Study On (sub) Sequentially Mesocompact Spaces And Hyperspaces

As we know that theory on covering property and hyperspaces have always been concerned with by scholars widely for a long time, which is studied initially in this paper.In the first chapter, the formation of topological space theory is simply introduced, as well as some knowledge about covering property and hyperspaces.The second chapter including four sections is devoted to (sub)sequentially mesocompactness. Section 2.1 which is brief gives some rules of symbols for chapter 2. In section 2.2, the most important in this chapter, sequentially mesocompactness is characterized in term of well-monotone cover, interior-preserving cover, suborthocompact and cushioned refinement. And we conclude two important theorems on characterization of sequentially mesocompact spaces as following:Theorem 2.2.8 The following conditions are mutually equivalent for a topological space X:1) The space X is sequentially mesocompact.2) Every interior-preserving and directed open cover of the space X has a closure-preserving closed refinement which is refined by the collection consisting of all convergent sequences (cs).3) Every interior-preserving open cover of the space X has an interior-preserving open cswise W-refinement.4) Every interior-preserving open cover of the space X has an interior-preserving open cswise star Fk -refinement.5) Every interior-preserving and directed open cover of the space X has an interior-preserving open cswise star refinement.Theorem 2.2.9 The following conditions are mutually equivalent for a topological space X:1) The space X is sequentially mesocompact.2) Every open cover of the space X has a semi-open cswise W-refinement.3) The space X is suborthocompact and every directed open cover of it has aSemi-open cs-wise star refinement.4) The space X is suborthocompact and every directed open cover of it has a cushioned refinement which is refined by the collection consisting of all convergent sequences .These two theorems help us get information about the structure of sequentially mesocompact spaces from some aspects. In this section the study of properties of sequentially mesocompact spaces is also started, and two conclusion are drawn as following:1) The space X is sequentially mesocompact if and only if it is sequentiallymeso-expandable and has property b*2. 2) Normal sequentially mesocompactness has Fσ hereditary property.In section 2.3, subsequentially mesocompactness is characterized in term of directed open covers. Section 2.3 gives almost the same results and proof as section2.2 gives us, which will not be stateded again.In short section 2.4, we know that the image of a (sub)sequentially mesocompact space under a closed sequential-covering mapping is (sub)sequentially mesocompact.Chapter 3 is devoted to hyperspaces. From the study in section 3.2 ,we found there are some relatives between the space X and hyperspace 2X on separation properties. For example, theorem 3.2.4 is that a topological space X is normal if and only if hyperspace 2X is regular, which is more helpful for us to study (countable) compactness and paracompactness deeply. In section 3.3 , we draw conclusion about countable compactness of hyperspaces as following :For a hausdorff (T2) space X, hyperspace 2X is locally countable compact ifand only if it is countable compact.The statement above is a simply introduction to this paper. I hope it will give us some help to study relevant theory in general topology.