A Study On Some Bitopological Properties Which Related To D-spaces

General topology has many directions,such as metric spaces,generalized metric spaces,covering properties and so on.Generalized metric spaces are the generalization of metric spaces,which include semi-stratifiable spaces,σ-spaces and Moore spaces.Covering properties include compact spaces,paracompact spaces,metacompact spaces,subparacompact spaces,θ-refinable spaces,meta-Lindelof spaces,weakly θ-refinable s-paces,weakly δθ-refinable spaces,irreducible spaces,etc.The notion of D-spaces is a topological property closely related to covering property.The property of D-spaces is a popular research direction in general topology in recent years.Up to now,topologists have made great progress in the study of D-spaces and have obtained that many gener-alized metric spaces have the properties of D-spaces.In recent years,some topologists have proposed notions which are weaker than the notion of D-spaces.For example,aD-spaces,bD-spaces.Every closed subspace of an aD-space is a bD-space.In 1963,J.C.Kelly proposed the concept of bitopological spaces.Since then,topologists have begun to study some concepts in bitopological spaces,such as separa-tion properties in bitopological spaces,generalization of concepts in generalized metric spaces in bitopological spaces,generalization of some covering properties in bitopolog-ical spaces,and so on.In this paper,we introduce notions of bitopological D-spaces,aD-spaces and bD-spaces.We get the following results:(1)Let(X,τ1,τ2)be a bitopological space such that τ1(?)τ2 and(X,τ2)is a T1-space.If(X,τ1,τ2)is τ1-semi-stratifiable with respect to τ2,then(X,τ1,τ2)is τ1-D with respect to τ2.(2)Let(X,τ1,τ2)be a bitopological space that satisfies τ1(?)τ2 and(X,τ2)is a T1-space.If X=X1 ∪ X2 such that for each i ∈ {1,2},(Xi,τ1|Xi,T2|Xi)is τ1|Xi-σwith respect to τ2|Xi,then(X,τ1,τ2)is τ1-D with respect to τ2.(3)Let(X,τ1,τ2)be a bitopological space such that(X,τ2)is a T1-space.If(X,τ1,τ2)is τ1-weakly δθ-refinable with respect to τ2,then(X,τ1,τ2)is τ1-bD with respect to τ2.In this thesis,we provide a useful conclusion:Let X be a T1-space and V be an open cover of X.If Y={x∈X:1≤ord(x,V)≤ω},then there exist a closed discrete in X subset D of Y and a mapping φ:D→V satisfying for every d∈D,d∈φ(d),such that Y C ∪{φ(d):d ∈ D}.By the conclusion,we can get Conclusion(3)and a known theorem that every T1 weakly δθ-refinable space is irreducible.Since weakly δθ-refinable space are closed hereditary and paracompact spaces,metacompact spaces,subparacompact spaces,θ-refinable spaces,meta-Lindelof spaces,weakly θ-refinable spaces are all weakly δθ-refinable space,some conclusions on aD-spaces in bitopological spaces can be obtained from conclusion(3).