Studies On The Properties Of Local-strongly Paracompact Space And Base-countably Paracompact Space

This paper discusses two classes of generalized paracompact spaces: local-strongly paracompact and base-countably paracompact spaces.The research is mainly focused on there properties under the closed Lindelof mapping,quasi-perfect mapping and base-countably paracompact mapping and equivalent characterization of base-countably paracompactnesse is gained.The main conclusions are as follows:1. Let X is i - local strongly paracompact space (i = 1,2,3).If X is regular, then three of them are equivalent.2. If X is i- local strongly paracompact space, they are hereditary for open subspaces and closed subspaces(i = 1,2,3).3. If X is regular, i - local strongly paracompact space is an inverse of closed Lindelof mapping(i = 1,2,3).4.I- local strongly paracompact space is preserved under closed Lindelof mapping(i = 1,2,3).5. Let X and Y are both i- local compact space . If X is regular,, then X×Y is i - local strongly paracompact space (i = 1,2,3).6. If X is i - local strongly paracompact space , Y is i - local compact space . The product space X×Y is i- local strongly paracompact space(i = 1,2,3).7. X is a Base-paracompact space iff X is a Base-countably paracompact space and every open cover of X has aσ- locally finite open refinement by members of the basis which witnesses Base-countably paracompact space.8. Let X is normal, X is a Base-countably paracompact space iff there exsists an open basis B for X with |B|=ω(X) such that every countably open cover of X has a locally finite shrinking by members of the basis B.9. Base-countably paracompact space is preserved under open quasi-perfect mapping.10. Base-countably paracompact space is an inverse invariant of quasi-perfect mapping.11. Base-countably paracompact space is an inverse invariant of base-countably paracompact mapping.12. Let f: X→Y is a closed Lindelof mapping. If X is regular, then f: X→Y is Base-countably paracompact mapping.13. Let X and Y are both Base-countably paracompact space. If Y is locally compact, then the product space X×Y is Base-countably paracompact space.There are four chapters in this paper.In Chapter 1, introduced the research background and main conclusions in this paper.In Chapter 2, preparation knowledge, defines some notions and gives some lemmas and theorems we will use in this paper.In Chapter 3, three kinds of local strongly paracompact spaces are defined.It is proved that three kinds of local strongly paracompactnesse are equivalent in the case of regular,and there properties are discussed respectively.It is shown that they are hereditary for both open subspaces and closed subspaces.versions of some other classical properties of local strongly paracompactnesse are obtained.In Chapter 4, firstly, The relations between base-paracompact spaces and base-countably paracompact spaces are studied, equivalent characterization of them under the especially conditions is proved. Secondly.We gives a equivalent characterization about base-countably paracompact spaces under the normal conditions. At last, Some properties of base-countably paracompactnesse are investigated. It is shown that they are hereditary for closed subspaces,preserved under open quasi-perfect mapping,inverse invariant of quasi-perfect mapping and base-countably paracompact mapping and so on.