The Convergence Properties Of Some Topological Space

Dikranjan and Giuli introduced a kind of Cech -closure operator—θ- closure, then get S(n)-θ- closed spaces which are feebly compact. Obviouslyθ-closure can not uniquely determine the topology of a space , then what is the influence ofθ- closure on the topology of a space? We study this problem in terms ofθ- convergence of nets and filters, with convergence theory as a tool . At the same time ,we introduceθ-sequential spaces,θ- Frechet spaces,θ-radial spaces and almostθ-radial spaces, then carry over some of the theory of sequential and Frechet spaces to these spaces . Finally, we introduce the concept ofθ* - convergence and characterize countable S(2) -θ- closed spaces .Main results:Theorem 1.12 Suppose (X,T) is a space, C={(Q,x): Q is a net in X whichθ-convergent to x },then T is the finest topology in which Q converges on x, (?) (Q, x)∈C .Theorem 2.4 (X,T_θ) is a sequential T₁ and {ω₁} - net space if and only if (X , T ) is discrete.Theorem 3.3 (X,T) is a T₂ space if and only if every constant sequence 9 -converges on only one point.Theorem 3.4 Suppose X is a space, then the following conditions are equivalent: (1) X is a Urysohn space. (2) Each net in X converges on at most one point. (3) Each filter in X converges on at most one point.Theorem 3.5 Suppose X is a space, then the following conditions are equivalent: (1) X is a regular space.(2) If ( S_n : n∈D ) is a net whichθ-converges on x,then ( S_n: n∈D) converges on x . (3) If F is a filter whichθ-converges on x , then F converges on x. Theorem 4.1 If X, Y areθ-sequential spaces, then X×Y is A-θ-sequential, where .Theorem 4.3 If X is aθ-radial space which satisfies countableθ-tightness, then it isθ-Frechet.Theorem 4.5 If X is aθ-almost radial space which satisfies countableθ-tightness, then it isθ-sequential.Theorem 5.2 X is countable S(2) -θ-closed if and only if each infinite subset has aθ* -ωcluster point.Theorem 5.3 X is countable S(2)-θ-closed if and only if each sequence in X has aθ* -limit point.Theorem 5.5 If X is a countable S(2) -θ-closed andθ*-pseudo-radial space ,then X isθ* -sequentially compact.