Study On The Properties Of Ortho-compact And Base-ortho-compact Spaces

This paper consists of four Chapters.Chapter 1 is preface.We introduce research background and the basis of topic choice and main results of this paper.In chapter 2 We introduce preliminaries in this paper.It puts great emphasis on the introduction of properties and results of base-paracompact spaces.In chapter 3 we give characterization of generalized paracompact spaces by base. The following results are obtained.1.A topological space X is subparacompact if and only if there exists a base(?) for X such that for any open cover by members of(?)has aσdiscrete closed refinement.2.Let(?)be a base for a topological space X.Then X is mesocompact (metacompact)if and only if for any open cover by members of(?)there exists a compact-finite(point-finite)open refinement.3.Let X be a topological space.Then the following assertions are equivalent:(1)X is ortho-compact space.(2)There exists a base(?)for X such that for any open cover by members of (?)has an interior-preserving open refinement.4.Let X be an ortho-compact space.Then there is a base(?)for every open cover(?)of X by members of(?),there exists a(?),such that(?)is an interior-preserving open refinement.Chapter 4 is the key of this paper.The properties and equivalent characterizations of base ortho-compact space are given by covering and mapping methods in this chapter.We have the following results.1.Let X be base-ortho-compact space.If M is a closed subset of X such thatω(X)=ω(M),then M is base-ortho-compact space.2.Let X be a countably compact space.If X is regular Lindel(?)f space,then X is base-ortho-compact space.3.Let X be a topological space.Then the following conditions are equivalent:(1)X is base-ortho-compact.(2)There is a base(?)with |(?)|=ω(X)for every open cover(?)of X by members of(?),there exists a(?),such that(?)is a interior-preserving open refinement of(?).(3)There is a base(?)with |(?)|=ω(X)for every open cover(?)of X by members of(?),there exists an open cover u of X such that u is a interior-preserving open refinement of(?).4.T₂-space X is hereditarily base-ortho-compact space if and only if for every open subspace of X is base-ortho-compact.5.Base-ortho-compact space is also base-ortho-compact under the finite-to-one open mapping.