Mathematika

Research Article

A generic factorization theorem

P. S. Kenderova1 and J. Orihuelaa2

a1 Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria.

a2 Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain.

Abstract

Let F:ZX 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:PX 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: Z1P1 such that F(Z)G(f(z)) for every zZ1. 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.

(Received September 01 1993)

Key Words:

  • 54C60: GENERAL TOPOLOGY; Maps and general types of spaces defined by maps; set-valued maps.