About extension of upper semicontinuous multi-valued maps and applications

arXiv:0810.3127v1 [math.GN] 17 Oct 2008 About extension of upper semicontinuous multi-valued maps and applications Y. Askoura October 29, 2018 Gretia, INRETS, 2 Avenue du G´en´eral Malleret-Joinville, 94114 Arcueil, France email : askoura@inrets.fr, Abstract We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Pre- cisely, 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 com- pletely 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. Mathematics Subject Classification: 54C60, 54C20 Keywords: Extension of multi-valued maps, Tietze-Urysohn extension theorem, Fixed points, Maximal elements, Qualitative equilibrium 1 Introduction The aim of this paper is to extend upper semicontinuous (usc in short) multi-valued maps over spaces as large as possible. In particular, we want to avoid metrizability in definition domains. We are interested by extending usc multi-valued maps defined on a closed subset of a given topological space to another. Similar results are already obtained by Cellina [6], Brodskii [4, 5], Tan and Wu [21] and Ma [15], using the metrizability of the domain. A more general result, in this direction, is that of Borges [2], which established extensions for usc maps defined on closed subsets of stratifiable spaces to any topological space. Cite also results of Drozdovskii and Filippov [8] and Shishkov [20] which extended usc maps defined on closed subsets of paracompact completely normal spaces to completely metrizable ones. Another type of extensions, which is not concerned here, is to extend maps defined on dense subsets [11, 14]. 1 In the sequel, when speaking about extension of maps, we signify extensions of the same type (single valued if the map is single valued and multi-valued if the map is multi-valued) and preserving the given continuity concept (continuity if the original map is single valued and continuous, upper semicontinuity if the original map is multi-valued and upper semi- continuous). If we try to compare the extension of usc multi-valued maps with continuous single-valued maps, two things appear. First, for usc maps, the extension need only be usc, then we are tempted to say that the first extension is easier. But, within the definition domains, for a map, to be continuous and single-valued is very constraining comparatively with the fact to be usc and multi-valued, then provides us additional properties. So the two problems are, a priori, quite different, and without evident comparison between them. The results of this paper (and some of the cited ones) prove, in fact, that the extension of multi- valued maps is more constraining. We consider an usc multi-valued map T : A ⊂E →I, where A is a closed subset of a topological space E and I the unit real interval. We obtain an extension of T when E is completely normal or E is normal and A is perfectly normal. As it is known, this type of results provides a characterization of completely normal spaces. An extension result (for multi-valued maps) in an infinite uncountable product of spaces gives directly some existence results of maximal elements and fixed points ([10], [19] and [12]) needed in game theory. The problem is to avoid in the proof, properties which are not satisfied in an uncountable products of usual (or a simply important class of) spaces (like metrizability). Unfortunately, the complete normality is not, in general, a property of an infinite uncountable product of spaces. So, the results characterizing the complete normality by extensions of usc multi-valued maps, suggest us to search maximal elements (resp. equilibrium for qualitative games) for an uncountable usc multi-valued maps (resp. with an uncountable set of players and usc preference correspondences) with additional requirements. Such applications are given in the second part of this paper. Our result can be seen as a multi-valued version of the Tietze-Urysohn extension theorem. As it is known, the original proof of this fundamental result uses the uniform convergence of a sequence of functions. However, some researchers ([1, 16, 17, 18, 22]) asked for a possibility to prove this result without the use of uniform convergence. The proof of our extension theorem is inspired particularly by the paper of Ossa [17], and proves that his technics are successful for usc multi-valued maps. It is direct and elementary. In this paper, usc means upper semicontinuous, lsc means lower semicontinuous, co(A) means the convex hull of A. If Y is a topological space and X a sub

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