Selections that characterize topological completeness

Jan van Mill; Jan Pelant; Roman Pol

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 2, page 127-141
  • ISSN: 0016-2736

Abstract

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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].

How to cite

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van Mill, Jan, Pelant, Jan, and Pol, Roman. "Selections that characterize topological completeness." Fundamenta Mathematicae 149.2 (1996): 127-141. <http://eudml.org/doc/212112>.

@article{vanMill1996,
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].},
author = {van Mill, Jan, Pelant, Jan, Pol, Roman},
journal = {Fundamenta Mathematicae},
keywords = {selection; Vietoris topology; completely metrizable; completely metrizable space},
language = {eng},
number = {2},
pages = {127-141},
title = {Selections that characterize topological completeness},
url = {http://eudml.org/doc/212112},
volume = {149},
year = {1996},
}

TY - JOUR
AU - van Mill, Jan
AU - Pelant, Jan
AU - Pol, Roman
TI - Selections that characterize topological completeness
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 2
SP - 127
EP - 141
AB - 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].
LA - eng
KW - selection; Vietoris topology; completely metrizable; completely metrizable space
UR - http://eudml.org/doc/212112
ER -

References

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  1. [1] M. M. Čoban and E. A. Michael, Representing spaces as images of zero-dimensional spaces, Topology Appl. 49 (1993), 217-220. 
  2. [2] R. Engelking, General Topology, Heldermann, Berlin, 1989. 
  3. [3] R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150-158. Zbl0167.20504
  4. [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. [5] V. Gutev, S. Nedev, J. Pelant, and V. Valov, Cantor set selectors, Topology Appl. 44 (1992), 163-168. Zbl0769.54020
  6. [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. [7] F. Hausdorff, Über innere Abbildungen, Fund. Math. 23 (1934), 279-291. Zbl60.0510.02
  8. [8] V. G. Kanoveĭ and A. V. Ostrovskiĭ, On non-Borel F I I -sets, Soviet Math. Dokl. 24 (1981), 386-389. 
  9. [9] K. Kunen, Set Theory. An Introduction to Independence Proofs, Stud. Logic Found. Math. 102, North-Holland, Amsterdam, 1980. 
  10. [10] K. Kuratowski, Topology I, Academic Press, New York, 1968. 
  11. [11] D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178. Zbl0222.02075
  12. [12] E. A. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238. Zbl0070.39502
  13. [13] E. A. Michael, A theorem on semi-continuous set-valued functions, Duke Math. J. 26 (1959), 656-674. 
  14. [14] J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), 601-605. Zbl0473.54010
  15. [15] A. H. Stone, On σ-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655-666. Zbl0117.40103

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