On Continuity and Generalized Continuity with Respect to Two Topologies

Abstract

Let $\tau _1$ and $\tau _2$ be topologies in $X$ and let $\tau = \tau _1 \cap \tau _2$. Some conditions concerning the topologies $\tau $, $\tau _1$ and $\tau _2$ and describing the relations between the $\tau $-continuity (quasicontinuity) [cliquishness] and the $\tau _i$-continuity (quasicontinuity) [cliquishness], $i = 1,2$, of functions defined on $X$ are considered.