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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 $$\mathbb{R}\times E$$.

##### MSC:
 54C65 Selections in general topology 34A60 Ordinary differential inclusions
##### Keywords:
directionally continuous selections
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