kappa _x}x$ . If $\zeta _0>\eta $ then by $(2)_\xi $ the set $A\cap d_{\zeta _0}(A_0)$ is $\zeta _0$ -s-strongly stationary in $P_{\kappa _x}x$ and thus $A\cap d_{\zeta _0}(A_0)\cap d_\eta (S)\cap d_\eta (T)\neq \emptyset $ .

Let us prove that if $(1)_{\leq \xi }$ , $(2)_{\leq \xi }$ , and $(3)_{\leq \xi +1}$ hold then $(1)_{\xi +1}$ holds (this argument is similar to that of Proposition 3.14). Suppose A is not $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ . Then by $(1)_{\leq \xi }$ , for some $\zeta \leq \xi $ there are sets S and T which are $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ such that $A\cap d_\zeta (S)\cap d_\zeta (T)=\emptyset $ . Thus $d_\zeta (S)\cap d_\zeta (T)\cap (0,x]$ is an open neighborhood of x in the $\tau _{\xi +1}$ topology that does not intersect A in some point other than x. Conversely, suppose A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ . To show that $x\in d_{\xi +1}(A)$ , let U be an arbitrary basic open neighborhood of x in the $\tau _{\xi +1}$ topology. By Lemma 3.6, we can assume that U is of the form

$$\begin{align*}U=I\cap d_\zeta(A_0)\cap\cdots\cap d_\zeta(A_{n-1}),\end{align*}$$

where $I\in {\mathcal B}_0$ , $n<\omega $ , $\zeta <\xi +1$ and $A_i\subseteq P_\kappa X$ for $i<n$ . Since $x\in U$ it follows from $(1)_\zeta $ that each $A_i$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ , and thus by $(3)_{\xi +1}$ we see that $A\cap d_\zeta (A_0)\cap \cdots \cap d_\zeta (A_{n-1})$ is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ , and thus U intersects A in some point other than x.

Finally, we prove that if $(1)_{\leq \xi +1}$ , $(2)_{\leq \xi }$ , and $(3)_{\leq \xi +1}$ hold, then $(2)_{\xi +1}$ must also hold. Suppose A is $\xi +2$ -s-strongly stationary in $P_{\kappa _x}x$ . By $(2)_{\leq \xi }$ , it suffices to show that whenever S and T are $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ , the set ${A\cap d_{\xi +1}(S)\cap d_{\xi +1}(T)}$ is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ . So, fix Y and Z which are $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ for some $\zeta \leq \xi $ . By $(3)_{\xi +1}$ , it follows that ${S\cap d_\zeta (Y)}$ and $T\cap d_\zeta (Z)$ are $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ , and hence by the $\xi +1$ -s-strong stationarity of A and $(1)_{\xi +1}$ we have $A\cap d_{\xi +1}(S\cap d_\zeta (Y))\cap d_{\xi +1}(T\cap d_\zeta (Z))\neq \emptyset $ . But

$$\begin{align*}d_{\xi+1}(S\cap d_\zeta(Y))\cap d_{\xi+1}(T\cap d_\zeta(Z))=d_{\xi+1}(S)\cap d_{\xi+1}(T)\cap d_\zeta(Y)\cap d_\zeta(Z)\end{align*}$$

by Lemma 3.5, and thus $A\cap d_{\xi +1}(S)\cap d_{\xi +1}(T)$ is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ . The backward direction of $(2)_{\xi +1}$ follows easily from $(1)_{\leq \xi }$ .

Corollary 3.17. Suppose $P_{\kappa _x}x$ is $\xi $ -s-strongly stationary where $\xi \leq \kappa _x$ and A is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ for some $\zeta < \xi $ . Then, for all $\zeta \leq \zeta ' \leq \xi $ , $d_\zeta (A)$ is a $\zeta '$ -s-weak club in $P_{\kappa _x}x$ .

Proof Fix $\zeta '$ with $\zeta \leq \zeta ' \leq \xi $ . It follows from Theorem 3.16(3) that $d_\zeta (A)$ is $\xi $ -s-strongly stationary and hence $\zeta '$ -s-strongly stationary in $P_{\kappa _x} x$ . Furthermore, $d_\zeta (A)$ is $\zeta '$ -s-closed below $P_{\kappa _x}x$ since $d_{\zeta '}(d_\zeta (A))\subseteq d_\zeta (d_\zeta (A))\subseteq d_\zeta (A)$ .

Corollary 3.18. Suppose that $x \in P_\kappa X$ and $\xi \leq \kappa _x$ . Then x is not isolated in $(P_\kappa X, \tau _\xi )$ if and only if $P_{\kappa _x} x$ is $\xi $ -s-strongly stationary.

Proof For the forward direction, suppose that $P_{\kappa _x} x$ is not $\xi $ -s-strongly stationary. Then there is $\zeta < \xi $ and sets $S,T \subseteq P_{\kappa _x}x$ such that S and T are both $\zeta $ -s-strongly stationary in $P_{\kappa _x} x$ but there is no $y \prec x$ such that S and T are both $\zeta $ -s-strongly stationary in $P_{\kappa _y} y$ . Then, by Theorem 3.16(1), we have $d_\zeta (S) \cap d_\zeta (T) = \{x\}$ , so x is isolated in $(P_\kappa X, \tau _\xi )$ .

For the converse, suppose that $P_{\kappa _x} x$ is $\xi $ -s-strongly stationary, and fix an interval $I \in {\mathcal B}_0$ , an $n < \omega $ , ordinals $\xi _0, \ldots , \xi _{n-1} < \xi $ , and sets $A_0, \ldots , A_{n-1} \subseteq P_{\kappa _x} x$ such that

$$\begin{align*}x \in U := I \cap d_{\xi_0}(A_0) \cap \dots \cap d_{\xi_{n-1}}(A_{n-1}). \end{align*}$$

Let $\zeta := \max \{\zeta _i \mid i < n\} < \xi $ . By Corollary 3.17, each of I, $d_{\xi _0}(A_0)$ , …, $d_{\xi _{n-1}}(A_{n-1})$ is a $\zeta $ -s-weak club in $P_{\kappa _x} x$ . By Corollary 3.19, U is also $\zeta $ -s-weak club in $P_{\kappa _x} x$ . In particular, $U \neq \{x\}$ ; hence, x is not isolated in $P_{\kappa _x} x$ .

Corollary 3.19. Suppose $P_{\kappa _x}x$ is $\xi $ -s-strongly stationary where $0<\xi \leq \kappa _x$ .

1. (1) A set A is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ if and only if for all $\zeta <\xi $ we have $A\cap C\neq \emptyset $ for every set $C\subseteq P_{\kappa _x}x$ which is a $\zeta $ -s-weak club in $P_{\kappa _x}x$ . Thus, the filter on $P_{\kappa _x}x$ generated by the collection of all sets which are $\zeta $ -s-weak clubs in $P_{\kappa _x}x$ for some $\zeta <\xi $ is the filter dual to ${\mathop{\mathrm{NS}}}_{\kappa _x,x}^\xi $ .

2. (2) A set A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ if and only if $A\cap C\neq \emptyset $ for every set $C\subseteq P_{\kappa _x}x$ which is a $\xi $ -s-weak club in $P_{\kappa _x}x$ . Thus the filter generated by the $\xi $ -s-weak club subsets of $P_{\kappa _x}x$ is the filter dual to ${\mathop{\mathrm{NS}}}_{\kappa _x,x}^{\xi +1}$ .

Proof We only provide a proof of (1) since the proof of (2) is essentially identical. Suppose A is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ . Fix $\zeta <\xi $ and assume that $C\subseteq P_{\kappa _x}x$ is a $\zeta $ -s-weak club in $P_{\kappa _x}x$ . Since C is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ there is some $y\in d_\zeta (C)\cap A$ , but since $d_\zeta (C)\subseteq C$ we have $y\in C\cap A$ . Conversely, suppose that for all $\zeta <\xi $ and every $C\subseteq P_{\kappa _x}x$ that is a $\zeta $ -s-weak club in $P_{\kappa _x}x$ we have $A\cap C\neq \emptyset $ . To show that A is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ , suppose S and T are $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ for some $\zeta <\xi $ . Then, since we are assuming that $P_{\kappa _x}x$ is $\xi $ -s-strongly stationary, it follows by Theorem 3.16(3) that $d_\zeta (S)\cap d_\zeta (T)$ is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ . Furthermore,

$$\begin{align*}d_\zeta(d_\zeta(S)\cap d_\zeta(T))\subseteq d_\zeta(d_\zeta(S))\cap d_\zeta(d_\zeta(T))\subseteq d_\zeta(S)\cap d_\zeta(T),\end{align*}$$

which implies that $d_\zeta (S)\cap d_\zeta (T)$ is a $\zeta $ -s-weak club in $P_{\kappa _x}x$ . Thus $A\cap d_\zeta (S)\cap d_\zeta (T)\cap P_{\kappa _x}x\neq \emptyset $ , and hence A is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ as desired.

Proposition 3.20. For $x\in P_\kappa X$ and $\xi \leq \kappa _x$ , the set $P_{\kappa _x}x$ is $\xi $ -s-strongly stationary if and only if ${\mathop{\mathrm{NS}}}_{\kappa _x,x}^\xi $ is a nontrivial ideal.

Proof Suppose $P_{\kappa _x}x$ is $0$ -s-strongly stationary. Then ${\mathop{\mathrm{NS}}}_{\kappa _x,x}^0$ is the ideal $I_{\kappa _x,x}$ consisting of all subsets A of $P_{\kappa _x}x$ such that there is some $y\in P_{\kappa _x}x$ with ${A\cap (y,x)=\emptyset} $ . Clearly this is a nontrivial ideal since $P_{\kappa _x}x\notin I_{\kappa _x,x}$ .

