# Continuity and separation axioms via infra-topological spaces

Volume 30, Issue 3, pp 213--225
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.

### Share and Cite

##### ISRP Style

T. M. Al-Shami, Z. A. Ameen, R. Abu-Gdairi, A. Mhemdi, Continuity and separation axioms via infra-topological spaces, Journal of Mathematics and Computer Science, 30 (2023), no. 3, 213--225

##### AMA Style

Al-Shami T. M., Ameen Z. A., Abu-Gdairi R., Mhemdi A., Continuity and separation axioms via infra-topological spaces. J Math Comput SCI-JM. (2023); 30(3):213--225

##### Chicago/Turabian Style

Al-Shami, T. M., Ameen, Z. A., Abu-Gdairi, R., Mhemdi, A.. "Continuity and separation axioms via infra-topological spaces." Journal of Mathematics and Computer Science, 30, no. 3 (2023): 213--225

### Keywords

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

•  54A05
•  54B99
•  54C08
•  54D99