A Graphic Presentation of Some Bitopological Spaces

By: Mabruk Sola , Osama Alshfeh

Department of Mathematics - Tripoli University

Issue: Vol 22 |First Issue | 2016

article language: English

Abstract:

Given a bitopological space , where both and belong to a certain class of topological spaces , we will show that there exist a graph which will give a graphic presentation of the bitopological space . Keywords: Graph; Bitopology; maps; Idempotent. 1. Preliminaries: 1-1. Definition: If is a set, a map is said to be an idempotent map if . 1-2. Lemma: If is any idempotent map, defined by for any , then is a closure operation in the set . Proof: see [1]. 1-3. Definition: If is a non-empty set, is an idempotent map. Let denotes the topology on such that: for any . We call the topology induced by the idempotent map . 1-4. Definition: A space is said to be space if and only if each one-point set is either open or closed in . The following theorem is 1.5.6 of [1]. 1-5. Theorem: If is the topology induced by an idempotent map , then the frontier of any one-point set is either empty or a one-point set. 1-6. Theorem: If is the topology induced by an idempotent map , then is a space.