A Study Of γ-metrizable Spaces And D-spaces

General topology has many branches,including metric spaces,generalized metric spaces,covering properties and so on.The notion of D-spaces is a topological property closely related to covering properties.The property of D-spaces is a popular research direction in general topology in recent years.So far,topologists have made great progress in the study of D-spaces and have proved that many generalized metric spaces have the D-property.In 2011,A.V.Arhangel’skii proposed concepts of compactly metrizable and countably metrizable,and discussed the related properties of these spaces.On this basis,in 2019,Shumrani put forward the concept of γ-metrizable.Let（X,τ）be a topological space,and let γ be an open covering of X.We say that X is γ-metrizable if there is a metric d on X which metrizes each member of γ,and he proved that every regularγ-metrizable space is metrizable.We illustrate by example that the above conclusion is wrong,therefore γ-metrizable spaces are weaker than metric spaces.Since metric spaces have good property and every metric space is a D-space,so in this paper we study some properties of γ-metrizable spaces,such as multiplicability,heredity,separability and so on,and we further discuss under what conditions a γ-metrizable space is metrizable,and whether every γ-metrizable space has D-property.The following results are obtained finally:（1）If Xn is a γn-metrizable space for every n ∈ N,then the product space X=（?）Xn is γ-metrizable for the natural open cover γ of X.（2）If X has a point-finite open cover by semi-stratifiable subspaces of X,then X is semi-stratifiable.Hence X is a semi-stratifiable space if X is a metacompactγ-metrizable space.（3）If X is a regular γ-metrizable space and the family γ is a σ-HCP open cover of X,then X is metrizable.（4）A space X is metrizable if and only if X is a paracompact γ-metrizable space.（5）Every T1-space with a σ-weakly hereditarily closure-preserving network is a D-space.（6）If X is a γ-metrizable space and the family γ is a σ-weakly hereditarily closurepreserving open cover of X,then X is a D-space.