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

• Published in: AIMS mathematics Vol. 7; no. 2; pp. 2220 - 2236
• Main Author: Ghour, Samer Al
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2022
• Subjects: irresolute function; omega _{s} $-continuous function; omega _{s} $-open sets; pre-semi-open function; semi-compact; semi-continuous function; semi-open function
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2022126

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.