Now suppose $\xi>0$ . Let us show that ${\mathop{\mathrm{NS}}}_{\kappa _x,x}^\xi $ is an ideal. Suppose A and B are both not $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ . By Corollary 3.19, there are sets $C_A,C_B\subseteq P_{\kappa _x}x$ such that $C_A$ is a $\zeta _A$ -s-weak club in $P_{\kappa _x}x$ for some $\zeta _A<\xi $ , $C_B$ is a $\zeta _B$ -s-weak club in $P_{\kappa _x}x$ for some $\zeta _B<\xi $ , such that $C_A\cap A=\emptyset $ and ${C_B\cap B=\emptyset }$ . Then $d_{\zeta _A}(C_A)\cap d_{\zeta _B}(C_B) \cap (A\cup B)=\emptyset $ where $d_{\zeta _A}(C_A)\cap d_{\zeta _B}(C_B)$ is a $\zeta $ -s-weak club in $P_{\kappa _x}x$ for $\zeta =\max \{\zeta _A,\zeta _B\}$ because $d_{\zeta _A}(C_A)\cap d_{\zeta _B}(C_B)\cap P_{\kappa _x}x$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ by Theorem 3.16(3) and furthermore

$$\begin{align*}d_\zeta(d_{\zeta_A}(C_A)\cap d_{\zeta_B}(C_B))\subseteq d_\zeta(C_A)\cap d_\zeta(C_B).\\[-34pt] \end{align*}$$

Theorem 3.21. Suppose that $0 < \xi < \kappa $ . Then the following are equivalent:

1. (1) ${\mathcal B}_\xi $ is a base for $\tau _\xi $ ;

2. (2) for every $\zeta \leq \xi $ , every $x \in P_\kappa X$ , and every $A \subseteq P_\kappa X$ , if A is $\zeta $ -strongly stationary in $P_{\kappa _x} x$ , then A is $\zeta $ -s-strongly stationary in $P_{\kappa _x} x$ .

Proof For the forward direction, suppose that (2) fails, and let $\zeta $ , x, and A form a counterexample, with $\zeta $ minimal among all such counterexamples. Note that we must have $\zeta> 0$ .

Claim 3.22. $P_{\kappa _x} x$ is not $\zeta $ -s-strongly stationary.

Proof Suppose otherwise. We will show that A is in fact $\zeta $ -s-strongly stationary, contradicting our choice of A. By Corollary 3.19, it suffices to show that, for all $\eta < \zeta $ and every $\eta $ -s-weak club C in $P_{\kappa _x} x$ , we have $A \cap C \neq \emptyset $ . Fix such $\eta $ and C. Then C is $\eta $ -s-strongly stationary in $P_{\kappa _x} x$ and hence, by the minimality of $\zeta $ , $\eta $ -strongly stationary in $P_{\kappa _x} x$ . Thus, since A is $\zeta $ -strongly stationary, there is $y \in A$ such that C is $\eta $ -strongly stationary in $P_{\kappa _y} y$ and hence, again by the minimality of $\zeta $ , $\eta $ -s-strongly stationary in $P_{\kappa _y} y$ . But then, since C is an $\eta $ -s-weak club in $P_{\kappa _x} x$ , we have $y \in C \cap A$ , as desired.

We can therefore fix an $\eta < \zeta $ and sets $S,T \subseteq P_{\kappa _x} x$ such that S and T are both $\eta $ -s-strongly stationary in $P_{\kappa _x} x$ but there is no $y \in P_{\kappa _x} x$ such that S and T are both $\eta $ -s-strongly stationary in $P_{\kappa _x} x$ . Then we have $d_\eta (S) \cap d_\eta (T) = \{x\}$ , and hence $\{x\} \in \tau _\xi $ . To show that (1) fails, it thus suffices to show that $\{x\} \notin {\mathcal B}_\xi $ .

Since $P_{\kappa _x} x$ is $1$ -strongly stationary, it follows that $\kappa _x$ is a limit cardinal, and hence $\{x\} \notin {\mathcal B}_0$ . Now suppose that $B \subseteq P_{\kappa _x} x$ , $\xi _0 < \xi $ , and $x \in d_{\xi _0}(B)$ . Since $P_{\kappa _x} x$ is not $\zeta $ -s-strongly stationary, it follows that $\xi _0 < \zeta $ and B is $\xi _0$ -s-stationary in $P_{\kappa _x} x$ . By minimality of $\zeta $ , B is $\xi _0$ -stationary in $P_{\kappa _x} x$ , so, since $P_{\kappa _x} x$ is $\zeta $ -strongly stationary, there is $y \in P_{\kappa _x} x$ such that B is $\xi _0$ -strongly stationary in $P_{\kappa _y} y$ . Again by minimality of $\zeta $ , B is $\xi _0$ -s-strongly stationary in $P_{\kappa _y} y$ , so $y \in d_{\xi _0}(B)$ . It follows that $\{x\} \notin {\mathcal B}_\xi $ .

For the backward direction, suppose that (2) holds, and fix $x \in P_\kappa X$ , $I \in {\mathcal B}_0$ , $0 < n < \omega $ , ordinals $\xi _0, \ldots , \xi _{n-1} < \xi $ , and sets $A_0, \ldots , A_{n-1} \subseteq P_{\kappa _x} x$ such that

$$\begin{align*}x \in I \cap d_{\xi_0}(A_0) \cap \dots \cap d_{\xi_{n-1}}(A_{n-1}). \end{align*}$$

Let $\zeta := \max \{\xi _0, \ldots , \xi _{n-1}\} < \xi $ . It follows that $P_{\kappa _x}$ is $\zeta $ -s-strongly stationary. If $P_{\kappa _x} x$ is not $(\zeta + 1)$ -strongly stationary, then there is $A \subseteq P_{\kappa _x} x$ such that $d_\zeta (A) = \{x\}$ . We can therefore assume that $P_{\kappa _x} x$ is $(\zeta + 1)$ -strongly stationary and hence, by (2), $(\zeta +1)$ -s-strongly stationary. But then it follows that $I \cap d_{\xi _0}(A_0) \cap \dots \cap d_{\xi _{n-1}}(A_{n-1})$ is a $\zeta $ -s-weak club in $P_{\kappa _x} x$ . In particular, it is $\zeta $ -s-strongly stationary in $P_{\kappa _x} x$ , so

$$\begin{align*}x \in d_\zeta(I \cap d_{\xi_0}(A_0) \cap \dots \cap d_{\xi_{n-1}}(A_{n-1})) \subseteq I \cap d_{\xi_0}(A_0) \cap \dots \cap d_{\xi_{n-1}}(A_{n-1}), \end{align*}$$

and $d_\zeta (I \cap d_{\xi _0}(A_0) \cap \dots \cap d_{\xi _{n-1}}(A_{n-1})) \in {\mathcal B}_\xi $ . Therefore, ${\mathcal B}_\xi $ is a base for $\tau _\xi $ .

3.5 Consequences of $\Pi ^1_\xi $ -indescribability

In this section we establish the consistency of the $\xi $ -s-strong stationarity of $P_{\kappa _x}x$ , for $\xi \leq \kappa _x$ , using a two-cardinal version of transfinite indescribability.

The classical notion of $\Pi ^m_n$ -indescribability studied by Levy [24] was generalized to the two-cardinal setting in a set of handwritten notes by Baumgartner (see [9, Section 4]). More recently, various transfinite generalizations of classical $\Pi ^1_n$ -indescribability, involving certain infinitary formulas have been studied in the cardinal context [2–4, 12, 13, 15] and in the two-cardinal context [11].

Let us review the definition of $\Pi ^1_\xi $ -indescribability in the two-cardinal context used in [11]. For the reader’s convenience, we review the notion of $\Pi ^1_\xi $ formula introduced in [2]. Recall that a formula of second-order logic is $\Pi ^1_0$ , or equivalently $\Sigma ^1_0$ , if it does not have any second-order quantifiers, but it may have finitely many first-order quantifiers and finitely many first- and second-order free variables. We use the standard convention that uppercase letters denote second-order variables, unless other specification is given. Bagaria inductively defined the notion of $\Pi ^1_\xi $ formula for any ordinal $\xi $ as follows. A formula is $\Sigma ^1_{\xi +1}$ if it is of the form

$$\begin{align*}\exists X_0\dots\exists X_k\varphi(X_0,\ldots,X_k),\end{align*}$$

where $\varphi $ is $\Pi ^1_\xi $ , and a formula is $\Pi ^1_{\xi +1}$ if it is of the form

$$\begin{align*}\forall X_0\dots\forall X_k\varphi(X_0,\ldots, X_k),\end{align*}$$

where $\varphi $ is $\Sigma ^1_\xi $ . If $\xi $ is a limit ordinal, we say that a formula is $\Pi ^1_\xi $ if it is of the form

$$\begin{align*}\bigwedge_{\zeta<\xi}\varphi_\zeta,\end{align*}$$

where $\varphi _\zeta $ is $\Pi ^1_\zeta $ for all $\zeta <\xi $ and the infinite conjunction has only finitely many free second-order variables. We say that a formula is $\Sigma ^1_\xi $ if it is of the form

$$\begin{align*}\bigvee_{\zeta<\xi}\varphi_\zeta, \end{align*}$$

where $\varphi _\zeta $ is $\Sigma ^1_\zeta $ for all $\zeta <\xi $ and the infinite disjunction has only finitely many free second-order variables.

