We discuss the question of extending homeomorphisms between closed subsets of the Cantor cube $D^{\tau }$. It is established that any homeomorphism between two closed negligible subsets of $D^{\tau }$ can be extended to an autohomeomorphism of $D^{\tau }$.

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of ${{Z}^{\omega }}$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of ${{\omega }_{1}}$ Cohen reals, viewed as a subspace of ${{2}^{\omega }}$.

In this note we partially answer a question of Cascales, Orihuela and Tkachuk [‘Domination by second countable spaces and Lindelöf ${\rm\Sigma}$-property’, Topology Appl.158(2) (2011), 204–214] by proving that under $CH$ a compact space $X$ is metrisable provided $X^{2}\setminus {\rm\Delta}$ can be covered by a family of compact sets $\{K_{f}:f\in {\it\omega}^{{\it\omega}}\}$ such that $K_{f}\subset K_{h}$ whenever $f\leq h$ coordinatewise.

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.

We study the existence of continuity points for mappings $f\,:\,X\,\times \,Y\,\to \,Z$ whose $x$-sections $Y\,\backepsilon \,y\,\to \,f\left( x,y \right)\,\in \,Z$ are fragmentable and $y$-sections $X\,\backepsilon \,x\,\to \,f\left( x,y \right)\,\in \,Z$ are quasicontinuous, where $X$ is a Baire space and $Z$ is a metric space. For the factor $Y$, we consider two infinite “point-picking” games ${{G}_{1}}\,\left( y \right)$ and ${{G}_{2}}\,\left( y \right)$ defined respectively for each $y\,\in \,Y$ as follows: in the $n$-th inning, Player I gives a dense set ${{D}_{n}}\,\subset \,Y$, respectively, a dense open set ${{D}_{n}}\,\subset \,Y$. Then Player II picks a point ${{y}_{n}}\,\in \,{{D}_{n}}$; II wins if $y$ is in the closure of $\left\{ {{y}_{n}}\,:\,n\,\in \,\mathbb{N} \right\}$, otherwise I wins. It is shown that (i) $f$ is cliquish if II has a winning strategy in ${{G}_{1}}\,\left( y \right)$ for every $y\,\in \,Y$, and (ii) $f$ is quasicontinuous if the $x$-sections of $f$ are continuous and the set of $y\,\in \,Y$ such that II has a winning strategy in ${{G}_{2}}\,\left( y \right)$ is dense in $Y$. Item (i) extends substantially a result of Debs and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of “small” compact spaces.

We prove, among other things, that if ${{C}_{p}}\left( X \right)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

For every infinite cardinal k we construct a space Ck universal for all compact T1 -spaces of weight ≤ k. It follows, in particular, that there are only 2k topologically different compact T1 -spaces. We show that Cw is universal for all second countable developable T1 -spaces. The existence of closely universal compact T1 -spaces is discussed.

Let A and A denote the classes of ordinal spaces with the order topology and Σ-product spaces of the two point discrete space respectively. Characterizations are given in terms of ultrafiIters of clopen sets of those O-dimensional Hausdorff topological spaces that can be embedded homeomorphically as a closed subspace of a topological product of either spaces from the class Λ or the class Δ. Both classes consist of spaces that are ω0-bounded. An example is given of a O-dimensional Hausdorff ω0-bounded space that cannot be homeomorphically embedded as a closed subset of a product of spaces from either Λ or Δ, answering a question of R. G. Woods.

It is known that if a topological property of Tychonoff spaces is closed-hereditary, productive and possessed by all compact Hausdorff spaces, then each (0-dimensional) Tychonoff space X is a dense subspace of a (0-dimensional) Tychonoff space with such that each continuous map from X to a (0-dimensional) Tychonoff space with admits a continuous extension over . In response to Broverman's question [Canad. Math. Bull. 19 (1), (1976), 13–19], we prove that if for every two 0-dimensional Tychonoff spaces X and Y, if and only if , then is contained in countable compactness.

If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.