A space X is discretely absolutely star-Lindelöf if for every open cover U of X and every dense subset D of X, there exists a countable subset F of D such that F is discrete closed in X and St(F, U) = X, where St(F, U) = {U ∈ U : U ∩F = ∅}. We show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed G δ-subspace.

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Abstract. It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

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QUESTION: Is there a ZFC example of a pseudocompact space that is perfect but not separablc?In [3], it is shown that perfect pseudocompact spaces have the countable chain condition and are first countable. Furthermore, pseudocompact Moore spaces arc separable [4]. Because Moore spaces are perfect, we are prompted to consider whether perfect pseudocompact spaces have to be separable. It is not a ZFC theorem: a compact Suslin line is a counterexample. But is there a ZFC example?Associated consistency results can be found in [ 2 , 5 ] : assuming MA + TCH, every perfect (countably) compact space is separable-in fact, hereditarily separable. The space ? of [l] is a pseudocompact Moore space that is not hereditarily separable.If indccd there were an example answering this question, the results just given imply that it would be neither a Moore space nor locally compact.