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ArticleOriginal scientific text

Title

Selections that characterize topological completeness

Authors 1, 2, 3

Affiliations

  1. Faculteit Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081 A, 1081 HV Amsterdam, The Netherlands
  2. epartment of Mathematics, Warsaw University, Banacha 2, 00-913 Warszawa 59, Poland
  3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 11567 Praha 1, Czech Republic

Abstract

We show that the assertions of some fundamental selection theorems for lower-semicontinuous maps with completely metrizable range and metrizable domain actually characterize topological completeness of the target space. We also show that certain natural restrictions on the class of the domains change this situation. The results provide in particular answers to questions asked by Engelking, Heath and Michael [3] and Gutev, Nedev, Pelant and Valov [5].

Keywords

ENG
selection, Vietoris topology, completely metrizable

Bibliography

  1. M. M. Čoban and E. A. Michael, Representing spaces as images of zero-dimensional spaces, Topology Appl. 49 (1993), 217-220.
  2. R. Engelking, General Topology, Heldermann, Berlin, 1989.
  3. R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150-158.
  4. W. G. Fleissner, Applications of stationary sets to topology, in: Surveys in Topology, G. M. Reed (ed.), Academic Press, New York, 1980, 163-193.
  5. V. Gutev, S. Nedev, J. Pelant, and V. Valov, Cantor set selectors, Topology Appl. 44 (1992), 163-168.
  6. R. W. Hansell, Descriptive topology, in: Recent Progress in General Topology, M. Hušek and J. van Mill (eds.), North-Holland, Amsterdam, 1992, 275-315.
  7. F. Hausdorff, Über innere Abbildungen, Fund. Math. 23 (1934), 279-291.
  8. V. G. Kanoveĭ and A. V. Ostrovskiĭ, On non-Borel FII-sets, Soviet Math. Dokl. 24 (1981), 386-389.
  9. K. Kunen, Set Theory. An Introduction to Independence Proofs, Stud. Logic Found. Math. 102, North-Holland, Amsterdam, 1980.
  10. K. Kuratowski, Topology I, Academic Press, New York, 1968.
  11. D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178.
  12. E. A. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238.
  13. E. A. Michael, A theorem on semi-continuous set-valued functions, Duke Math. J. 26 (1959), 656-674.
  14. J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), 601-605.
  15. A. H. Stone, On σ-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655-666.
Fundamenta Mathematicae
1996/149/2
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Pages:
127-141
Main language of publication
English
Received
1995-01-23
Accepted
1995-11-17
Published
1996
Exact and natural sciences