Rocky Mountain Journal of Mathematics

New characterizations of minimal cusco maps

Ľubica Holá and Dušan Holý

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We give new characterizations of minimal cusco maps in the class of all set-valued maps extending results from \cite{BZ1, GM}. Let $X$ be a topological space and $Y$ a Hausdorff locally convex linear topological space. Let $F: X \to Y$ be a set-valued map. The following are equivalent: (1)~$F$ is minimal cusco; (2)~$F$ has nonempty compact values, there is a quasicontinuous, subcontinuous selection $f$ of $F$ such that $F(x) = \overline{co\,}\overline f(x)$ for every $x \in X$; (3)~$F$ has nonempty compact values, there is a densely defined subcontinuous, quasicontinuous selection $f$ of $F$ such that $F(x) = \overline{co}\,\overline f(x)$ for every $x \in X$; (4)~$F$ has nonempty compact convex values, $F$ has a closed graph, every extreme function of $F$ is quasicontinuous, subcontinuous and any two extreme functions of $F$ have the same closures of their graphs. Some applications to known results are given.

Article information

Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1851-1866.

First available in Project Euclid: 2 February 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
Secondary: 54B20: Hyperspaces

Minimal cusco map quasicontinuous function sub continuous function set-valued mapping selection extreme function


Holá, Ľubica; Holý, Dušan. New characterizations of minimal cusco maps. Rocky Mountain J. Math. 44 (2014), no. 6, 1851--1866. doi:10.1216/RMJ-2014-44-6-1851.

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