The Characteristics Of S-mesocompact Spaces

This paper focuses on studies and researches of equivalent characterizations and the mapping properties of S-mesocompact and some other properties of αS-mesocompact sets. Main achievements as follows:Theorem 1 If(X,T) is an S-mesocompact T2-space, then for every closed subset A of X and x A there exist U∈T and V∈SO(X,T)such that 得 x∈U,A U and ?, this is equivalent to say that for every open subset U of X and there exists T such that.Theorem 2 Every S-mesocompact T2-space is semiregular, i.e., T=TS.Theorem 3 Every extremally disconnected S-mesocompact T2-space is regular.Theorem 4 Let(X,T) be extremally disconnected. Then the follwwing are equivalent:a)(X,T) is S-mesocompact and S-closed.b)(X,T) is compact.Theorem 5 If(,T α)is S-mesocompact then(,T)is S-mesocompact.Theorem 6 Let(X,T) be an extremally disconnected space. Then(,T)is S-mesocompact if(,TSO)is S-mesocompact.Theorem 7 Let(X,T) be a T2-space. Then(,T)is S-mesocompact if only if each open cover U of X has a locally finite semi-closed refinement V(that is SC(X,T) for every V).Theorem 8 Let(,T)be a regular space. Then(,T)is S-mesocompact if only if each open cover U of X has a locally finite regular closed refinement V(that is RC(,T)for every V).Theorem 9 If(X,T) is S-mesocompact then every regular open subspace of X is S-mesocompact subspace.Theorem 10 Let(X,T) is S-mesocompact, : are countable regular open set,then is S-mesocompact subspace of X.Theorem 11 Let A be a clopen subspace of a space(,T).then A is αS-mesocompact if and only if it is S-mesocompact.Theorem 12 If(X,T) is a T2-space and A is αS-mesocompact,then A is θS-closed.Theorem 13 The topological sum is S-mesocompact if and only if the space(,T α) is S-mesocompact, for each.Theorem 14 let(,T) be compact and let(,M) be S-mesocompact. then the product space(,T)×(,M) is S-mesocompact.Theorem 15 Let X be a normal S-mesocompact space, A is open F-subspace of X, is closed set of X, A= :, for each.Theorem 16 Let X be a normal S-mesocompact space, A is open F-subspace of X, then A is S-mesocompact.Theorem 17 Th image if a S-mesocompact space under a closed and compact-covering mapping is S-mesocompact.Theorem 18 Let X、Y are compact space, : is perfect mapping,if Y is S-mesocompact space, then X is S-mesocompact space.