A study on D-spaces and the metrizability of compact spaces with property (σ-A)

Abstract

In this article, we introduce two notions which are called property (σ-A) and property (σ-B). They are generalizations of property (A) and property (B), respectively. Every space with a point-countable base (or σ-NSR pair-base) satisfies property (σ-A). Every space with the Collins-Roscoe property satisfies property (σ-B). We show that every compact Hausdorff space with property (σ-A) is metrizable. Thus some known conclusions can be generalized. This shows that property (σ-A) plays a key role in the metrizability of compact Hausdorff spaces.We show that the properties of property (σ-A) and property (σ-B) are closed under finite products. Every finite product of T1-spaces which satisfy property (σ-B) (property (σ-A), σ-sheltering (F), σ-well-ordered (F)) is hereditarily a D-space. If (X,T) satisfies ω1-sheltering (F), then (X,Tω) is hereditarily a D-space. We show that if a space X satisfies ω1-sheltering (F) and every countable discrete subspace of X is closed, then X is hereditarily a D-space. This gives a partial answer to a question posed by Z.Q. Feng and J.E. Porter in 2015.We finally give examples to show that there exists a space which has property (σ-A) but it does not have a point-countable base and there exists a space which has property (C) but it does not have property (σ-A).