Connectedness and covering properties via infra topologies with application to fixed point theorem

Tareq M. Al-shami; Amani Rawshdeh; Heyam H. Al-jarrah; Abdelwaheb Mhemdi; Tareq M. Al-shami; Amani Rawshdeh; Heyam H. Al-jarrah; Abdelwaheb Mhemdi

doi:10.3934/math.2023447

AIMS Mathematics

2023, Volume 8, Issue 4: 8928-8948. doi: 10.3934/math.2023447

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Research article

Connectedness and covering properties via infra topologies with application to fixed point theorem

1.
Department of Mathematics, Sana'a University, Sana'a, Yemen
2.
Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Jordan
3.
Department of Mathematics, Faculty of Science, Yarmouk University, Jordan
4.
Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia

Received: 03 October 2022 Revised: 18 January 2023 Accepted: 19 January 2023 Published: 09 February 2023
MSC : 54A05, 54D99

A new generalization of classical topology, namely infra topology was introduced. The importance of studying this structure comes from two matters, first preserving topological properties under a weaker condition than topology, and second, the possibility of applying infra-interior and infra-closure operators to study rough-set concepts. Herein, we familiarize new concepts in this structure and establish their master properties. First, we introduce the notions of infra-connected and locally infra-connected spaces. Among some of the results we obtained, the finite product of infra-connected spaces is infra-connected, and the property of being a locally infra-connected space is an infra-open hereditary property. We successfully describe an infra-connected space using infra-open sets, which helps to study concepts given in this section under certain functions. Then, we determine the condition under which the number of infra-components is finite or countable. Second, we define the concepts of infra-compact and infra-Lindelöf spaces and study some of their basic properties. With the help of a counterexample, we elucidate that the infra-compact subset of an infra-$ T_2 $ space is not infra-closed, in general. We end this work by one of the interesting topics in mathematics "fixed point theorem", we show that when the infra-continuous function defined on an infra-compact space has a unique fixed point. To elucidate the topological properties that are invalid in the frame of infra topology, we provide some counterexamples.
- infra topology,
- infra-connected space,
- locally infra-connected space,
- infra-cut points,
- infra-compact space,
- infra-Lindelöf space,
- infra-fixed point
Citation: Tareq M. Al-shami, Amani Rawshdeh, Heyam H. Al-jarrah, Abdelwaheb Mhemdi. Connectedness and covering properties via infra topologies with application to fixed point theorem[J]. AIMS Mathematics, 2023, 8(4): 8928-8948. doi: 10.3934/math.2023447

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Abstract

A new generalization of classical topology, namely infra topology was introduced. The importance of studying this structure comes from two matters, first preserving topological properties under a weaker condition than topology, and second, the possibility of applying infra-interior and infra-closure operators to study rough-set concepts. Herein, we familiarize new concepts in this structure and establish their master properties. First, we introduce the notions of infra-connected and locally infra-connected spaces. Among some of the results we obtained, the finite product of infra-connected spaces is infra-connected, and the property of being a locally infra-connected space is an infra-open hereditary property. We successfully describe an infra-connected space using infra-open sets, which helps to study concepts given in this section under certain functions. Then, we determine the condition under which the number of infra-components is finite or countable. Second, we define the concepts of infra-compact and infra-Lindelöf spaces and study some of their basic properties. With the help of a counterexample, we elucidate that the infra-compact subset of an infra-$ T_2 $ space is not infra-closed, in general. We end this work by one of the interesting topics in mathematics "fixed point theorem", we show that when the infra-continuous function defined on an infra-compact space has a unique fixed point. To elucidate the topological properties that are invalid in the frame of infra topology, we provide some counterexamples.

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