A Study On The Property Of S-weak θ Refinable And S-weak (?) Refinable Space

This paper studies the S-weak θ refinable space and related properties of S-weakθ refinable space. The main results are as follows:Theoreml If T2- space (X, J) is S-Weak θ-refinable, then for each closed subset U of X and x(?)U, there exist A ∈ X and B ∈ SO(X,J) such that x∈A, U(?)B and A∩B=(?). This is equivalent to say that for every open subset A of X and there exists B∈J such that x ∈ B(?)Scl (B)î–‡.Theorem 2 Every extremally disconnected(e.d.) S-Weak θ-refinable T2-space is Weak θ-refinable.Theorem 3 Every S-Weak θ-refinable regular S-closed space is Weak θ-refinable.Theorem 4 If J α is topology and (X, Jα) is S-Weak θ-refinable then (X,J) is S-Weak θ-refinable.Theorem 5 Let (X, J) is a T2-space. Then (X, J) is S-Weak θ-refinable if and only if each open cover A of X has a semi-closed refinement B=∪n∈N Bn, for each x∈X, there exists n∈N such that 1≤ord(x,Bn)<ω(Bn={Bna:α∈I,n∈ N}).Theorem 6 Let (X,J) be a regular space. Then (X,J) is S-Weak θ-refinable if and only if each open cover A of X has a regular closed refinement B= U n∈NBn, for each x∈X, there exists n∈N such that 1≤ord(x,B)<ω(Bn={Bna:α∈I n∈ N}).Theorem 7 Every g-closed subset of an S-Weak θ-refinable space is aS-Weak θ-refinable subset.Theorem 8 Every open aS-Weak θ-refinable subspace of a space (X,J) is S-Weak θ-refinable.Theorem 9 Let U be a clopen subspace of a topological space. Then U is aS-Weak θ-refinable if and only if it is S-Weak θ-refinable.Theorem 10 Let A is αS-Weak θ-refinable subset of T2-space(X, J). Then A is θS-closed.Proposition 1 Let U be an Sg-closed subset of a space (X, J) and Vbe any subset of X. If U is aS-Weak θ-refinable and U(?)Scl(U), then Vis an αS-Weak θ-refinable subset in (X, J).Proposition 2 Let U and V be subsets of a space (X, J) such that U(?)V and B ∈ J. Then U is αS-Weak θ-refinable in (V, Jv) if and only if U is aS-Weak θ-refinable in (X,J).Theorem 11 The topological sum ⊕ α∈I Xα is S-Weak θ-refinable if and only if the space (Xα, Jα) is S-Weak θ-refinable, for each α ∈ I.Theorem 12 Every open Fσ-subspace of S-Weak θ-refinable regular space is S-Weak θ-refinable.Theorem 13 Let space{X,J) is S-θ-refinable, then{X,J) is S-Weak θ-refinable.Theorem 14 Let A is a closed subspace of S-Weak θ-refinable space(X,J). Then A is S-Weak θ-refinable.Theorem 15 If an space (X,J) is the union of countabely αS-Weak θ-refinable closed subsets, then X is S-Weak θ-refinable.Theorem 16 Every open Fa-subspace of S-Weak θ-refinable space is S-Weak θ-refinable.Theorem 17 Each subspace of S-Weak θ-refinable perfect space is S-Weak θ-refinable.Theorem 18 Let f:Xâ†'Y is double-continuous perfect mapping, if Y is S-Weak θ-refinable, then X is S-Weak θ-refinable.Theorem 19 Let f:Xâ†'Y is double-continuous closed-Lindelof mapping, if X is regular and Y is S-Weak θ-refinable, then X is S-Weak θ-refinable.