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A generic factorization theorem

Part of: Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  26 February 2010

P. S. Kenderov  and
J. Orihuela
P. S. Kenderov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria.
J. Orihuela
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain.
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Abstract

Let F:Z→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→X can be constructed so that for every dense Gδ-subset P1 of P there exist a dense Gδ Z1 of Z and a (single-valued) continuous map f: Z1→P1 such that F(Z)⊂G(f(z)) for every z∈Z1. In particular, if G is single valued on a dense Gδ-subset of P, then F is also single-valued on a dense Gδ-subset of its domain. The above theorem remains valid if Z is Čech complete space and X is an arbitrary completely regular space.

These factorization theorems show that some generalizations of a theorem of Namioka 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 classification

Secondary: 54C60: Set-valued maps

Information

Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 56 - 66
Copyright
Copyright © University College London 1995

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