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A generic factorization theorem
Author(s) -
Kenderov P. S.,
Orihuela J.
Publication year - 1995
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011359
Subject(s) - mathematics , metrization theorem , linear subspace , complete metric space , space (punctuation) , factorization , metric space , discrete mathematics , domain (mathematical analysis) , weierstrass factorization theorem , combinatorics , pure mathematics , separable space , mathematical analysis , linguistics , philosophy , algorithm
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 G δ ‐subset P 1 of P there exist a dense G δ Z 1 of Z and a (single‐valued) continuous map f : Z 1 → P 1 such that F ( Z )⊂ G ( f ( z )) for every z ∈Z 1 . In particular, if G is single valued on a dense G δ ‐subset of P , then F is also single‐valued on a dense G δ ‐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.