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