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D-spaces and thick covers. (English) Zbl 1244.54056

This paper gives over 15 results relating nearly a dozen subtly defined properties. For readability, this review mentions only a few of these results relating only a few of these properties. The results mentioned in this review all end in “then X is a D-space,” but many of the results in this paper do not have this format.

Recall that X is a D-space iff, given an open cover {U x :xX} where each xU x , there is a closed discrete DX so that X= xD U x .

To give the flavor of this paper, here are three results:

If X has an almost thick cover by closed Lindelöf D-sets, then X is a D-space.

Here a cover is almost thick iff H[X] <ω there is L H a finite union of elements of so that if AX and A is not closed then there is H[A] <ω with L H cl(AA).

If X is t-metrizable, then X is a D-space.

Here a space (X,τ) is t-metrizable iff it has a finer metrizable topology π and a function J:[X] <ω [X] <ω so that if AX then cl τ A cl π H[A] <ω J(H).

If X is a union of finitely many screenable σ-spaces, then X is a D-space.

Here a σ-space is one with a σ-discrete closed network; a space is screenable iff each open cover has a σ-pairwise disjoint open refinement.

54D20Noncompact covering properties (paracompact, Lindelöf, etc.)
54A25Cardinality properties of topological spaces
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