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)



Vietoris continuous selections and disconnectedness-like properties

Authors: Valentin Gutev and Tsugunori Nogura
Journal: Proc. Amer. Math. Soc. 129 (2001), 2809-2815
MSC (2000): Primary 54C65, 54B20, 54F45
Published electronically: February 9, 2001
MathSciNet review: 1838807
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Suppose that $X$ is a Hausdorff space such that its Vietoris hyperspace $({\mathcal{F}}(X),\tau _{V})$ has a continuous selection. Do disconnectedness-like properties of $X$ depend on the variety of continuous selections for $({\mathcal{F}}(X),\tau _{V})$ and vice versa? In general, the answer is ``yes'' and, in some particular situations, we were also able to set proper characterizations.

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

  • 1. G. Artico, U. Marconi, R. Moresco and J. Pelant, Selectors and Scattered Spaces, Topology Appl., to appear.
  • 2. D. Bertacchi and C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl. 88 (1998), 179-197. MR 99g:54003
  • 3. M. Choban, Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc. 22 (1970), 258-280.
  • 4. C. Costantini and V. Gutev, Recognizing special metrics by topological properties of the ``metric''-Proximal hyperspace, preprint.
  • 5. R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150-158. MR 39:6272
  • 6. V. Gutev and T. Nogura, Selections for Vietoris-like hyperspace topologies, Proc. London Math. Soc. 80 (1) (2000), 235-256. CMP 2000:04
  • 7. E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13:54f
  • 8. J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), 601-605. MR 82i:54038
  • 9. T. Nogura and D. Shakhmatov, Characterizations of intervals via continuous selections, Rendiconti del Circolo Matematico di Palermo, Serie II 56 (1997), 317-328. MR 99d:54012
  • 10. T. Nogura and D. Shakhmatov, Spaces which have finitely many continuous selections, Bollettino U. M. I. (7) 11-A (1997), 723-729. MR 99a:54011
  • 11. T. C. Przymusinski, On the dimension of product spaces and an example of M. Wage, Proc. Amer. Math. Soc. 76 (1979), 315-321. MR 80f:54033
  • 12. M. L. Wage, The dimension of product spaces, preprint (1977).
  • 13. M. L. Wage, The dimension of product spaces, Proc. Mat. Acad. Sci. USA 75 (1978), 4671-4672. MR 80a:54064

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C65, 54B20, 54F45

Retrieve articles in all journals with MSC (2000): 54C65, 54B20, 54F45

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

Tsugunori Nogura
Affiliation: Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790 Japan

Keywords: Selections, hyperspaces, zero-dimensionality
Received by editor(s): November 17, 1999
Received by editor(s) in revised form: January 17, 2000
Published electronically: February 9, 2001
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society