Relations between some topologies

Abstract

Generalizations of openness, such as semi-open, preopen,semi-pre-open, $\alpha$-open, etc\. are important in topologicalspaces and in particular in topological spaces on which ideals aredefined. $\alpha$-equivalent topologies and $\ast$-equivalenttopologies with respect to an ideal have some common properties.Relations between these aforementioned notions of openness areinvestigated within the framework of $\alpha$-equivalence and$\ast$-equivalence.