A Topological Space ¦È-complex

Dikranjan and Giuli introduced the notion of S(n)-θ-closed spaces which are characterized by means of special covers and filter, and raised six open problems, four of which have been solved in the negative by providing three counterexamples in [2]. In this paper we introduce the concept of θ- complex and discuss three questions raised by Dikranjan and Giuli in [1] as follows:Problem 1. Does S(n)-θ-closed imply Katetov - S(n) ?Problem 2. Can every S(n) - closed space be embedded in an S(n) -θ-closed space?Problem 3. Characterize the spaces (X, Ï„) such that (X, Ï„_θ) is T₂. In θ -complex, we give the answer to Problem 1 and Problem 2 in the positive, and also show a sufficient and necessary condition under which (X,Ï„_θ) is T₂.In Section 1 introduction and preliminaries are presented. In Section 2 We give the concept of θ- complex. In section 3 six examples are provided to explain the definition of θ- complex. In section 4, in order to describe the relations about vertex, open filter and closed filter, we give the definition of θ -complex's graph , and show a few properties about θ -complex's graph;ln section 5, we study the topological properties of θ- complex concerning minimality, products and embedding. In order to facilitate the research , we give three lemmas as follows :Lemma 5.1 Let T be a Tychonoff plank which doesn't have vertexes, then T is a locally compact regular space.Lemma 5.2 Let B be an ultrafilter in T, adB= φ, then: (l)foreach B∈B ,and each a<ω, B∩[a,ω)×ω₁≠φ;or (2) for each B ∈B , and each β <ω₁ ,B∩ω×[β,ω)≠φ.Lemma 5.3. Let T be a Tychonoff plank which doesn't have vertexes, il be an ultrafilter in T, adtt = ^, then{(a,a))x(j3,l.Theorem5.2 A 8-complex K is S(n)-6- closed if and only if for each limbic filter point 53, N(% 2n)>l.Theorem5.3 Let K be a 6- complex and its graph G, then K is semiregular if and only if there doesn't exist limbic filter point fU such thatTheorem 5.4 Let (K> T)be a 0 - complex,then (/sT,Te)is T2 if and only if (K,t)is 5(3)-space.Theorem 5.5 A 0-complex ^ is minimal S(n)- space if and only if(1) A" is an S(n)- space;(2) There doesn't exist limbic filter 27 such that(3) Let il be a central filter point and AT(H,1) = O, thenTheorem 5.6 A 0-complex K is a Katetov - S{n) space if and only if ï¿¡ is an S(w)- space.Theorem 5.7 Let K be an S(n) - closed 0-complex, then iCcan be embedded in an S(n)-6- closed space.