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

This paper discusses two classes of generalized paracompact spaces: local-strongly subparacompact and base-countably subparacompact spaces.The research is mainly focused on there properties under the closed Lindelof mapping、 quasi-perfect mapping and base-countably subparacompact mapping and equivalent characterization of base-countably subparacompactnesse is gained.The main conclusions are as follows:(1) Let X is i-local strongly subparacompact space (i=1,2,3). If X is regular, then three of them are equivalent.(2) If X is i-local strongly subparacompact space, they are hereditary for open subspaces and closed subspaces(i=1,2,3).(3) If X is regular,i-local strongly subparacompact space is an inverse of closed Lindelof mapping (i=1,2,3).(4)I-local strongly subparacompact space is preserved under open and quasi-perfect mapping (i=1,2,3).(5) Let X is i-local strongly subparacompact space, Y is compact space. If X is regular, then X×Yis i-local strongly subparacompact space(i=1,2,3).(6) If X is i-local strongly subparacompact space, Y is i-local compact space. The product space X x Y is i-local strongly subparacompact space (i=1,2,3).(7) X is a Base-subparacompact space if X is a Base-countably subparacompact space and every open cover of X has σ-discrete closed refinement by members of the basis which witnesses Base-countably subparacompact space.(8) Let X is normal, X is a Base-countably subparacompact space if X there exsists an open basis B for X with B|=ω(X) such that every countably open cover of X has σ-discrete closed refinement by members of the basis B(9) Base-countably subparacompact space is preserved under open quasi-perfect mapping.(10) Base-countably subparacompact space is an inverse invariant of quasi-perfect mapping.(11) Base-countably subparacompact space is an inverse invariant of base-countably subparacompact mapping.(12) Let f:X→Y is a closed Lindelof mapping. If X is regular and σ-discrete space, then/:X→Y is Base-countably subparacompact mapping.(13) Let X and Y are both Base-countably subparacompact space. If Y is locally compact, then the product space X x Y is Base-countably subparacompact space.There are four chapters in this paper.In Chapter1, introduced the research background and main conclusions in this paper.In Chapter2,preparation knowledge,defines some notions and gives some lemmas and theorems we will use in this paper.In Chapter3,three kinds of local strongly subparacompact spaces are defined.It is proved that three kinds of local strongly subparacompactnesse 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 subparacompactnesse are obtained.In Chapter4,firstly, The relations between base-paracompact spaces and base-countably subparacompact spaces are studied, equivalent characterization of them under the especially conditions is proved. Secondly,We gives a equivalent characterization about base-countably subparacompact spaces under the normal conditions. At last,Some properties of base-countably subparacompactnesse 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 subparacompact mapping and so on.