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A non-separable Christensen’s theorem and set tri-quotient maps. (English) Zbl 1179.54043

Let \(X\), \(Y\) be metrizable spaces and \(F:{\mathcal K}(X)\to{\mathcal K}(Y)\) a monotone map acting between the hyperspaces of all compact sets of \(X\), \(Y\) resp., such that any countable \(L\in{\mathcal K}(Y)\) is covered by \(F(Y)\) for some \(K\in{\mathcal K}(X)\). Then \(Y\) is proved to be completely metrizable provided \(X\) is completely metrizable. Generalizations to the case of sieve-complete spaces and set tri-quotient maps are also investigated and possible applications treated.

MSC:

54E50 Complete metric spaces
54C60 Set-valued maps in general topology
54B20 Hyperspaces in general topology
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