The two-cardinal definition of $\Pi ^1_\xi $ -indescribability below uses the following two-cardinal version of the usual $V_\alpha $ -hierarchy below a fixed cardinal $\kappa $ . Suppose $\kappa $ is an uncountable regular cardinal and X is a set of ordinals with $|X|\geq \kappa $ . For $\alpha \leq \kappa $ we define

$$ \begin{align*} V_0(\kappa,X)&=X,\\ V_{\alpha+1}(\kappa,X)&=P_\kappa(V_\alpha(\kappa,X))\cup V_\alpha(\kappa,X),\text{ and}\\ V_\alpha(\kappa,X)&=\bigcup_{\eta<\alpha} V_\alpha(\kappa,X)\ \text{if}\ \alpha\ \text{is a limit.} \end{align*} $$

Clearly $V_\kappa \subseteq V_\kappa (\kappa ,X)$ and if X is transitive then so is $V_\alpha (\kappa ,X)$ for $\alpha \leq \kappa $ . Furthermore, both $P_\kappa X$ and $P_\kappa X\times P_\kappa X$ are subsets of $V_\kappa (\kappa ,X)$ . For more regarding the expressive power of $\Pi ^1_\xi $ formulas over structures of the form $(V_\kappa (\kappa ,X),\in ,R_0,\ldots ,R_{n-1})$ , where $R_0,\ldots ,R_{n-1}\subseteq V_\kappa (\kappa ,X)$ , one may consult [1, Section 3] or [11].

Definition 3.23 [11].

For $\xi <\kappa $ we say that $S\subseteq P_\kappa X$ is $\Pi ^1_\xi $ -indescribable in $P_\kappa X$ if for any $R_0,\ldots ,R_{n-1}\subseteq V_\kappa (\kappa ,X)$ and any $\Pi ^1_\xi $ sentence $\varphi $ such that

$$\begin{align*}(V_\kappa(\kappa,X),\in,R_0,\ldots,R_{n-1})\models\varphi,\end{align*}$$

there is an $x\in S$ such that $x\cap \kappa =\kappa _x$ and

$$\begin{align*}(V_{\kappa_x}(\kappa_x,x),\in,R_0\cap V_{\kappa_x}(\kappa_x,x),\ldots,R_{n-1}\cap V_{\kappa_x}(\kappa_x,x))\models\varphi.\end{align*}$$

The collection

$$\begin{align*}\Pi^1_\xi(\kappa,X)=\{A\subseteq P_\kappa X\mid\ A\ \textrm{is not}\ \Pi^1_\xi\text{-}\textrm{indescribable in}\ P_\kappa X\}\end{align*}$$

is called the $\Pi ^1_\xi $ -indescribability ideal on $P_\kappa X$ .

Standard arguments, which we omit, establish the consistency of two-cardinal indescribability from supercompactness (see, for example, [1, Theorem D], [11, Corollary 5.5] and [15, Proposition 3.11]).

Proposition 3.24. Suppose $\kappa $ is $\lambda $ -supercompact where $\kappa \leq \lambda $ and $\lambda ^{<\kappa }=\lambda $ . Then $P_\kappa \lambda $ is $\Pi ^1_\xi $ -indescribable for all $\xi <\kappa $ . Furthermore, the set

$$\begin{align*}\{x\in P_\kappa\lambda\mid\ \kappa_x=x\cap\kappa\ \textrm{and}\ P_{\kappa_x}x \textrm{is}\ \Pi^1_\xi\text{-}\textrm{indescribable for all} \xi<\kappa_x\}\end{align*}$$

is in any normal measure U on $P_\kappa \lambda $ .

Abe [1, Lemma 4.1] showed that if $P_\kappa X$ is $\Pi ^1_n$ -indescribable then $\Pi ^1_n(\kappa ,X)$ is a strongly normal ideal on $P_\kappa X$ . As pointed out in [16], a straightforward application of the arguments for [1, Lemma 4.1] and [2, Proposition 4.4], which is left to the reader, establishes the following.

Proposition 3.25. For $\xi <\kappa $ , if $P_\kappa X$ is $\Pi ^1_\xi $ -indescribable then $\Pi ^1_\xi (\kappa ,X)$ is a strongly normal ideal on $P_\kappa X$ .

Next we show that the $\xi $ -s-strong stationarity of a set S in $P_{\kappa _x}x$ can be expressed by a $\Pi ^1_\xi $ formula.

Lemma 3.26. For all $\xi <\kappa $ there is a $\Pi ^1_\xi $ formula $\Phi _\xi (R,S,T)$ with three free second-order variables such that for $x\in P_\kappa X$ , a set $A\subseteq P_{\kappa _x}x$ is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ if and only if

$$\begin{align*}(V_{\kappa_x}(\kappa_x,x),\in,A,P_{\kappa_x}x,\prec_x)\models\Phi_\xi[A,P_{\kappa_x}x,\prec_x],\end{align*}$$

where $\prec _x$ denotes the usual strong subset ordering $\prec $ restricted to $P_{\kappa _x}x$ .

Proof We proceed by induction on $\xi $ . We let $\Phi _0(R,S,T)$ be the $\Pi ^1_0$ formula

$$\begin{align*}(\forall y\in S) (\exists x\in R)\ (y,x)\in T\end{align*}$$

so that $\Phi _0[A,P_{\kappa _x}x,\prec ]$ expresses that A is $0$ -s-strongly stationary (i.e., $\prec $ -cofinal) in $P_{\kappa _x}x$ over the structure $(V_{\kappa _x}(\kappa _x,x),\in ,A,P_{\kappa _x}x,\prec )$ .

Suppose $\xi $ is a limit ordinal. It is easy to see that $\Phi _\xi = \bigwedge _{\zeta <\xi }\Phi _\zeta $ is as desired.

Suppose $\xi =\zeta +1$ . Let $\Phi _\zeta $ be the $\Pi ^1_\zeta $ -formula obtained from the induction hypothesis. Then for all $x\in P_\kappa X$ a set $A\subseteq P_{\kappa _x}x$ is $\zeta $ -s-strongly stationary in $P_{\kappa _x}x$ if and only if

$$\begin{align*}(V_{\kappa_x}(\kappa_x,x),\in,A,P_{\kappa_x}x,\prec_x)\models\Phi_\zeta[A,P_{\kappa_x}x,\prec_x].\end{align*}$$

For $x\in P_\kappa X$ we see that $A\subseteq P_{\kappa _x}x$ is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ if and only if

$$\begin{align*}\Phi_\zeta[A,P_{\kappa_x}x,\prec_x] \land (\forall S\subseteq P_{\kappa_x}x)(\forall T\subseteq P_{\kappa_x}x)[\Phi_\zeta[S,P_{\kappa_x}x,\prec_x]\land\Phi_\zeta[T,P_{\kappa_x}x,\prec_x]\longrightarrow\end{align*}$$

$$\begin{align*}(\exists y\in A) \Phi_\zeta[S\cap P_{\kappa_y}y,P_{\kappa_y}y,\prec_y]\land \Phi_\zeta[T\cap P_{\kappa_y}y,P_{\kappa_y}y,\prec_y]]\end{align*}$$

holds in $(V_{\kappa _x}(\kappa _x,x),\in ,A,P_{\kappa _x}x,\prec _x)$ . It is easy to check that the previous formula is equivalent to a $\Pi ^1_\xi $ formula, hence the desired formula $\Phi _\xi (R,S,T)$ exists.

Corollary 3.27. For $x\in P_\kappa X$ with $\kappa _x=x\cap \kappa $ , if $A\subseteq P_{\kappa _x}x$ is $\Pi ^1_\xi $ -indescribable in $P_{\kappa _x}x$ then A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ .

Proof To show that A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ , fix sets $S,T\subseteq P_{\kappa _x}x$ that are $\zeta $ -s-strongly stationary where $\zeta \leq \xi $ and let $\Phi _\zeta $ be the $\Pi ^1_\zeta $ -formula obtained from Lemma 3.26. Since A is $\Pi ^1_\xi $ -indescribable, it is $\Pi ^1_\zeta $ -indescribable and the fact that

$$\begin{align*}(V_{\kappa_x}(\kappa_x,x),\in,S,T,P_{\kappa_x}x,\prec_x)\models\Phi_\zeta[S,P_{\kappa_x}x,\prec_x]\land\Phi_\zeta[T,P_{\kappa_x}x,\prec_x]\end{align*}$$

implies that there is some $y\in A\cap P_{\kappa _x}x$ with $\kappa _y=y\cap \kappa $ such that the structure

$$\begin{align*}(V_{\kappa_y}(\kappa_y,y),\in,S\cap P_{\kappa_y}y,T\cap P_{\kappa_y}y,P_{\kappa_y}y,\prec_y)\end{align*}$$

satisfies

$$\begin{align*}\Phi_\zeta[S\cap P_{\kappa_y}y,P_{\kappa_y}y,\prec_y]\land\Phi_\zeta[T\cap P_{\kappa_y}y,P_{\kappa_y}y,\prec_y],\end{align*}$$

and hence S and T are $\zeta $ -s-strongly stationary in y. Therefore A is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ .

Corollary 3.28. For $\xi <\kappa $ , if there is an $x\in P_\kappa X$ such that $P_{\kappa _x}x$ is $\Pi ^1_\xi $ -indescribable then the $\tau _{\xi +1}$ -topology on $P_\kappa X$ is not discrete.

Proposition 3.29. Suppose $P_\kappa X$ is $\Pi ^1_1$ -indescribable. Then a set $A\subseteq P_\kappa X$ is $2$ -s-strongly stationary in $P_\kappa X$ if and only if for every pair $S,T$ of strongly stationary subsets of $P_\kappa X$ there is an $x\in A$ such that $x\cap \kappa =\kappa _x$ is a Mahlo cardinal and the sets S and T are both strongly stationary in $P_{\kappa _x}x$ .

