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 | References | Similar Articles | Additional Information
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