Continuity and separation axioms via infra-topological spaces

Volume 30, Issue 3, pp 213--225 https://doi.org/10.22436/jmcs.030.03.03

Publication Date: December 29, 2022 Submission Date: November 13, 2022 Revision Date: November 27, 2022 Accteptance Date: December 07, 2022

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Authors

T. M. Al-Shami - Department of Mathematics, Sana'a University, P.O.Box 1247 Sana'a, Yemen. - Future University, Egypt. Z. A. Ameen - Department of Mathematics, College of Science, University of Duhok, Duhok 42001, Iraq. R. Abu-Gdairi - Department of Mathematics, Faculty of Science, Zarqa University, P.O. Box 13110 Zarqa, Jordan. A. Mhemdi - Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia.

Abstract

In order to investigate a particular topic in mathematics, more specifically, general topology, it is always desirable to find a weaker condition. This work is planned to study a weak (topological) structure named infra-topological space. An infra-topological space is the collection of subsets of a universe that includes the empty set and is closed under finite intersections. The continuity, openness, and homeomorphism of mappings between infra-topological spaces are explored. Through the use of some examples, analogous properties and characterizations of ordinary mappings cannot be hopped on infra-topological structures. Then, the concepts of product and coproduct of infra-topological spaces are analyzed. Furthermore, the notion of infra-quotient topologies, which are inspired by infra-continuity, is introduced. The essential properties indicate that infra-quotient topologies and ordinary quotient topologies act in parallel. The final part of this paper is devoted to the investigation of infra separation axioms (infra \(T_i\)-spaces, \(i=0,1,\ldots, 4\)). The behaviour of ordinary separation axioms cannot be translated to an infra-topological structure. More precisely, infra-\(T_3\) and infra-\(T_4\)-spaces are independent, and singletons need not be infra-closed in infra-\(T_1\)-spaces.

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Keywords

• Infra-topology
• infra-continuity
• infra-open
• infra separation axioms

MSC

•  54A05
•  54B99
•  54C08
•  54D99

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