Proof Suppose A is $2$ -s-strongly stationary in $P_\kappa X$ . Fix sets S and T that are strongly stationary in $P_\kappa X$ . The fact that $\kappa $ is Mahlo and the sets S and T are strongly stationary in $P_\kappa X$ can be expressed by a $\Pi ^1_1$ sentence:

$$\begin{align*}(V_\kappa(\kappa,X),\in,P_\kappa X,S,T)\models\varphi.\end{align*}$$

The set

$$\begin{align*}C=\{x\in P_\kappa X\mid (V_{\kappa_x}(\kappa_x,x),\in,P_{\kappa_x}x,S\cap V_{\kappa_x}(\kappa_x,x),T\cap V_{\kappa_x}(\kappa_x,x))\models\varphi\}\end{align*}$$

is in the filter $\Pi ^1_1(\kappa ,X)^*$ . Thus C is, in particular, strongly stationary in $P_\kappa X$ and so by Lemma 3.10 we see that C is $1$ -s-strongly stationary in $P_\kappa X$ . Since A is $2$ -s-strongly stationary in $P_\kappa X$ , there is an $x\in A\cap C$ and it follows that $\kappa _x$ is Mahlo and the sets S and T are strongly stationary in $P_{\kappa _x}x$ .

Conversely, to show that A is $2$ -s-strongly stationary in $P_\kappa X$ , fix sets Q and R that are $1$ -s-strongly stationary in $P_\kappa X$ . By Lemma 3.10, Q and R are strongly stationary in $P_\kappa X$ . Thus, by assumption, there is an $x\in A$ such that $x\cap \kappa =\kappa _x$ is Mahlo and the sets Q and R are both strongly stationary in $P_{\kappa _x}x$ . By Lemma 3.10, Q and R are both $1$ -s-strongly stationary in $P_{\kappa _x}x$ . Hence A is $2$ -s-strongly stationary in $P_\kappa X$ .

Proposition 3.30. For $x\in P_\kappa X$ with $x\cap \kappa =\kappa _x$ , if $P_{\kappa _x}x$ is $\Pi ^1_\xi $ -indescribable where $\xi <\kappa _x$ , then the ideal ${\mathop{\mathrm{NS}}}_{\kappa _x,x}^{\xi +1}$ (see Definition 3.9) is strongly normal.

Proof For each $z\in P_{\kappa _x}X$ choose $C_z\in ({\mathop{\mathrm{NS}}}^{\xi +1}_{\kappa _x,x})^*$ . Without loss of generality, by Corollary 3.19, we may assume that each $C_z$ is a $\xi $ -s-weak club in $P_{\kappa _x}x$ .

Since each $C_z$ is in the filter $\Pi ^1_\xi (\kappa _x,x)^*$ and $\Pi ^1_\xi (\kappa _x,x)$ is strongly normal, it follows that the set $C=\bigtriangleup _\prec \{C_z\mid z\in P_{\kappa _x}x\}$ is in the filter $\Pi ^1_\xi (\kappa _x,x)^*$ and thus C is $\xi +1$ -s-strongly stationary in $P_{\kappa _x}x$ by Corollary 3.27. By Theorem 3.16(2), it follows that $d_\xi (C)$ is $\xi $ -s-strongly stationary in $P_{\kappa _x}x$ , and since $d_\xi $ is the Cantor derivative of the space $(P_\kappa X,\tau _\xi )$ , it follows that $d_\xi (d_\xi (C))\subseteq d_\xi (C)$ and hence $d_\xi (C)$ is a $\xi $ -s-weak club in $P_\kappa X$ . Thus it will suffice to show that $d_\xi (C)\subseteq C$ .

Let us verify that $d_\xi (C)\subseteq \bigtriangleup _\prec \{d_\xi (C_z)\mid z\in P_{\kappa _x}x\}$ . Suppose $y\in d_\xi (C)$ , then the set $\bigtriangleup _\prec \{C_z\mid z\in P_{\kappa _x}x\}$ is $\xi $ -s-strongly stationary in $P_{\kappa _y}y$ . To show that ${y\in \bigtriangleup _\prec \{d_\xi (C_z)\mid z\in P_{\kappa _x}x\}}$ we must verify that $y\in \bigcap _{z\prec y}d_\xi (C_z)$ . Fix $z_0\prec y$ , then $(z_0,y)\cap \bigtriangleup _\prec \{C_z\mid z\in P_{\kappa _x}x\}\subseteq C_{z_0}$ and since $(z_0,y)\cap \bigtriangleup _\prec \{C_z\mid z\in P_{\kappa _x}x\}$ is $\xi $ -s-strongly stationary in $P_{\kappa _y}y$ we see that $y\in d_\xi (C_{z_0})$ . Thus $d_\xi (C)\subseteq \bigtriangleup _\prec \{d_\xi (C_z)\mid z\in P_{\kappa _x}x\}$ .

Since each $C_z$ is a $\xi $ -s-weak club in $P_{\kappa _x}x$ , it follows that $d_\xi (C_z)\subseteq C_z$ and thus

$$\begin{align*}d_\xi(C)\subseteq\bigtriangleup_\prec\{d_\xi(C_z)\mid z\in P_{\kappa_x}x\}\subseteq \bigtriangleup_\prec\{C_z\mid z\in P_{\kappa_x}x\}=C.\\[-34pt] \end{align*}$$

3.6 Variations

In this subsection, we investigate a couple of variations on the sequence of derived topologies considered above. First, we show that by restricting our attention to a certain natural club subset of $P_\kappa X$ , certain questions about the resulting spaces become more tractable.

Let $P^{\prime }_\kappa X$ be the set of $x \in P_\kappa X$ for which $\kappa _x = x \cap \kappa $ . Similarly, if $x \in P^{\prime }_\kappa X$ , then $P^{\prime }_{\kappa _x} x = P^{\prime }_\kappa X \cap P_{\kappa _x} x$ . If $\kappa $ is weakly inaccessible, then $P^{\prime }_\kappa X$ is evidently a club, and hence a weak club, in $P_\kappa X$ . It follows that, if $\xi < \kappa $ , $x \in P_\kappa X$ , and $\kappa _x$ is weakly inaccessible, then

(1) $$ \begin{align} (P_{\kappa_x} x\ \text{is}\ \xi\text{-}\text{s-stationary}) \Longleftrightarrow (P^{\prime}_{\kappa_x} x\ \text{is}\ \xi\text{-}\text{s-stationary in } P_{\kappa_x} x). \end{align} $$

For each $\xi < \kappa $ , let $\tau ^{\prime }_\xi $ be the subspace topology on $P^{\prime }_\kappa X$ induced by $\tau _\xi $ , and let ${\mathcal B}^{\prime }_\xi = \{U \cap P^{\prime }_\kappa X \mid U \in {\mathcal B}_\xi \}$ ; it follows that $\tau ^{\prime }_\xi $ is the topology on $P^{\prime }_\kappa X$ generated by ${\mathcal B}^{\prime }_\xi $ .

Proposition 3.31. Suppose that $x \in P^{\prime }_\kappa X$ . Then the following are equivalent:

1. (1) $\kappa _x$ is weakly inaccessible;

2. (2) x is not isolated in $(P^{\prime }_\kappa X, \tau ^{\prime }_0)$ .

Proof If $\kappa _x$ is weakly inaccessible and $y \prec x$ , with $y \in P_\kappa X$ , then, letting $\lambda $ be the least cardinal with $|y| < \lambda $ , we have $y \cup \lambda \in (y,x] \cap P^{\prime }_\kappa X$ . The implication (1) $\implies $ (2) follows immediately.

For the converse, suppose first that $\kappa _x = \lambda ^+$ is a successor cardinal, and let $y \prec x$ be such that $|y| = \lambda $ . Then $(y,x] = \{x\}$ , so x is isolated in $\tau _0$ , and hence also in $\tau ^{\prime }_0$ . Suppose next that $\kappa _x$ is singular, and let $y \subseteq \kappa _x$ be a cofinal subset such that $|y| = \mathop{\mathrm{cf}}(\kappa _x)$ . Then $(y,x] \cap P^{\prime }_\kappa X = \{x\}$ , so x is isolated in $\tau ^{\prime }_0$ .

Using this proposition, we can establish the following characterization of when ${\mathcal B}^{\prime }_\xi $ forms a base for $\tau ^{\prime }_\xi $ . Since the proof is essentially the same as that of Theorem 3.21, we leave it to the reader.

Theorem 3.32. Suppose that $0 < \xi < \kappa $ . Then the following are equivalent:

1. (1) ${\mathcal B}^{\prime }_\xi $ is a base for $\tau ^{\prime }_\xi $ ;

2. (2) for every $\zeta \leq \xi $ , every $x \in P^{\prime }_\kappa X$ for which $\kappa _x$ is weakly inaccessible, and every $A \subseteq P_\kappa X$ , if A is $\zeta $ -strongly stationary in $P_{\kappa _x} x$ , then A is $\zeta $ -s-strongly stationary in $P_{\kappa _x} x$ .

Corollary 3.33. ${\mathcal B}^{\prime }_1$ is a base for $\tau ^{\prime }_1$ .

Proof This is immediate from Proposition 3.10 and Theorem 3.32.

