An extension theorem for multifunctions and a characterization of complete metric spaces.

*(English)*Zbl 0649.54007The 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’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

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