Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance

Abstract:

Atsuji has internally characterized those metric spaces $X$ for which each real-valued continuous function on $X$ is uniformly continuous as follows: (1) the set $X’$ of limit points of $X$ is compact, and (2) for each $\varepsilon > 0$, the set of points in $X$ whose distance from $X’$ exceeds $\varepsilon$ is uniformly discrete. We obtain these new characterizations: (a) for each metric space $Y$, the Hausdorff metric on $C\left ( {X,Y} \right )$, induced by a metric on $X \times Y$ compatible with the product uniformity, yields the topology of uniform convergence; (b) there exists a metric space $Y$ containing an arc for which the Hausdorff metric on $C\left ( {X,Y} \right )$ yields the topology of uniform convergence; (c) the Hausdorff metric topology on ${\text {CL}}\left ( X \right )$ is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space.