Some Properties Of S (2m + 1) - Space

In topology, compact spaces are a kind of very important topological space, compact and T₂ space is a perfect unity of separate property and compact property. In this paper, we combine S(2m + 1)-separate property with feebly compact property such for S(m + 1)-closed, which make some new results, m is a nature number, and m≥2. In the same time, we set up superfilter space and connect the kind of space with compact spaces , then we make some good results of paper[1], papcr[2] extend to 5(2m+ 1)-space and study some other properties of S(2m+ 1)-space.In chapter 1, we defineθ^m+1-absolute in S(2m+1)-space, then we combine S(2m+ 1)-separate property with S(m+ 1)-closed property, and prove that S(2m+1)-space is S(m+1)-closed if and only if TX is a compact space; In chapter 2, we study S(m,+1)-sets of 5(2m +1)-space, we apply filter language to define S(m + l)-scts, onθ^m+1-absolute (TX,k), k : TXâ†'X, we discuss some internal properties of S(m + 1)-sets; In chapter 3, we introduce functional cardinalityχ_m+1(X), and give some cardinal inequalities of S(2m + 1)-space. In the following, we get main results:Theorem 1.3.2 S(2m+ 1)-spacc is S(m+ 1)-closed if and only if TX is a compact space.Theorem 1.3.3 Let X be an S(2m+1)-spacc, (TX, k) isθ^m+1-absolute,k: TXâ†'X, A (?) X, then (?) = cl_θm+1A.Theorem 1.3.5 Let X be S(m + 1)-closed S(2m +1)-space, A(?) X, then there is a closed set B (?) TX such that k(B)= A if and only if there is an open filter U such that ad_gm+1U = A .Theorem 2.2.1 Let X be an S(m+ l)-closed space, then cl_θm+1 A is an S(m+ 1)-set. Theorem 2.2.2 S(2m + 1)-spacc X is S(m + 1)-closed, then the following are equivalent:(1) A is aθ^m+1-set in X;(2) A is an S(m + l)-set in X;(3) k^-1{A) is a compact subset;Theorem 2.2.3 Let (?) be an open filter of X, X is an S(m + 1)-closed space, then ad_θm+1(?) is an S(m + 1)-set.Theorem 3.2.3 About S(m + 1)-closed space, the following cardinal inequalities hold:(?)_m+1(X)=χ_m+1(X),andχ_m+1(X)≤χ(X)Theorem 3.2.4 Let S(2m+ 1)-space be S(m+ 1)-closed, then |X| <2^χ(X).