Topological Sensitivity on Hyperspaces

Abstract

In 2001, Arhangel'skii proved that a topological group that is factorizable over the class of (strong) $PT$-groups, is a (strong) $PT$-group. Wegeneralize this result to Tychonoff quasitopological groups and prove that a quasitopological group $G$ that is factorizable over the class of quasi-$PT$-groups, is a quasi-$PT$-group. We define the notion of $\mathcal{P}$-sz-factorizability, generalizing $\mathcal{P}$-factorizability and prove that a (quasi)topological group $G$ that is sz-factorizable over the class of (quasi-)$PT$-groups, is a (quasi-)$PT$-group. A topological group $G$ that is sz factorizable over the class of strong $PT$-groups, is a strong $PT$-group. We also present some other results on quasi-$PT$-groups and $PT$-groups.