We saw above that the topology $(P_\kappa X, \tau _1)$ can be characterized by specifying that, if $x \in P_\kappa X$ and $A \subseteq P_\kappa X$ , then x is a limit point of A if and only if A is strongly $1$ -s-stationary in $P_{\kappa _x} x$ . By Proposition 3.10, if $\kappa _x$ is regular, then this is equivalent to A being $1$ -strongly stationary in $P_{\kappa _x} x$ , and if $\kappa _x$ is Mahlo, it is in turn equivalent to A being strongly stationary in $P_{\kappa _x} x$ . One can ask if there is a variant on this topology in which limit points are characterized by stationarity in the sense of [23] (recall the discussion at the end of Section 2). We now show that the answer is positive as long as $\kappa $ is weakly inaccessible and one only requires this of $x \in P_\kappa X$ for which $\kappa _x$ is weakly inaccessible. We first establish the following proposition.

Proposition 3.34. Suppose that $\kappa $ is weakly inaccessible, $A \subseteq P_\kappa X$ and the set

$$\begin{align*}\{x \in P_\kappa X \mid \kappa_x \text{ is regular and } A \cap P_{\kappa_x} x \text{ is stationary in } P_{\kappa_x}x\} \end{align*}$$

is stationary in $P_\kappa X$ . Then A is stationary.

Proof Fix a club C in $P_\kappa X$ . Since $\kappa $ is regular and uncountable, by [27, Theorem 1.5], we can find a function $f : [X]^2 \rightarrow P_\kappa X$ such that $B_f \subseteq C$ , where

$$\begin{align*}B_f := \{x \in P_\kappa X \mid f"[x]^2 \subseteq P(x)\}. \end{align*}$$

We actually get slightly more. Namely, let $C_f$ be the set of $x \in P_\kappa X$ for which $f"[x]^2 \subseteq P_{\kappa _x}x$ . Then clearly $C_f \subseteq B_f \subseteq C$ , and, moreover, $C_f$ is a club in $P_\kappa X$ . To see this, simply note that $C_f$ is clearly closed and, if $\langle y_n \mid n < \omega \rangle $ is a $\prec $ -increasing sequence of elements of $B_f$ , then $\bigcup \{y_n \mid n < \omega \} \in C_f$ , so $C_f$ is cofinal in $P_\kappa X$ . (This is where we use the fact that $\kappa $ is weakly inaccessible, and hence a limit cardinal.)

By assumption, we can find $x \in P_\kappa X$ such that:

1. (1) $\kappa _x$ is regular;

2. (2) $A \cap P_{\kappa _x} x$ is stationary in $P_{\kappa _x} x$ ;

3. (3) $x \in C_f$ .

Since $x \in C_f$ , we know that $g := f \restriction [x]^2$ satisfies $g:[x]^2 \rightarrow P_{\kappa _x}x$ . Since $\kappa _x$ is regular, it follows that $B_g$ is a club in $P_{\kappa _x}x$ . Then item (2) above implies that $B_g \cap A \cap P_{\kappa _x} x \neq 0$ . Since $B_g \subseteq B_f \subseteq C$ , it follows that $C \cap A \neq 0$ . The choice of C was arbitrary, and hence A is stationary in $P_\kappa X$ .

Note that Proposition 3.34 fails if $\kappa> \aleph _1$ is a successor cardinal. Indeed, if $\kappa = \nu ^+> \aleph _1$ , then $A = P_\nu X$ satisfies the hypothesis of Proposition 3.34 but is not stationary in $P_\kappa X$ .

Now, if $\kappa $ is weakly inaccessible, define a function $c:P(P_\kappa X) \rightarrow P(P_\kappa X)$ by letting

$$\begin{align*}c(A) = A \cup \{x \in P_\kappa X \mid \kappa_x \text{ is weakly inaccessible and } A \cap P_{\kappa_x} x \text{ is stationary in } P_{\kappa_x} x\}. \end{align*}$$

Proposition 3.34 implies that c is a closure operator. If $\tau $ is the topology

$$\begin{align*}\{U\subseteq P_\kappa X\mid c(P_\kappa X\setminus U)=P_\kappa X\setminus U\}\end{align*}$$

on $P_\kappa X$ generated by c, then, clearly $\tau $ is a witness to the following.

Corollary 3.35. If $\kappa $ is weakly inaccessible and X is a set of ordinals with $\kappa \subseteq X$ , then there is a topology $\tau $ on $P_\kappa X$ such that for $A\subseteq P_\kappa X$ , x is a limit point of A if and only if $\kappa _x$ is weakly inaccessible and $A\cap P_{\kappa _x}x$ is stationary in $P_{\kappa _x}x$ . In particular, x is a nonisolated point of the space $(P_\kappa X,\tau )$ if and only if $\kappa _x$ is weakly inaccessible.

4 On Ramseyness and indescribability

In this section we answer questions concerning the relationship between Ramseyness and indescribability, which were raised by the first author and Holy [15] in the context of cardinals, and by the first author and White [16] in the two-cardinal context. We provide detailed arguments in the cardinal context and simply state definitions and results in the two-cardinal context since the proofs are similar.

Let us review the definition and some basic properties of canonical functions. We follow the definitions and notation given in [18]. The sequence of canonical functions $\langle f_\alpha \mid \alpha <\lambda ^+\rangle $ is a sequence of canonical representatives of the ordinals less than $\lambda ^+$ in the generic ultrapower obtained by forcing with any normal ideal I on $Z\subseteq P(\lambda )$ . We recursively define $\langle f_\alpha \mid \alpha <\lambda ^+\rangle $ as follows. For $\alpha <\lambda $ we let

$$\begin{align*}f_\alpha(z)=\mathop{\mathrm{ot}}\nolimits(z\cap\alpha)\end{align*}$$

for all $z\in Z$ . For $\lambda <\alpha <\lambda ^+$ define

$$\begin{align*}f_\alpha(z)=\sup\{f_{b_{\lambda,\alpha}(\eta)}(z)+1\mid \eta\in z\},\end{align*}$$

where $b_{\lambda ,\alpha }:\lambda \to \alpha $ is a bijection. Let us note that if we take $Z=\lambda $ , then each $f_\alpha $ represents the ordinal $\alpha $ in any generic ultrapower obtained by forcing with a normal ideal on $\lambda $ . Whereas, in the two-cardinal setting, if we take $Z=P_\kappa \lambda $ , the function $f_\alpha $ represents $\alpha $ in any generic ultrapower obtained by forcing with a normal ideal on $P_\kappa \lambda $ .

Let us review some basic definitions concerning ineffable and Ramsey operators on cardinals. For $S\subseteq \kappa $ , we say that $\vec {S}=\langle S_\alpha \mid \alpha \in S\rangle $ is an S-list if $S_\alpha \subseteq \alpha $ for all $\alpha \in S$ . Given an S-list $\vec {S}$ , a set $H\subseteq S$ is said to be homogeneous for $\vec {S}$ if whenever $\alpha ,\beta \in H$ with $\alpha <\beta $ we have $S_\alpha =S_\beta \cap \alpha $ . If I is an ideal on $\kappa $ , we define another ideal ${\mathcal I}(I)$ such that for $S\subseteq \kappa $ we have $S\in {\mathcal I}(I)^+$ if and only if for every S-list $\vec {S}=\langle S_\alpha \mid \alpha \in S\rangle $ there is a set $H\in P(S)\cap I^+$ which is homogeneous for $\vec {S}$ . We say that $\kappa $ is almost ineffable if $\kappa \in {\mathcal I}([\kappa ]^{<\kappa })^+$ and $\kappa $ is ineffable if $\kappa \in {\mathcal I}({\mathop{\mathrm{NS}}}_\kappa )^+$ . The function $\mathcal {I}$ is referred to as the ineffable operator on $\kappa $ .

Recall that for a cardinal $\kappa $ and a set $S\subseteq \kappa $ , a function $f:[\kappa ]^{<\omega }\to \kappa $ is called regressive on S if $f(x)<\min (x)$ for all $x\in [S]^{<\omega }$ . Given a function $f:[\kappa ]^{<\omega }\to \kappa $ , a set $H\subseteq \kappa $ is said to be homogeneous for f if $f\upharpoonright [H]^n$ is constant for every $n<\omega $ . If I is an ideal on a cardinal $\kappa $ , we define another ideal ${\mathcal R}(I)$ such that for $S\subseteq \kappa $ we have $S\in {\mathcal R}(I)^+$ if and only if for every function $f:[\kappa ]^{<\omega }\to \kappa $ that is regressive on S, there is a set $H\in P(S)\cap I^+$ which is homogeneous for f. We say that a set $S\subseteq \kappa $ is Ramsey in $\kappa $ if $S\in {\mathcal R}([\kappa ]^{<\kappa })^+$ . Let us note that the definition of Ramsey set and, more generally, the definition of ${\mathcal R}(I)$ given above are standard and have many equivalent formulations (see [12, Proposition 2.8 and Theorem 2.10] for details). The function $\mathcal {R}$ is called the Ramsey operator on $\kappa $ .

The ineffable operator ${\mathcal I}$ and the Ramsey operator ${\mathcal R}$ on $\kappa $ are examples of what are called ideal operators, which have been studied in a broader context by several authors [15, 21, 22], and which are discussed in slightly more detail below. For a given ideal I and ideal operator ${\mathcal O}$ , such as ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ , we inductively define new ideals by letting

$$ \begin{align*} {\mathcal O}^0(I)&=I,\\ {\mathcal O}^{\alpha+1}(I)&={\mathcal O}({\mathcal O}^\alpha(I)), \textrm{ and}\\ {\mathcal O}^{\alpha}(I)&=\bigcup_{\beta<\alpha}{\mathcal O}^\beta(I). \end{align*} $$

So, for example, $S\in {\mathcal R}([\kappa ]^{<\kappa })^2$ if and only if for every function $f:[\kappa ]^{<\omega }\to \kappa $ that is regressive on S there is a set H that is Ramsey in $\kappa $ and homogeneous for f.

