Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T01:07:04.927Z Has data issue: false hasContentIssue false
  • English
  • Français

The Cone = Hyperspace Property

Published online by Cambridge University Press:  20 November 2018

James T. Rogers Jr.
James T. Rogers Jr.*
Affiliation:
Tulane University, New Orleans, Louisiana
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The author has recently shown [11] that the hyperspace of subcontinua of a solenoid is homeomorphic to the cone over that solenoid. This is an interesting result, for it is the first time that the hyperspace of subcontinua of a complicated space has been recognized. This homeomorphism, moreover, is the expected map; it maps the singletons onto the base of the cone and the point corresponding to the whole space onto the vertex of the cone. We say that spaces for which such natural homeomorphisms exist have the cone = hyperspace property. In the first section we prove the following theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 279 - 285
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bing, R. H., The elusive fixed-point property, Amer. Math. Monthly 76 (1969), 119–132.Google Scholar
2. Bing, R. H., Snake-like continua, Duke Math. J. 18 (1951), 653–663.Google Scholar
3. Dugundji, J., Topology (Allyn and Bacon, Boston, 1966).Google Scholar
4. Tom, Ingram, Decomposable circle-like continua, Fund. Math. 63 (1968), 193–198.Google Scholar
5. Kelley, J. L., Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), pp. 22–36.Google Scholar
6. Knill, R. J., Cones, products and fixed points, Fund. Math. 60 (1967), 35–46.Google Scholar
7. Kuratowski, K., Topology, Volume II (Academic Press, New York, 1968).Google Scholar
8. Sam B., Nadler, Jr., Multicoherence techniques applied to inverse limits, Trans. Amer. Math. Soc. 157 (1971), 227–234.Google Scholar
9. Rhee, C. J., On dimension of hyperspaces of a metric continuum, Bull. Soc. Roy. Sci. Liège 38 (1969), 602–604.Google Scholar
10. James T., Rogers, Jr., Dimension of hyperspaces, Bull. Pol. Acad. Sci. 19 (1971) 25–27.Google Scholar
11. James T., Rogers, Jr., Embedding the hyper space of a circle-like plane continua, Proc. Amer. Math. Soc. 29 (1971), 165–168.Google Scholar
12. Young, G. S., Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880–884.Google Scholar