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


Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 86 (1982), 75-80
MSC: Primary 26A15; Secondary 54B20, 54C35
DOI: https://doi.org/10.1090/S0002-9939-1982-0663870-7
MathSciNet review: 663870
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0663870-7
Keywords: Semicontinuous function, Dini's theorem, Hausdorff metric
Article copyright: © Copyright 1982 American Mathematical Society

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