S-paracompact Spaces

In this paper, we introduce the conceptions of S-paracompact space and its a S-paracompact subset, whose covering characters, a S-paracompact subsets, mapping properties and product spaces(including Tychonoff product, inverse limit and σ-product) will be discussed. The following main results are obtained.Theorem 1 Let X be a S-paracompact space and V be a open cover of it, then V has a(1) σ locally finite semi-closed refinement.(2) σ closure-preserving semi-closed refinement.(3) refinable sequence{Vn}n∈N and for every point x of X, there exists a natural number n such that ord{x,Vn)= 1.(4) refinable sequence{Vn}n∈N and for every point x of X, there exist a natural number n and an open set in V such that ord{x,Vn)(?) U.Theorem 2 Every S-paracompact T2 space is S-subparacompact.Theorem 3 Let X be a countably S-closed space and (?) be a S-locally finite semi-closed collection of it, then (?) is finite.Theorem 4 Every S-subparacompact countably S-closed space is compact.Theorem 5 Let X be a topological space. (1) If(X,(?)α) is S-subparacompact and each closed set of X is regular closed, then X is also S-subparacompact.(2) If X is S-subparacompact and A is a clopen set of it, then the subspace (S,(?|A)) of X is S-subparacompact.Theorem 6 Let X be a S-subparacompact space and A is a g-closed set of it, then A is an a S-subparacompact of it.Theorem 7 Let X be a topological space and A is an open α S-subparacompact set of it, then the subspace(A,(?)|A) of X is α S-subparacompact.Theorem 8 Let X be a topological space and A is a clopen set of it.Then A is an α S—subparacompact set of X if and only if the subspace (A,J|A)of X is α S—subparacompact.Theorem 9 Let X and Y be two topological spaces and the continuous mapping f of X onto Y A is open.If Y is S-subparacompact,then X is also S-subparacompact.Theorem 10 Let X and Y be two topological spaces and the continuous closed mapping f of X onto Y A is nearly open.If X is S-subparacompact, then Y is also S-subparacompact.Theorem 11 Let X be compact and Y be S-subparacompact,then the Tychonoff product space X×Y is S-subparacompact.Theorem 12 Let X=lim｛Xσ,πρσ,∧｝be a |∧|-paracompact space.If Xσ is S-subparacompact for each σ∈∧,then X is S-subparacompact.Theorem 13 Let X=lim｛Xσ,πρσ,∧｝ be countably S-paracompact and the mapping πσ of X onto Xσ is open for each σ∈∧.If Xσ is S-subparacompact for each σ∈∧,then X is S-subparacompact.Theorem 14 Let X=σ｛Xσ|σ∈∧｝be a |∧|-paracompact space.If every finite subproduct of X is S-subparacompact for each B∈[∧]＜w,then X is S-subparacompact.