### A Priestley view of spatialization of frames

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As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize $\sigma $-compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize $\sigma $-compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between $\sigma $-compact and Menger spaces.

Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T₅ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω₁-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ ℵ₁. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact...

In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property $\left(a\right)$. It follows that it is consistent that closed discrete subsets of a separable space $X$ which are also relatively normal (relatively countably paracompact, relatively $\left(a\right)$) in $X$ are necessarily countable. There are, however, consistent examples of...

In some sense, a dual property to that of Valdivia compact is considered, namely the property to be embedded as a closed subspace into a complement of a $\Sigma $-subproduct of a Tikhonov cube. All locally compact spaces are co-Valdivia spaces (and only those among metrizable spaces or spaces having countable type). There are paracompact non-locally compact co-Valdivia spaces. A possibly new type of ultrafilters lying in between P-ultrafilters and weak P-ultrafilters is introduced. Under Martin axiom and...

We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.

Let X be a connected locally compact metric space. It is known that if X is locally connected, then the space of ends (Freudenthal ends), EX, can be represented as the inverse limit of the set (space) S(X C) of components of X C, i.e., as the inverse limit of the inverse system $$EX=\underleftarrow{lim}\left(S(X\setminus C)\right),inclusions,CcompactinX)$$ . In this paper, the above result is significantly improved. It is shown that for a space which is not locally connected, we can replace the space of components by the space of quasicomponents Q(X C) of X C. The following...