Some Properties And Characterizations Of Hereditarilyσ-Bounded Mesocompact Spaces

This paper firstly presents the concepts of the hereditarily a-bounded mesocompact spaces, and then expounds the equivalent characterizations and other some properties of the hereditarily a-bounded mesocompact spaces, The main results were as follows:Theorem one:A space X is a hereditarily a-bounded mesocompact spaces if and only if every scattered partition of X has a a-compact bounded of open expansions.Theorem two:The following are equivalent:(1) A space X is a hereditarily a-bounded mesocompact spaces;(2) Every monotone decreasing family {Fα:α<γ} of close sets of X has a a-compact bounded open family (?)=∪n∈ω(?)n such that,for every α<γ, X-Fα=∪{V∈(?):V∩Fα=(?)};(3) Every monotone increasing family (?)={Uα:α∈γ} of open sets of X has a a-compact bounded open refinement (?)=∪n∈ω(?)n such that,for every α<γ, Uα=∪{V∈(?):V(?)Uα};(4) Every monotone increasing family (?)={Uα:α∈γ} of open cover of X has a a-compact bounded open refinement (?)=∪n∈ω(?)n such that,for every α<γ, Uα=∪{V∈(?):V(?)Uα};Theorem three:IfY has a bace of a-compact bounded,thenX×Y is hereditarily a-bounded mesocompact spaces if and only if X is hereditarily σ-bounded mesocompact spaces.Deduction one:If Y space is metrizable,then X x Y is hereditarily σ-bounded mesocompact spaces if and only if X is hereditarily a-bounded mesocompact spaces.Theorem four:If X=∏iεωXi is Tychonoff products of topological spaces {Xi:i∈ω], The following are equivalent:(1) A spact X is a hereditarily a-bounded mesocompact spaces;(2)(？)α∈[ω]<ω,∏i∈ω Xi is a hereditarily a-bounded mesocompact spaces;(3)(?)n∈ω,∏i<ω Xi is a hereditarily a-bounded mesocompact spaces.Theorem five:If f:X→Y is a-bounded mesocompact maps and Y is and normal hereditarily σ-bounded mesocompact spaces, then X is a hereditarily a-bounded mesocompact spaces.