About extension of upper semicontinuous multi-valued maps and applications

Details About extension of upper semicontinuous multi-valued maps and applications About extension of upper semicontinuous multi-valued maps and applications

2008-10-17

arxiv.org/abs/0810.3127v1

Mathematics

General Topology

Scientific paper

We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal subset) of a completely normal (resp. of a normal) space $X$ into the unit interval $[0,1]$ can be extended to the whole space $X$. The extension is upper semicontinuous with nonempty closed convex values. We apply this result for the extension of real semicontinuous functions, the characterization of completely normal spaces, the existence of Gale-Mas-Colell and Shafer-Sonnenschein type fixed point theorems and the existence of equilibrium for qualitative games.

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