On Dini’s theorem and a metric on $C(X)$ topologically equivalent to the uniform metric

Abstract:

Let $X$ be a compact metric space and let $UC(X)$ denote the u.s.c. real valued functions on $X$. Let $\tau$ be a topology on $UC(X)$. $\Omega \subset UC(X)$ is called a Dini class of functions induced by $\tau$ if (1) $\Omega$ is $\tau$-closed, (2) $C(X) \subset \Omega$, (3) for each $h \in \Omega$ whenever $\{ {h_n}\}$ is a decreasing sequence of u.s.c. functions convergent pointwise to $h$, then $\{ {h_n}\} \tau$-converges to $h$. By Dini’s theorem the topology of uniform convergence on $UC(X)$ induces $C(X)$ as its Dini class of functions. As a main result, when $X$ is locally connected we show that the hyperspace topology on $UC(X)$ obtained by identifying each u.s.c. function with the closure of its graph induces a larger Dini class of functions than $C(X)$, even though the restriction of this topology to $C(X)$ agrees with the topology of uniform convergence.