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Let X be a compact metric space and let UC(X) denote the u.s.c. real valued functions on X. Let r be a topology on UC(X). Q C UC(X) is called a Dini class of functions induced by T if (1) Q is T-closed, (2) C(X) C 0, (3) for each /iÇil whenever {hn) is a decreasing sequence of u.s.c. functions convergent pointwise to h, then {hn} r-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. Let X be a compact metric space and let C(X) and UC(X) denote the continuous and upper semicontinuous real valued functions on X, respectively. If / G C(X) and {hn} is a decreasing sequence of u.s.c. functions on X convergent pointwise to /, then Dini's theorem [10] says that the convergence must be uniform. Moreover, if {/„} is a sequence in C(X) convergent uniformly to an u.s.c. function h, then h must be in C(X). Thus, the class C(X), viewed as a subclass of UC(X), is induced by the topology of uniform convergence on UC(X) in the following sense. DEFINITION. Let X be a compact metric space and let fi C UC(X). If r is a topology on UC(X), then fi is called a Dini class of functions induced by t if (1) fi is T-closed, (2) C(X) C fi, (3) for each /i£il whenever {hn} is a decreasing sequence of u.s.c. functions convergent pointwise to h, then {hn} r-converges to h. It follows from the definition that t can induce at most one Dini class. To see this, suppose that fi and fi' are both Dini classes of r. Let / G fi be arbitrary. Since each u.s.c. function on X is a pointwise limit of a decreasing sequence of continuous functions [6], / G clT C(X) C clT fi' = fi'. Standard topologies on UC(X) seem to induce nice Dini classes of functions. For example, let p be a regular Borel measure on X. We first consider the convergence in measure topology on UC(X). Recall that {hn} is said to converge in measure to Received by the editors June 4, 1981 and, in revised form, November 11, 1981. 1980 Mathematics Subject Classification. Primary 26A15, 54B20, 54C35.

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