Metric Spaces on Which Continuous Functions Are Uniformly Continuous and Hausdorff Distance

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 £ > 0, the set of points in X whose distance from X' exceeds e is uniformly discrete. We obtain these new characterizations: (a) for each metric space V, the Hausdorff metric on C(X, Y), induced by a metric on X x 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(X, Y) yields the topology of uniform convergence; (c) the Hausdorff metric topology on CL(X) is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space. Let {W, d) be a metric space. If K C W and e is positive, let SE [K] denote the union of all open e-balls whose centers run over K. If Ki and K2 are nonempty subsets of W, and for some e > 0 both S£[Äi] D K2 and Se[Ä2] D Kx, then the Hausdorff distance h¿ between them is given by the formula h¿{Kx,K-í) = inf{e: Se[Äi] D K2 and Se[K2] D Kx}. Otherwise, we write h¿(Kx,K2) = 00. If we restrict hd to the closed nonempty subsets CL(W) of W, then h<¡ defines an infinite-valued metric. Basic facts about this metric can be found in Aubin [2], Castaing and Valadier [3], or Engelking [4]. Now let C(X, Y) denote the continuous functions from a metric space (X, dx) to a metric space (Y, dy). We denote uniform distance in C(X, Y) by dx, that is, di(f,g) = sup{dY(f(x),g(x)): x e X}. Alternatively, we can identify members of C(X, Y) with their graphs in X x Y, and define the distance between functions to be the Hausdorff distance between their graphs, as induced by some metric compatible with the product uniformity. For definiteness and computational simplicity, we choose the 601 metric: P[(xi,yi), (x2,2/2)] = max{dx(xi,x2), dY(yi,y2)}. In the sequel, whenever / and g are continuous, we shall write d2(f, g) for hp(f, g). It is easy to check that d2(f,g) < di{f,g), so that the topology of uniform convergence is always at least as strong as the one defined by d2. Furthermore, if X is compact and V is arbitrary, then di and d2 are equivalent metrics on C(X, Y) [7]. This equivalence has been the basis for numerous papers in approximation theory by Bl. Sendov, V. Popov, B. Penkov, V. Veselinov and their associates in Sofia (see, e.g, [9, 10 and 12]). However, <fi and d2 can be equivalent somewhat Received by the editors December 14, 1984 and, in revised form, February 7, 1985. 1980 Mathematics Subject Classification. Primary 54B20; Secondary 54C35, 54E45.