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