Recall that a set $S\subseteq \kappa $ is $\Pi ^1_n$ -indescribable in $\kappa $ if $(V_\kappa ,\in ,R)\models \varphi $ implies there is an $\alpha \in S$ with $(V_\alpha ,\in ,R\cap V_\alpha )\models \varphi $ whenever $S\subseteq V_\kappa $ and $\varphi $ is a $\Pi ^1_n$ sentence. Recall that $\varphi $ is $\Pi ^1_0$ if it is first order with finitely many second-order free variables. When $\kappa $ is $\Pi ^1_n$ -indescribable, the collection $\Pi ^1_n(\kappa )$ of all subsets of $\kappa $ which are not $\Pi ^1_n$ -indescribable in $\kappa $ forms a normal ideal on $\kappa $ [24]. Baumgartner studied ideals on $\kappa $ of the form ${\mathcal I}^\gamma (\Pi ^1_n(\kappa ))$ for $\gamma <\kappa ^+$ and $n\in \omega \cup \{-1\}$ where for notational convenience we take $\Pi ^1_{-1}(\kappa )=[\kappa ]^{<\kappa }$ (see [5, Section 7] and [6]). Ideals of the form ${\mathcal R}^\gamma ([\kappa ]^{<\kappa })$ and ${\mathcal R}^\gamma ({\mathop{\mathrm{NS}}}_\kappa )$ were introduced by Feng [17]; note that if $\kappa $ is inaccessible then $\Pi ^1_0(\kappa )={\mathop{\mathrm{NS}}}_\kappa $ .

Bagaria [2] introduced a notion of $\Pi ^1_\xi $ -indescribability of a cardinal $\kappa $ for $\xi <\kappa $ . The first author [13] extended Bagaria’s definition and introduced a notion of $\Pi ^1_\xi $ -indescribability of $\kappa $ for $\xi <\kappa ^+$ . ² Instead of reviewing the rather lengthy definition, we refer the reader to [13] for the definition of the $\Pi ^1_\xi $ -indescribability of a subset S of $\kappa $ for $\xi <\kappa ^+$ . The $\Pi ^1_\xi $ -indescribability ideal on $\kappa $ is then

$$\begin{align*}\Pi^1_\xi(\kappa)=\{S\subseteq\kappa\mid\ S\ \text{is not}\ \Pi^1_\xi\text{-}\text{indescribable in}\ \kappa\}.\end{align*}$$

Let us note that, in some sense, the definition of $\Pi ^1_\xi $ -indescribability does not play a large role in what follows because it is being “black boxed” by Lemma 4.5 and Theorem 4.2 (see the proof of Corollary 4.6).

Ideals of the form ${\mathcal R}^\gamma (\Pi ^1_\xi (\kappa ))$ were studied by the first author [12], and more generally, ideals of the form ${\mathcal I}^\gamma (\Pi ^1_\xi (\kappa ))$ and ${\mathcal R}^\gamma (\Pi ^1_\xi (\kappa ))$ for $\gamma <\kappa ^+$ and $\xi \in \kappa \cup \{-1\}$ were studied by the first author and Holy [15] (in fact the framework presented in [15] and [21] handles many ideal operators other than ${\mathcal I}$ and ${\mathcal R}$ ).

Notice that for a cardinal $\kappa $ , to each ideal of the form ${\mathcal O}^\gamma (\Pi ^1_\xi (\kappa ))$ where ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ , $\gamma <\kappa ^+$ and $\xi \in \kappa ^+\cup \{-1\}$ , there is a corresponding large cardinal hypothesis, namely $\kappa \in {\mathcal O}^\gamma (\Pi ^1_\xi (\kappa ))^+$ .

Definition 4.1. Suppose $\kappa $ is a cardinal, $\gamma <\kappa ^+$ and $\xi \in \kappa ^+\cup \{-1\}$ . Let $\Pi ^1_{-1}(\kappa )=[\kappa ]^{<\kappa }$ . We say that $\kappa $ is $\gamma\text{-}\Pi ^1_\xi $ -ineffable if $\kappa \in {\mathcal I}^\gamma (\Pi ^1_\xi (\kappa ))^+$ , and $\kappa $ is $\gamma \text{-}\Pi ^1_\xi $ -Ramsey if $\kappa \in {\mathcal R}^\gamma (\Pi ^1_\xi (\kappa ))^+$ .

For example, $\kappa $ is $1$ - $\Pi ^1_1$ -Ramsey if and only if every regressive function $f{\kern-1.5pt}:[\kappa ]^{<\kappa }{\kern-1pt}\to{\kern-1pt} \kappa $ has a homogeneous set which is $\Pi ^1_1$ -indescribable in $\kappa $ .

Recall that Baumgartner proved [5, Theorem 4.1] that when $\kappa $ is a subtle cardinal the set

$$\begin{align*}\{\alpha<\kappa\mid\ \alpha\ \textrm{is}\ \Pi^1_n\text{-}\textrm{indescribable for all}\ n<\omega\}\end{align*}$$

is in the subtle filter on $\kappa $ . More generally, the first author and Holy [15, Corollary 3.5] proved that when $\kappa $ is subtle the set

$$\begin{align*}\{\alpha<\kappa\mid\ \alpha\ \textrm{is}\ \Pi^1_\xi\text{-}\textrm{indescribable for all}\ \xi<\alpha^+\}\end{align*}$$

is in the subtle filter on $\kappa $ . Since whenever a cardinal $\kappa $ is almost ineffable it must also be subtle, it follows that the existence of an almost ineffable cardinal is strictly stronger in consistency strength than the existence of a cardinal $\kappa $ such that $\kappa $ is $\Pi ^1_\xi $ -indescribable for all $\xi <\kappa ^+$ . Furthermore, as shown in [15], this result can be pushed up the almost ineffability hierarchy. For example, the existence of an uncountable cardinal such that $\kappa \in {\mathcal I}^2([\kappa ]^{<\kappa })^+$ is strictly stronger than the existence of an uncountable cardinal $\kappa $ such that for all $\xi <\kappa ^+$ we have $\kappa \in {\mathcal I}(\Pi ^1_1(\kappa ))^+$ .

Theorem 4.2 [15, Theorem 3.8].

Suppose $\gamma <\kappa ^+$ , $S\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ and ${\vec {S}=\langle S_\alpha \mid \alpha \in S\rangle} $ is an S-list. Let A be the set of all ordinals $\alpha $ such that

$$\begin{align*}\exists X\subseteq S\cap\alpha\left[(\forall\xi<\alpha^+\ X\in{\mathcal I}^{f^\kappa_\gamma(\alpha)}(\Pi^1_\xi(\alpha))^+)\land(X\cup\{\alpha\}\ \textrm{ is hom. for}\ \vec{S})\right].\end{align*}$$

Then, $S\setminus A\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })$ .

Corollary 4.3 [15, Corollary 3.9].

Suppose $\kappa \in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ where $\gamma <\kappa ^+$ . Then the set

$$\begin{align*}\{\alpha<\kappa\mid(\forall\xi<\alpha^+)\ \alpha\in{\mathcal I}^{f^\kappa_\gamma(\alpha)}(\Pi^1_\xi(\alpha))^+\}\end{align*}$$

is in the filter ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^*$ . In other words, if $\kappa $ is $\gamma +1$ -almost ineffable then the set of $\alpha <\kappa $ which are $f^\kappa _\gamma (\alpha )$ - $\Pi ^1_\xi $ -ineffable for all $\xi <\alpha $ is in the filter ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^*$ .

4.1 New results on Ramseyness and indescribability

Now let us address the following question, and its generalizations, which were originally posed in [15].

Question 4.4. Is the existence of an uncountable cardinal $\kappa $ with $\kappa \in {\mathcal R}^2([\kappa ]^{<\kappa })^+$ strictly stronger than the existence of a cardinal $\kappa $ such that $\kappa \in {\mathcal R}(\Pi ^1_\xi (\kappa ))^+$ for all $\xi <\kappa ^+$ ?

The following lemma is standard and is an easy consequence of Feng’s characterization of Ramsey sets in terms of $(\omega ,S)$ -sequences [17, Theorem 2.3].

Lemma 4.5. Suppose $\kappa $ is a Ramsey cardinal. Then

$$\begin{align*}{\mathcal I}([\kappa]^{<\kappa})\subseteq{\mathcal R}([\kappa]^{<\kappa}).\end{align*}$$

Corollary 4.6 [15, Theorem 10.3].

Suppose $S\in {\mathcal R}([\kappa ]^{<\kappa })^+$ and let

$$\begin{align*}T=\{\alpha\in S\mid (\forall\xi<\alpha^+)\ S\cap\alpha\in \Pi^1_\xi(\alpha)^+\}.\end{align*}$$

Then $S\setminus T\in {\mathcal R}([\kappa ]^{<\kappa })$ .

Proof Suppose $S\in {\mathcal R}([\kappa ]^{<\kappa })^+$ and let T be as in the statement of the corollary. By Lemma 4.5 we see that $S\in {\mathcal I}([\kappa ]^{<\kappa })^+$ and by Theorem 4.2 we have ${S\setminus A\in {\mathcal I}([\kappa ]^{<\kappa })\subseteq {\mathcal R}([\kappa ]^{<\kappa })}$ . But $A\subseteq T$ so $S\setminus T\subseteq S\setminus A$ and hence $S\setminus T\in {\mathcal R}([\kappa ]^{<\kappa })$ .

