Some Properties Research On S-metacompact Spaces

This paper mainly studis the heredity,the product and the mapping properties of S-metacompact spaces. The following results are mainly proved:Theorem 1 The following thesis equivalent(1) X is countable S-metacompact space,(2) Every countable open coverU ={Ui}i∈N of X has a point finite semi-open collectionV ={Vi}i∈N which refines U,(3) Every increasing open coverW ={Wi}i∈N of X has a semi-closed sequenceF ={Fi}i∈N such that FiWi(i∈N) and ∪i∈N FiX,(4) Every decreasing open coverF ={Fi}i∈N of X and∩i∈NFi=? has a semi-open sequence W ={Wi}i∈N such that Fi Wi(i∈N) and∩i∈NWi=?.Theorem 2 If(X,T) is an S-metacompact Hausdorff space, then for every closed subset A of X and x A there exist U∈T and V∈SO(X,T) such that x∈U, A∈V and U∩V=?. This is equivalent to say that for every open subset U of X and x ∈U there exists V∈T such that x∈V scl(V) U.Theorem 3 Every extremally disconnected Hasdorff S-metacompct space is metacompact.Theorem 4 If(X,T α) is S-metacompact then(X,T) is S-metacompact.Theorem 5 Let(X,T) be a extremally disconnected space. If(X, T SO) is S-metacompact then(X,T) is S-metacompact.Theorem 6 Let(X,T) be a T2-space. Then(X,T) is S-metacompact if and only if each open cover U of X has a point finite semi-closed refinement V.(V∈SC(X,T), V∈V)Theorem 7 Let(X,T) be a extremally disconnected T2-space. Then(X,T)is S-metacompact if and only if each open cover U of X has a point finite regular-closed refinement. V∈RC(X,T).Theorem 8 Under the condition of regular space, metacompact space, S-metacom-pact space, nearly metacompact space are equivalent.Theorem 9 If topological space X has a point finite cover U ={Uβ:β∈T} which composed by S-metacompact subspace,for every β∈T, Uβ are mutually disjoint clopen sets,then X is S-metacompact space.Theorem 10 Every open αS-metacompact subset of a topological space(X, T) is S-metacompact.Theorem 11 {Xα}α∈A are series of disjoint of spaces, if every Xα(α∈A) is S-metacompact space, then the topological sum X=⊕α∈AXα is S-metacompact space.Theorem 12 Let A be a clopen subspace of a space(X, T). Then A is αS-metacompact if and only if it is S-metacompact.Theorem 13(X, T) is S-metacompact space, {Ak:k∈N} are countable open sets, then W=∪∞k=1Ak is S-metacompact subspace.Theorem 14 Let(X, T) be compact and let(Y,M) be S-metacompact. Then the product space(X,T)×(Y,M) is S-metacompact.Theorem 15 X, Y are two topological spaces. If X is S-metacompact space and f:Xâ†'Y is one to one continuous semi-open mapping, then Y is S-metacompact space.Theorem 16 f:Xâ†'Y is continuous closed Lindel?ff mapping which from regular space X to space Y, if Y is S-metacompact space then X is S-metacompact.