ON SOME TYPES OF COMPACT SPACES AND NEW CONCEPTS IN TOPOLOGICAL GRAPH THEORY

Abstract

In this thesis, we proceed to special properties of topological spaces. We know that compactness and connectedness are two of these. Compactness and connectedness are playing an important role in all branches of mathematics. On the other hand, we introduced new study in topological graph theory so we constructed a topology on the set of vertices of a graph G(V, E) and we studied some properties of this topological space via properties of the graph. In 1906, the term of "compact" was used for the first time by Frechet. From that time, many sorts of compactness were introduced by different topologists. In 1968, Asha Mathur [50] described compactness and its weaker forms through a table containing 72 properties. In 1985, M.E. Abd El-Monsef and A.M. Kozae [2] introduced a property Pαßȣ for generalizing 1920 types of compactness and closeness. This thesis is in continuation to the study of the property Pαßȣ of three variables which generalize the notions of compactness, paracompactness, closeness and many of their corresponding weaker forms, via the property Pαßȣ we generalized 15456 types of compactness and closeness. We further study some properties of these types and the relationship between various types of compactness was summarized, also. The other important property of topological spaces which studied in this thesis is connectedness. In this thesis, we introduced some types of connectedness in ideal topological spaces. The subject of ideals on a nonempty set X has been studied by Kuratowski in 1933 [38]. An ideal I on a set X is a nonempty collection of subsets of X which satisfies; (1) A∈ I and B⊂ A implies - 2 - B∈ I, (2) A∈ I and B∈ I implies A∪B ∈ I. In 1945, Vaidyanathaswamy [72] introduced the concept of ideal on topological spaces. Given a topological space (X, ) with an ideal I on X and if P(X) is the set of all subsets of X, a set operator ( )*: P(X)→ P(X), called a local function of A with respect to  and I, is defined as follows: for A⊂ X, A*(I,)= {x∈ X: A∩ U∉ I for every U∈ (x)} where (x)= {U∈ : x∈ U}. A Kuratowski closure operator cl*( ) for a topology *(I, ), called the ⋆-topology, finer than  is defined by cl*(A) = A∪ A*(I, ).