The next result shows that Corollary 4.6 can, in a sense, be pushed up the Ramsey hierarchy, and provides an affirmative answer to Questions 10.4–10.6 and 10.9 in [15]; it is at present the best known generalization of Theorem 4.2 from the context of the ineffable operator to that of the Ramsey operator.

Theorem 4.7. Suppose $\gamma <\kappa ^+$ , $S\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ and let

$$\begin{align*}T=\{\alpha\in S\mid (\forall \xi<\alpha^+)\ S\cap\alpha\in {\mathcal R}^{f^\kappa_\gamma(\alpha)}(\Pi^1_\xi(\alpha))^+\}.\end{align*}$$

Then $S\setminus T\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })$ .

Proof If $\gamma =0$ the result follows directly from Corollary 4.6.

Suppose $\gamma =\delta +1<\kappa ^+$ is a successor ordinal, and suppose for a contradiction that $S\setminus T{\kern-1pt}\in{\kern-1pt} {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ . Recall that the set $C{\kern-1pt}={\kern-1pt}\{\alpha {\kern-1pt}<{\kern-1pt}\kappa \mid f^\kappa _{\delta +1}(\alpha ){\kern-1pt}={\kern-1pt}f^\kappa _\delta (\alpha )+1\}$ is a club in $\kappa $ and thus the set

$$\begin{align*}E=(S\setminus T)\cap C\end{align*}$$

is in ${\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ . For each $\alpha \in E$ fix $\xi _\alpha <\alpha ^+$ such that

$$\begin{align*}S\cap\alpha\in{\mathcal R}^{f^\kappa_{\delta}(\alpha)+1}(\Pi^1_{\xi_\alpha}(\alpha))={\mathcal R}({\mathcal R}^{f^\kappa_\delta(\alpha)}(\Pi^1_{\xi_\alpha}(\alpha)),\end{align*}$$

and fix a regressive function $g_\alpha :[S\cap \alpha ]^{<\omega }\to \alpha $ which has no homogeneous set in ${\mathcal R}^{f^\kappa _\delta (\alpha )}(\Pi ^1_{\xi _\alpha }(\alpha ))^+$ . Let $f:[E]^{<\omega }\to \kappa $ be a regressive function such that

$$\begin{align*}f(\alpha_0,\ldots,\alpha_n)=g_{\alpha_n}(\alpha_0,\ldots,\alpha_{n-1})\end{align*}$$

for $n<\omega $ and $(\alpha _0,\ldots ,\alpha _n)\in [E]^{n+1}$ . Since $E\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ , there is a set ${H\in P(E)\cap {\mathcal R}^{\delta +1}([\kappa ]^{<\kappa })^+}$ homogeneous for f. By the inductive hypothesis it follows that if we let

$$\begin{align*}T_H=\{\alpha\in H\mid(\forall\xi<\alpha^+)\ H\cap\alpha\in{\mathcal R}^{f^\kappa_\delta(\alpha)}(\Pi^1_\xi(\alpha))^+\}\end{align*}$$

then $H\setminus T_H\in {\mathcal R}^{\delta +1}([\kappa ]^{<\kappa })$ . Thus we can fix an $\alpha \in T_H$ . It follows that ${H\cap \alpha \in {\mathcal R}^{f^\kappa _\delta (\alpha )}(\Pi ^1_{\xi _\alpha }(\alpha ))^+}$ and $H\cap \alpha \subseteq E\cap \alpha \subseteq S\cap \alpha $ is homogeneous for $g_\alpha $ , a contradiction.

Now suppose $\gamma <\kappa ^+$ is a limit ordinal, and suppose again for a contradiction that $S\setminus T\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ . Recall that $C=\{\alpha <\kappa \mid f^\kappa _\gamma (\alpha )\text { is a limit ordinal}\}$ is a club subset of $\kappa $ and thus

$$\begin{align*}E=(S\setminus T)\cap C\end{align*}$$

is in $R^{\gamma +1}([\kappa ]^{<\kappa })^+$ . For each $\alpha \in E$ , using the fact that $\alpha \notin T$ , let $\xi _\alpha <\alpha ^+$ be such that

$$\begin{align*}S\cap\alpha\in {\mathcal R}^{f^\kappa_\gamma(\alpha)}(\Pi^1_{\xi_\alpha}(\alpha)).\end{align*}$$

Since $f^\kappa _\gamma (\alpha )=\sup \{f^\kappa _{b_{\kappa ,\gamma }(\eta )}(\alpha )+1\mid \eta \in \alpha \}<\alpha ^+$ is a limit ordinal, we can choose an ordinal $r(\alpha )<\alpha $ such that

$$\begin{align*}S\cap\alpha\in {\mathcal R}^{f^\kappa_{b_{\kappa,\gamma}(r(\alpha))}(\alpha)+1}(\Pi^1_{\xi_\alpha}(\alpha)).\end{align*}$$

This defines a regressive function $r:E\to \kappa $ , and by normality of ${\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })$ (see [17, Theorem 2.1]), there is an $E^*\in P(E)\cap {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ and some $\beta _0<\kappa $ such that $g(\alpha )=\beta _0$ for all $\alpha \in E^*$ . Let $\nu =b_{\kappa ,\gamma }(\beta _0)$ and notice that for all $\alpha \in E^*$ ,

$$\begin{align*}S\cap\alpha\in{\mathcal R}^{f^\kappa_\nu(\alpha)+1}(\Pi^1_{\xi_\alpha}(\alpha)).\end{align*}$$

For each $\alpha \in E^*$ , we fix a regressive function $g_\alpha :[S\cap \alpha ]^{<\omega }\to \kappa $ which has no homogeneous set in ${\mathcal R}^{f^\kappa _\nu (\alpha )}(\Pi ^1_{\xi _\alpha }(\alpha ))^+$ . Let $f:[E^*]^{<\omega }\to \kappa $ be a regressive function such that

$$\begin{align*}f(\alpha_0,\ldots,\alpha_n)=g_{\alpha_n}(\alpha_0,\ldots,\alpha_{n-1})\end{align*}$$

for $n<\omega $ and $(\alpha _0,\ldots ,\alpha _n)\in [E]^{n+1}$ . Since $E^*\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ there is a set ${H\in P(E^*)\cap {\mathcal R}^\gamma ([\kappa ]^{<\kappa })^+}$ homogeneous for f. Since $\nu <\gamma $ we have

$$\begin{align*}H\in{\mathcal R}^\gamma([\kappa]^{<\kappa})^+\subseteq{\mathcal R}^{\nu+1}([\kappa]^{<\kappa})^+,\end{align*}$$

and we may apply the inductive hypothesis to see that the set

$$\begin{align*}T_H=\{\alpha\in H\mid(\forall\xi<\alpha^+)\ H\cap\alpha\in {\mathcal R}^{f^\kappa_\nu(\alpha)}(\Pi^1_\xi(\alpha))^+\}\end{align*}$$

satisfies $H\setminus T_H\in {\mathcal R}^{\nu +1}([\kappa ]^{<\kappa })$ . Thus we can fix an $\alpha \in T_H$ . But then the set

$$\begin{align*}H\cap\alpha\subseteq E^*\cap\alpha\subseteq E\cap\alpha\subseteq S\cap\alpha\end{align*}$$

is in ${\mathcal R}^{f^\kappa _\nu (\alpha )}(\Pi ^1_{\xi _\alpha }(\alpha ))^+$ and is homogeneous for $g_\alpha $ , which is a contradiction.

Corollary 4.8. Suppose $\gamma <\kappa ^+$ . If $\kappa \in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ then the set of $\alpha <\kappa $ which are $f^\kappa _\gamma (\alpha )$ - $\Pi ^1_\xi $ -Ramsey for all $\xi <\alpha ^+$ is in the filter ${\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^*$ .

4.2 New results on two-cardinal Ramseyness

Let us now discuss two-cardinal versions of the ineffable and Ramsey operator, which are defined using the strong subset ordering $\prec $ . Suppose $\kappa $ is a cardinal and X is a set of ordinals with $\kappa \subseteq X$ . For $S\subseteq P_\kappa X$ , we say that $\vec {S}=\langle S_x\mid x\in P_\kappa X\rangle $ is an $(S,\prec )$ -list if $S_x\subseteq P_{\kappa _x}x$ for all $x\in S$ . Given an $(S,\prec )$ -list, a set $H\subseteq S$ is said to be homogeneous for $\vec {S}$ if whenever $x,y\in H$ with $x\prec y$ we have $S_x=S_y\cap P_{\kappa _x}x$ . If I is an ideal on $P_\kappa X$ , we define another ideal ${\mathcal I}_\prec (I)$ such that for $S\subseteq P_\kappa X$ we have $S\in {\mathcal I}_\prec (I)^+$ if and only if for every $(S,\prec )$ -list $\vec {S}$ there is a set $H\in P(S)\cap I^+$ which is homogeneous for $\vec {S}$ . We say that $P_\kappa X$ is strongly ineffable if $P_\kappa X\in {\mathcal I}_\prec ({\mathop{\mathrm{NSS}}}_{\kappa ,X})^+$ and almost strongly ineffable if $P_\kappa X\in {\mathcal I}_\prec (I_{\kappa ,X})^+$ . Here $I_{\kappa ,X}$ is the ideal on $P_\kappa X$ consisting of all subsets of $P_\kappa X$ which are not $\prec $ -cofinal in $P_\kappa X$ .

