A Class Of Topological Space ¦Ò-¦È-complex-shaped

This paper introduces a kind of new topological space-σ-θ- complex . In this paper, the concept ofσ-θ- complex's graph and its properties are studied, then the topological properties ofσ-θ- complex's graph are discussed . Theσ-θ-complex's graph are infinite , so the connection of infinite graph and topological space are built up in this paper, which offeres a new approach for the study of infinite graph .In section 1 introduction and preliminaries are presented. In section 2 ,we give the concept ofσ-θ- complex , and give some examples of three non-σ-θ-complexes and fourσ-θ-complexes to explain the concept ofσ-θ-complex. In section 3, in order to describe the relations of vertex ,open filter and closed filter,the definition ofσ-θ-complex's graph are given, and show some properties as follows:Property 3.1 Let K be aσ-θ- complex its central filter point u, and its edge filter point or vertex v ,then d(u, v) = 2m-1,m∈ω.Property 3.2 Let K be aσ-θ-complex and its central filter point u₁,u₂,then d(u₁,u₂) = 2m.Property 3.3 Let K be aσ-θ-complex and its edge filter point or vertex v₁,v₂,then d(v₁,v₂)=2m.Property 3.4 Let K be aσ-θ-complex and its graph G, then for each central filter U, 2≤d_G (U)≤3.In section 4 ,the topological properties ofσ-θ-complex are studied , and the main results are as follows:Theorem 4.1 Let K| be aσ-θ- complex with infinite degree. And its graph has infinite degree at edge filter point. If there is only one vertex in K|^,then K| is a S(n) - space,for (?)_n<ω.Theorem 4.2σ-θ-complex cann't be S(n) - closed.Theorem 4.3 Let K be aσ-θ-complex with finite degree, K' is infinite compactification of K, then K' is S(n) - closed iff for each central filter point U ,N(U,2n-1)≥1.Theorem 4.4 Let K be aσ-θ-complex with finite degree, K' is infinite compactification of K ,then K' is S(n)-θ- closed iff for each edge filter point B, N(B,2n)≥2.Theorem 4.5 Let K be aσ-θ- complex with finite degree, K' is infinite compactification of K, then (K',Ï„_θ) is T₂ iff (K',Ï„) is S{3) - space.Theorem 4.6 Let K be aσ-θ- complex with finite degree, K' is infinite compactification of K. If K' is S(n) - closed ,then K' can be embedded in a S(n)-θ-closed space.Theorem 4.7 Let K be aσ-θ- complex with finite degree, K' is infinite compactification of K. K' is a minimal S(n) - space iff(1) K' is a S(n)- space;(2) There is no edge filter point D such that N(D,2) = 1,d_G(D)=1(3) If central filter point U satisfies N(U,1) = 0, then N(U,2n-1)≥2;...