On some types of functions and a form of compactness via $ \omega _{s} $-open sets

Samer Al Ghour; Samer Al Ghour

doi:10.3934/math.2022126

AIMS Mathematics

2022, Volume 7, Issue 2: 2220-2236. doi: 10.3934/math.2022126

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Research article

On some types of functions and a form of compactness via $ \omega _{s} $-open sets

Samer Al Ghour ^,

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

Received: 03 June 2021 Accepted: 02 November 2021 Published: 10 November 2021
MSC : 54A10, 54A20, 54C08, 54C10

In this paper, $ \omega _{s} $-irresoluteness as a strong form of $ \omega _{s} $-continuity is introduced. It is proved that $ \omega _{s} $-irresoluteness is independent of each of continuity and irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.
- $ \omega _{s} $-open sets,
- semi-continuous function,
- $\omega _{s} $-continuous function,
- irresolute function,
- semi-open function,
- pre-semi-open function,
- semi-compact
Citation: Samer Al Ghour. On some types of functions and a form of compactness via $ \omega _{s} $-open sets[J]. AIMS Mathematics, 2022, 7(2): 2220-2236. doi: 10.3934/math.2022126

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Abstract

In this paper, $ \omega _{s} $-irresoluteness as a strong form of $ \omega _{s} $-continuity is introduced. It is proved that $ \omega _{s} $-irresoluteness is independent of each of continuity and irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.

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