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

摘要

In this article, we introduce two notions which are called property (sigma-A) and property (sigma-B). They are generalizations of property (A) and property (B), respectively. Every space with a point-countable base (or sigma-NSR pair-base) satisfies property (sigma-A). Every space with the Collins-Roscoe property satisfies property (sigma-B). We show that every compact Hausdorff space with property (sigma-A) is metrizable. Thus some known conclusions can be generalized. This shows that property (sigma-A) plays a key role in the metrizability of compact Hausdorff spaces. We show that the properties of property (sigma-A) and property (sigma-B) are closed under finite products. Every finite product of T-1-spaces which satisfy property (sigma-B) (property (sigma-A), sigma-sheltering (F), sigma-well-ordered (F)) is hereditarily a D-space. If (X, T) satisfies omega(1)-sheltering (F), then (X, T-omega) is hereditarily a D-space. We show that if a space Xsatisfie s omega(1)-sheltering (F) and every countable discrete subspace of Xis closed, then Xis 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 (sigma-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 (sigma-A). (C) 2020 Elsevier B.V. All rights reserved.