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An extension theorem for multifunctions and a characterization of complete metric spaces. (English) Zbl 0649.54007
The following theorem is proved: A metrizable space Y is topologically complete if and only if each upper semi-continuous compact-valued multifunction $$F: A\to Y$$, where A is dense in a given space X, has an upper semi-continuous compact-valued extension to a $$G_{\delta}$$-subset of X containing A.
{Reviewer’s remark. The above theorem remains valid if Y is completely regular. Indeed, define a multifunction $$F_ 1: X\to \beta Y$$ by $$F_ 1(x)=\cap \{cl_{\beta Y}(F(U\cap A)):$$ U is a neighbourhood of x in $$X\}$$. It is easily seen that $$F_ 1$$ is upper semi-continuous and compact-valued and $$F_ 1(x)=F(x)$$ for all $$x\in A$$. Now, let $$A_ 1=\{x\in X:$$ $$F_ 1(x)\subset Y\}$$. If Y is Čech-complete, then $$A_ 1$$ is a $$G_{\delta}$$-subset of X containing A. To prove the inverse implication it is enough to consider Y as a subset of $$\beta$$ Y and the identity mapping $$i: Y\to Y.\}$$
Reviewer: V.Valov
##### MSC:
 54C20 Extension of maps 54C60 Set-valued maps in general topology 54E50 Complete metric spaces
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##### References:
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