Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Continuous selections and $C$-spaces

Authors: Valentin Gutev and Vesko Valov
Journal: Proc. Amer. Math. Soc. 130 (2002), 233-242
MSC (2000): Primary 54C60, 54C65, 55M10
Published electronically: May 22, 2001
MathSciNet review: 1855641
Full-text PDF

Abstract | References | Similar Articles | Additional Information


A characterization of paracompact $C$-spaces via continuous selections avoiding $Z_\infty$-sets is given. The result is applied to prove a countable sum theorem for paracompact $C$-spaces, and to obtain a new partial solution of a question raised by E. Michael.

References [Enhancements On Off] (What's this?)

  • 1. D. Addis and J. Gresham, A class of infinite-dimensional spaces. Part I: Dimension theory and Alexandroff's Problem, Fund. Math. 101 (1978), 195-205. MR 80b:54041
  • 2. A. Chigogidze, Inverse Spectra, North-Holland, Amsterdam, 1996. MR 97g:54001
  • 3. J. Dieudonne, Une généralisation des espaces compacts, J. de Math. Pures et Appl. 23 (1944), 65-76. MR 7:134f
  • 4. C. H.Dowker, On countably paracompact spaces, Canad. J. of Math. 3 (1951), 219-224. MR 13:264c
  • 5. V. Gutev, Continuous selections, $G_{\delta}$-subsets of Banach spaces and usco mappings, Comment. Math. Univ. Carolinae 35 (1994), no. 3, 533-538. MR 96i:54011
  • 6. W. Haver, A covering property for metric spaces, Lecture Notes in Math. 375, Springer Verlag, New York, 1974. MR 51:1756
  • 7. M. Katetov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85-91. MR 14:304a
  • 8. E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361-382. MR 17:990e
  • 9. -, Continuous selections II, Ann. of Math. 64 (1956), 562-580. MR 18:325e
  • 10. -, Continuous selections avoiding a set, Top. Appl. 28 (1988), 195-213. MR 90h:54025
  • 11. -, Some problems, Open problems in Topology, J. van Mill and J. M. Reed (Editors), Chapter 17, 271-278, North-Holland, Amsterdam, 1990. CMP 91:03
  • 12. J. van Mill, Infinite-dimensional Topology Prerequisites and Introduction, North-Holland, Amsterdam, 1989. MR 90a:57025
  • 13. J. Munkres, Topology: a first course, Prentice Hall, Englewood Cliffs, NY, 1975. MR 57:4063
  • 14. R. Pol, A weakly infinite-dimensional compactum which is not countable dimensional, Proc. Amer. Math. Soc. 82 (1981), 634-636. MR 82f:54059
  • 15. H. Torynczyk, Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds, Fund. Math. 101 (1978), 93-110. MR 80g:57019
  • 16. V. Uspenskij, A selection theorem for $C$-spaces, Top. Appl. 85 (1998), 351-374. MR 99d:54013

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C60, 54C65, 55M10

Retrieve articles in all journals with MSC (2000): 54C60, 54C65, 55M10

Additional Information

Valentin Gutev
Affiliation: School of Mathematical and Statistical Sciences, Faculty of Science, University of Natal, King George V Avenue, Durban 4041, South Africa

Vesko Valov
Affiliation: Department of Mathematics, Nipissing University, 100 College Drive, P. O. Box 5002, North Bay, Ontario, Canada P1B 8L7

Keywords: Continuous selection, $C$-space, $Z_\infty$-set
Received by editor(s): November 17, 1999
Received by editor(s) in revised form: May 9, 2000
Published electronically: May 22, 2001
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society