A Study Of ?-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 ?-metrizable.Let(X,?)be a topological space,and let ? be an open covering of X.We say that X is ?-metrizable if there is a metric d on X which metrizes each member of ?,and he proved that every regular?-metrizable space is metrizable.We illustrate by example that the above conclusion is wrong,therefore ?-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 ?-metrizable spaces,such as multiplicability,heredity,separability and so on,and we further discuss under what conditions a ?-metrizable space is metrizable,and whether every ?-metrizable space has D-property.The following results are obtained finally:(1)If Xn is a ?n-metrizable space for every n ? N,then the product space X=(?)Xn is ?-metrizable for the natural open cover ? 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 metacompact?-metrizable space.(3)If X is a regular ?-metrizable space and the family ? is a ?-HCP open cover of X,then X is metrizable.(4)A space X is metrizable if and only if X is a paracompact ?-metrizable space.(5)Every T1-space with a ?-weakly hereditarily closure-preserving network is a D-space.(6)If X is a ?-metrizable space and the family ? is a ?-weakly hereditarily closurepreserving open cover of X,then X is a D-space.