Let $[S]_\prec ^{<\omega }$ be the collection of all tuples $\vec {x}=(x_0,\ldots ,x_{n-1})\in S^n$ such that $n<\omega $ and $x_0\prec \cdots \prec x_{n-1}$ . A function $f:[P_\kappa X]_\prec ^{<\omega }\to P_\kappa X$ is called $\prec $ -regressive on S if $f(x_0,\ldots ,x_{n-1}) \prec x_0$ for all $(x_0,\ldots ,x_{n-1})\in [S]_\prec ^{<\omega }$ . Given a function $f:[P_\kappa X]_\prec ^{<\omega }\to P_\kappa X$ , a set $H\subseteq P_\kappa X$ is said to be homogeneous for f if $f\upharpoonright [H]^n$ is constant for all $n<\omega $ . For $S\subseteq P_\kappa X$ , let $S\in {\mathcal R}_\prec (I)^+$ if and only if for every function ${f:[P_\kappa X]^{<\omega }\to P_\kappa X}$ that is $\prec $ -regressive on S, there is a set $H\in P(S)\cap I^+$ which is homogeneous for f. We say that $P_\kappa X$ is strongly Ramsey if $P_\kappa X\in {\mathcal R}_\prec (I_{\kappa ,X})^+$ .

The first author and White [16] showed that many results from the literature [5, 6, 12, 15, 17] on the ineffable operator ${\mathcal I}$ and the Ramsey operator ${\mathcal R}$ , and their relationship with indescribability, can be extended to ${\mathcal I}_\prec $ and ${\mathcal R}_\prec $ . For example, by iterating the ideal operators ${\mathcal I}_\prec $ and ${\mathcal R}_\prec $ , one obtains hierarchies in the two-cardinal setting which are analogous to the classical ineffable and Ramsey hierarchies. One question left open by [16] is that which is analogous to Question 4.4 for the two-cardinal context. For example, if $P_\kappa X\in {\mathcal R}_\prec ^2(I_{\kappa ,X})^+$ , does it follow that the set

$$\begin{align*}\{x\in P_\kappa X\mid(\forall \xi<\kappa_x)\ x\in{\mathcal R}_\prec(\Pi^1_\xi(\kappa_x, x))^+\}\end{align*}$$

is in the filter ${\mathcal R}_\prec (I_{\kappa ,X})^*$ ?

The proof of Theorem 4.7 generalizes in a straight-forward way to establish the following.

Theorem 4.9. Suppose $\gamma <|X|^+$ , $S\in {\mathcal R}_{\prec }^{\gamma +1}(I_{\kappa ,X})^+$ and let

$$\begin{align*}T=\{x\in S\mid(\forall \xi<\kappa_x)\ S\cap P_{\kappa_x}x\in {\mathcal R}_{\prec}^{f_\gamma(x)}(\Pi^1_\xi(\kappa_x,x))^+\}.\end{align*}$$

Then $S\setminus T\in {\mathcal R}_{\prec }(I_{\kappa ,X})$ .

Corollary 4.10. Suppose $\gamma <|X|^+$ . If $P_\kappa X\in {\mathcal R}_{\prec }(I_{\kappa ,X})^+$ , then the set

$$\begin{align*}\{x\in P_\kappa X\mid (\forall\xi<{\kappa_x})\ P_{\kappa_x}x\in{\mathcal R}_\prec^{f_\gamma(x)}(\Pi^1_\xi(\kappa_x,x))^+\}\end{align*}$$

is in the filter ${\mathcal R}_{\prec }^{\gamma +1}(I_{\kappa ,X})^*$ .

5 Questions and ideas

Let us formulate a few open questions relevant to the topics of this article. For this section, let us assume $\kappa $ is some regular uncountable cardinal and $X\supseteq \kappa $ is a set of ordinals. First, we consider the following questions regarding the consistency strength of various principles considered above.

Question 5.1. What is the consistency strength of “whenever $S\subseteq P_\kappa X$ is strongly stationary there is some $x\in P_\kappa X$ for which $S\cap P_{\kappa _x}x$ is strongly stationary in $P_{\kappa _x}x$ ”? Is this similar to the situation for cardinals? Is the strength of this kind of reflection of strong stationary sets strictly between the “great Mahloness” of $P_\kappa X$ and the $\Pi ^1_1$ -indescribability of $P_\kappa X$ ?

Question 5.2. What is the consistency strength of the $2$ -s-strong stationarity of $P_\kappa X$ ? What is the consistency strength of the hypothesis that whenever S and T are strongly stationary in $P_\kappa X$ there is some $x\in P_\kappa X$ such that S and T are both strongly stationary in $P_{\kappa _x}x$ ?

The following questions regarding separation of various properties considered in this article remain open.

Question 5.3. Can we separate reflection of strongly stationary sets from pairwise simultaneous reflection of strongly stationary sets? In other words, is it consistent that whenever S is strongly stationary in $P_\kappa X$ there is some $x\in P_\kappa X$ such that S is strongly stationary in $P_{\kappa _x} x$ , but at the same time, pairwise reflection fails in the sense that there exists a pair $S,T$ of strongly stationary subsets of $P_\kappa X$ such that for every $x\in P_\kappa X$ both S and T are not strongly stationary in $P_{\kappa _x}x$ ?

It is conceivable that some two-cardinal $\Box (\kappa )$ -like principle could be used to address Questions 5.3. For example, $\Box (\kappa )$ implies that every stationary subset of $\kappa $ can be partitioned into two disjoint stationary sets that do not simultaneously reflect (see [20, Theorem 2.1] as well as [8, Theorem 3.50] and [14, Theorem 7.1] for generalizations).

Question 5.4. Is some two-cardinal $\Box (\kappa )$ -like principle formulated using weak clubs (defined in Section 2) consistent? Does it deny pairwise simultaneous reflection of strongly stationary subsets of $P_\kappa X$ ?

It is also natural to ask whether the various reflection properties introduced here can be separated from the large cardinal notions that imply them.

Question 5.5. Can we separate $\xi +1$ -strong stationarity or $\xi +1$ -s-strong stationarity in $P_\kappa X$ from:

1. (1) $\Pi ^1_\xi $ -indescribability in $P_\kappa X$ similar to what was done in [3]; or

2. (2) $\Pi ^1_1$ -indescribability in $P_\kappa X$ similar to what was done in [7]?

In [3], it was shown that consistently ${\mathop{\mathrm{NS}}}^{\xi +1}_\kappa $ can be non-trivial while $\kappa $ is not $\Pi ^1_{\xi }$ -indescribable. In [7, Definition 0.7], a normal version of the ideal ${\mathop{\mathrm{NS}}}_\kappa ^\xi $ was introduced, ${\mathop{\mathrm{NS}}}^{\xi , d}_\kappa $ . It was shown that consistently, ${\mathop{\mathrm{NS}}}^{\xi , d}_\kappa $ can be non-trivial for all $\xi <\omega $ while $\kappa $ is not even $\Pi ^1_1$ -indescribable.

Question 5.6. Is it consistent that $\kappa \in {\mathcal I}(\Pi ^1_{\xi }(\kappa ))$ and $\kappa \not \in {\mathcal I}({\mathop{\mathrm{NS}}}^{\xi +1}_\kappa )$ . Is it consistent that $\kappa \in {\mathcal I}(\Pi ^1_1(\kappa ))$ and $\kappa \notin {\mathcal I}({\mathop{\mathrm{NS}}}_\kappa ^{\xi ,d})$ for all $\xi <\omega $ ?

Finally, let us consider some questions that arise by considering Proposition 3.8 and [13]. Bagaria noticed that, using the definitions of [2], no ordinal $\alpha $ is $\alpha +1$ -stationary (see the discussion after Definition 2.6 in [2]) and no cardinal $\kappa $ is $\Pi ^1_\kappa $ -indescribable (see the discussion after Definition 4.2 in [2]). The first author showed that Bagaria’s definitions of $\xi $ -s-stationarity and derived topologies $\langle \tau _\xi \mid \xi <\delta \rangle $ on an ordinal $\delta $ , can be modified in a natural way so that a regular cardinal $\mu $ can carry a longer sequence of derived topologies $\langle \tau _\xi \mid \xi <\mu ^+\rangle $ , such that, for each $\xi <\mu $ there is a club $C_\xi $ in $\delta $ such that $\alpha \in C_\xi $ is not isolated in the $\tau _\xi $ topology if and only if $\alpha $ is $f^\mu _\xi (\alpha )$ -s-stationary (see [13, Theorem 6.15]). The first author also generalized Bagaria’s notion of $\Pi ^1_\xi $ -indescribability so that a cardinal $\kappa $ can be $\Pi ^1_\xi $ -indescribable for all $\xi <\kappa ^+$ , and that the $\Pi ^1_\xi $ -indescribability of $\kappa $ implies the $\xi +1$ -s-stationarity of $\kappa $ for all $\xi <\kappa ^+$ (see [13, Proposition 6.18]). It is natural to ask whether similar techniques can be used to generalize the results in Section 3.2 of the present article. For example, can one modify the definition of $\xi $ -strong stationarity so that Proposition 3.8 can fail for the modified notion?

Question 5.7. Can one use canonical functions to modify the definition of $\xi $ -s-strong stationarity so that it is possible for $x\in P_\kappa X$ to be $\xi $ -strongly stationary or $\xi $ -s-strongly stationary for some $\xi>\kappa _x$ ?

Question 5.8. Can the definitions of two-cardinal $\Pi ^1_\xi $ -indescribability (Definition 3.23), $\xi +1$ -s-strong stationarity (Definition 3.7), and the two-cardinal derived topologies (see Section 3.2) be modified using canonical functions so that Corollary 3.27 might generalize to values of $\xi $ for which $\kappa _x<\xi <|x|^+$ and Theorem 3.16 might generalize to values of $\xi $ for which $\kappa <\xi <|X|^+$ ?