For many years, there have been conducting research (e.g., by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance, Ramsey’s theorem for coloring graphs, Hindman’s finite sums theorem, and van der Waerden’s arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence, and ordinary convergence.

The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems.

The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an $\mathcal {I}_{1/n}$-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey.

The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Katětov order that has been extensively examined for many years so far.

This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory, and number theory.

Let $X,\,Y$ be Banach spaces and fix a linear operator $T \in \mathcal {L}(X,\,Y)$ and ideals $\mathcal {I},\, \mathcal {J}$ on the nonnegative integers. We obtain Silverman–Toeplitz type theorems on matrices $A=(A_{n,k}: n,\,k \in \omega )$ of linear operators in $\mathcal {L}(X,\,Y)$, so that

\[ \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\mathcal{I}\text{-}\lim \boldsymbol{x}) \]

$X$

$\boldsymbol {x}=(x_0,\,x_1,\,\ldots )$

$\mathcal {I}$

In this paper, the turnpike property is established for a nonconvex optimal control problem in discrete time. The functional is defined by the notion of the ideal convergence and can be considered as an analogue of the terminal functional defined over infinite-time horizon. The turnpike property states that every optimal solution converges to some unique optimal stationary point in the sense of ideal convergence if the ideal is invariant under translations. This kind of convergence generalizes, for example, statistical convergence and convergence with respect to logarithmic density zero sets.

We introduce natural strengthenings of sequential compactness, the r-Ramsey property for each natural number $r\geq 1$ . We prove that metrizable compact spaces are r-Ramsey for all r and give examples of compact spaces that are r-Ramsey but not $(r+1)$ -Ramsey for each $r\geq 1$ (assuming Continuum Hypothesis (CH) for all $r>1$ ). Productivity of the r-Ramsey property is considered.

We study the Rudin–Keisler pre-order on Fréchet–Urysohn ideals on $\omega $ . We solve three open questions posed by S. García-Ferreira and J. E. Rivera-Gómez in the articles [5] and [6] by establishing the following results:

• • For every AD family $\mathcal {A},$ there is an AD family $\mathcal {B}$ such that $\mathcal {A}^{\perp } <_{{\textsf {RK}}}\mathcal {B}^{\perp }.$

• • If $\mathcal {A}$ is a nowhere MAD family of size $\mathfrak {c}$ then there is a nowhere MAD family $\mathcal {B}$ such that $\mathcal {I}\left (\mathcal {A}\right ) $ and $\mathcal {I}\left ( \mathcal {B}\right ) $ are Rudin–Keisler incomparable.

• • There is a family $\left \{ \mathcal {B}_{\alpha }\mid \alpha \in \mathfrak {c}\right \} $ of nowhere MAD families such that if $\alpha \neq \beta $ , then $\mathcal {I}\left ( \mathcal {B}_{\alpha }\right ) $ and $\mathcal {I}\left ( \mathcal {B}_{\beta }\right ) $ are Rudin–Keisler incomparable.

Here $\mathcal {I}(\mathcal {A})$ denotes the ideal generated by an AD family $\mathcal {A}$ .

In the context of hyperspaces with the Vietoris topology, for a Fréchet–Urysohn-filter $\mathcal {F}$ we let $\mathcal {S}_{c}\left ( \mathcal {\xi }\left ( \mathcal {F}\right ) \right ) $ be the hyperspace of nontrivial convergent sequences of the space consisting of $\omega $ as discrete subset and only one accumulation point $\mathcal {F}$ whose neighborhoods are the elements of $\mathcal {F}$ together with the singleton $\{\mathcal {F}\}$ . For a FU-filter $\mathcal {F}$ we show that the following are equivalent:

• • $\mathcal {F}$ is a FUF-filter.

• • $\mathcal {S}_{c}\left ( \mathcal {\xi }\left ( \mathcal {F} \right ) \right ) $ is Baire.

We study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics: Xr(t,y) tracking the number of customers in the system at time t that have residual service times strictly greater than y. The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the process Xr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻, L1)) endowed with the Skorokhod M1 topology. The limit processes in the FCLT possess a stochastic decomposition property.

We give easy proofs that (a) the Continuum Hypothesis implies that if the product of $X$ with every Lindelöf space is Lindelöf, then $X$ is a $D$-space, and (b) Borel's Conjecture implies every Rothberger space is Hurewicz.

It is shown that it follows from PFA that there is no compact scattered space of height greater than $\omega $ in which the sequential order and the scattering heights coincide.

This paper introduces the concepts of fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity in intuitionistic fuzzy topological spaces. After defining the fundamental concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces, we present intuitionistic fuzzy $\gamma$-open sets and intuitionistic fuzzy $\gamma$-continuity and other results related topological concepts.

Let B be a complete ccc Boolean algebra and let $\tau_s$ be the topology on B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonempty open set in $(B,\tau_s)$ is topologically dense.

2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.

3. The topological space $(B,\tau_s)$ is sequentially compact if and only if the generic extension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but not independent reals.

A basic theory for probabilistic convergence spaces based on filter convergence is introduced. As in Florescu's previous theory of probabilistic convergence structures based on nets, one is able to assign a probability that a given filter converges to a given point. Various concepts and theorems pertaining to convergence spaces are extended to the realm of probabilistic convergence spaces, and illustrated by means of examples based on convergence in probability and convergence almost everywhere. Diagonal axioms due to Kowalsky and Fischer are also studied, first for convergence spaces and then in the setting of probabilistic convergence spaces.

The notion of pointwise cleavability is introduced. We clarify those results concerning cleavability which can be or can not be generalized to the case of pointwise cleavability.

The importance of compactness in this theory is shown. Among other things we prove that t, ts, πx, the property to be Fréchet-Urysohn, radiality, biradiality, bisequentiality and so on are preserved by pointwise cleavability on the class of compact Hausdorff spaces.

A general procedure is introduced for identifying categories of Cauchy spaces which have completions which possess a certain convergence property P, and for constructing completion functors on such categories. These results are applied to obtain a characterization of Cauchy spaces which allow T3 completions, and to construct T3 completions for categories of Cauchy groups and Cauchy lattices.

Let L be a lattice and q a convergence structure (or a topology) finer than the interval topology of L. In case of compact maximal chains and continuous lattice translations, the connected components of the space (L,q) are characterized using lattice conditions only. Moreover, lattice conditions of L are related to connectedness conditions of the order convergence space (L, o). Throughout this note, maximal chain conditions and maximal chain techniques play an important role.

Compactifications are constructed for convergence ordered spaces and topological ordered spaces with extension properties that resemble those of the Stone-Čech compactification.

Let X and Y be metric spaces. This paper considers the relationship between uniform convergence in C(X, Y) and topological convergence of functions in C(X, Y), viewed as subsets of X×Y. In general, uniform convergence in C(X, Y) implies Hausdorff metric convergence which, in turn, implies topological convergence, but if X and Y are compact, then all three notions are equivalent. If C([0, 1], Y) is nontrivial arid topological convergence in C(X, Y) implies uniform converger ce then X is compact. Theorem: Let X be compact and Y be loyally compact but noncompact. Then topological convergence in C(X, Y) implies uniform convergence if and only if X has finitely many components. We also sharpen a related result of Naimpally.