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UBC Theses and Dissertations

Generalization of topological spaces Lim, Kim-Leong

Abstract

Given a set X , let P(X) be the collection of all subsets of X . A nonempty sub-collection u, of P(X) is called a generalized topology for X and the ordered pair (X, u) is called a generalized topological space or an abstract space or simply a space. Elements of u are said to be u-open and their complements are said to be u-closed. We define u-closure, u-limit point, ….. and so on in the natural way. Most of the basic notions in point set topology are defined analogously. It is expected that many important results in point set topology will not be carried over and a number of interesting properties will be lost or weakened. Nevertheless, some of them will still hold true despite the absence of the finite intersection axiom and the arbitrary union axiom for the collection of subsets. The primary objective of this thesis is to investigate which theorems in point set topology still remain valid in our more general setting. A secondary objective is to provide some counterexamples showing certain basic results in point set topology turn out to be false in the setting. It should be noted that other basic notions which are not discussed here at all can be defined similarly. However, in order to attain desirable and interesting conclusions, additional conditions must be imposed.

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