WHEN TWO COMPATIBLE METRICS GENERATE THE SAME TOPOLOGIES OF UNIFORM CONVERGENCES ON FUNCTIONAL SPACES

Abstract

Let $X$ be a Tychonoff space, $Y$ be a metrizable one and $C(X,Y)$ be the space of continuous functions from $X$ to $Y$. It is a classical result that two compatible metrics on $Y$ generate the same topologies of uniform convergence on compacta on $C(X,Y)$. We extend the result for the space $U(X,Y)$ of upper semicontinuous nonempty compact-valued maps from $X$ to $Y$. We also present characterizations of uniform equivalence of metrics via uniform convergence on functional spaces, as well as functional characterizations of locally compact and $k$-spaces. We give a partial answer to a question posed in Topological properties of spaces of continuous functions (1988), after Example 1.2.7, when two compatible metrics on $Y$ generate the same topologies of uniform convergence on $C(X,Y)$.