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A generic factorization theorem. (English) Zbl 0827.54012

Summary: Let \(F : Z \to X\) be a minimal usco map from the Baire space \(Z\) into the compact space \(X\). Then a complete metric space \(P\) and a minimal usco \(G : P \to X\) can be constructed so that for every dense \(G_\delta\)- subset \(P_1\) of \(P\) there exist a dense \(G_\delta\)-subset \(Z_1\) of \(Z\) and a (single-valued) continuous map \(f : Z_1 \to P_1\) such that \(F(z) \subset G(f (z))\) for every \(z \in Z_1\). In particular, if \(G\) is single-valued on a dense \(G_\delta\)-subset of \(P\), then \(F\) is also single-valued on a dense \(G_\delta\)-subset of its domain. The above theorem remains valid if \(Z\) is a Čech complete space and \(X\) is an arbitrary completely regular space.
These factorization theorems show that some generalizations of a theorem of I. Namioka [Pac. J. Math. 51, 515-531 (1974; Zbl 0294.54010)] concerning generic single-valuedness and generic continuity of mappings defined in more general spaces can be derived from similar results for mappings with complete metric domains.
The theorems can be used also as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.

MSC:

54C60 Set-valued maps in general topology

Citations:

Zbl 0294.54010
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References:

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