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

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.