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Selections and representations of multifunctions in paracompact spaces. (English) Zbl 0807.54020
Summary: Let \((X,{\mathcal T})\) be a paracompact space, \(Y\) a complete metric space, \(F: X\to 2^ Y\) a lower semicontinuous multifunction with nonempty closed values. We prove that if \({\mathcal T}^ +\) is a (stronger than \({\mathcal T}\)) topology on \(X\) satisfying a compatibility property, then \(F\) admits a \({\mathcal T}^ +\)-continuous selection. If \(Y\) is separable, then there exists a sequence \((f_ n)\) of \({\mathcal T}^ +\)-continuous selections such that \(F(x)= \overline {\{f_ n (x); n\geq 1\}}\) for all \(x\in X\). Given a Banach space \(E\), the above result is then used to construct directionally continuous selections on arbitrary subsets of \(\mathbb{R}\times E\).

54C65 Selections in general topology
34A60 Ordinary differential inclusions
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