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

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References

Bou.Bouziad, Ahmed. Une class d'espace co-Namioka. C.R. Acad. Sci. Paris, 310 (1990), 779–782.Google Scholar
Chr.Christensen, J. P. R.. Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings. Proc. Amer. Math. Soc, 86 (86), 649–655.CrossRefGoogle Scholar
ChrK.Christensen, J. P. R. and Kenderov, P. S.. Dense strong continuity of mappings and the Radon-Nykodym property. Math. Scand., 54 (54), 70–78.CrossRefGoogle Scholar
CK 1.Coban, M. M. and Kenderov, P. S.. Dense Gateaux differentiability of the sup-norm in C(T) and the topological properties of T. Compt. Rend. Acad. Bulg. Sci., 38 (38), 1603–1604.Google Scholar
CK 2.Coban, M. M. and Kenderov, P. S.. Generic Gateaux differentiability of convex functionals in C(T) and the topological properties of T. Math, and Education in Math., Proc. of the XV-th Spring Conf. of the Union of Bulg. Mathematicians, 1986, op. 141–149.Google Scholar
CKR 1.Čoban, M. M., Kenderov, P. S. and Revalski, J. P.. Densely defined selections of multivalued mappings. Trans. Amer. Math. Soc, 344 (344), 533–552.CrossRefGoogle Scholar
ČKR 2.Čoban, M. M., Kenderov, P. S. and Revalski, J. P.. Characterizations of topological spaces with almost completeness properties. Preprint.Google Scholar
Deb 1.Debs, G.. Fonctions separement continues et de premier class sur un espace produit. Math. Scand., 59 (59), 122–130.CrossRefGoogle Scholar
Deb 2.Debs, G.. Points de continuite d'une fonction séparément continue. Proc. Amer. Math. Soc, 97 (97), 167–176.Google Scholar
Deb 3.Debs, G.. Points de continuite d'une fonction séparément continue II. Proc. Amer. Math. Soc, 99 (99), 777–782.Google Scholar
Dev.Deville, R.. Espaces de Baire et espaces de Namioka. Séminaire d'Initation á Analyse (G. Choquet-G. Godefroy-M. Rogalski-J. Saint-Raymond), 23éme Annee, 1983/84, No. 2, 9pp.Google Scholar
Fro 1.Frolik, Z.. Generalizations of the Gδ-property of complete metric spaces. Czech. Math. J., 10 (10), 359–379.CrossRefGoogle Scholar
Fro 2.Frolík, Z.. Baire spaces and some generalizations of complete metric spaces. Czech. Math.J., 11 (11), 237–247.CrossRefGoogle Scholar
KeNa.Kelley, J. and Namioka, I.. Linear Topological Spaces (Princeton University Press, 1963).CrossRefGoogle Scholar
K 1.Kenderov, P. S.. Most of the optimization problems have unique solution. Compt. rend. Acad. Bulg. Set, 37 (37), 297–299.Google Scholar
K2.Kenderov, P. S.. Most of the optimization problems have unique solution. ISNM Birkhauser, Basel, 72 (72), 203–216.Google Scholar
KR 1.Kenderov, P. S. and Revalski, J. P.. Residually defined selections of set-valued mappings. Séminaire d'Initation a I'Analyse (G. Choquet-G. Godefroy-M. Rogalski-J. Saint-Raymond), 30éme Année, 104 1990/91, No. 17, 7pp.Google Scholar
KR 2.Kenderov, P. S. and Revalski, J. P.. The Banach-Mazur game and generic existence of solutions to optimization problems. Proc. Amer. Math. Soc, 118 (1993), 911–917.CrossRefGoogle Scholar
M. E.Michael, . A note on completely metrizable spaces. Proc. Amer. Math. Soc, 96 (1986), 513–522.CrossRefGoogle Scholar
Na. I.Namioka, . Separate continuity and joint continuity. Pacific J. Math., 51 (1974), 515–531.CrossRefGoogle Scholar
SR. J.Saint-Raymond, . Jeux topologique et espaces de Namioka. Proc. Amer. Math. Soc, 87 (87), 499–504.CrossRefGoogle Scholar
St 1.Stegall, C.. A class of topological spaces and differentiation of functions in Banach spaces. Vorlesungen aus dem Fachbereich Malhematik der Universitdt Essen, 10 (10), 63–67.Google Scholar
St 2.Stegall, C.. Generalization of a theorem of Namioka. Proc. Amer. Math. Soc, 102 (102), 559–564.CrossRefGoogle Scholar
St 3.Stegall, C.. The topology of certain spaces of measures. Topology Appl, 41, n.1-2 (2), 73–112.CrossRefGoogle Scholar
Tal.Talagrand, M.. Espaces de Baire et espace de Namioka. Math. Ann., 270 (270), 159–164.CrossRefGoogle Scholar
Tel.Telgarsky, R.. Topological games: on the 50-th anniversary of the Banach-Mazur game. Rocky Mount. J. Math., 17 (17), 227–276.Google Scholar
Tr.Troallic, J.-P.. Fonctions à valeurs dans espaces fonctionels generaux: theorems de R. Ellis et de I. Namioka, C.R. Acad. Sci. Paris, 287, serie A (1978), 63–66.Google Scholar