Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T00:27:51.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Concerning the Cone = Hyperspace Property

Published online by Cambridge University Press:  20 November 2018

Dorothy D. Sherling
Dorothy D. Sherling*
Affiliation:
Smith College, Northampton, Massachusetts
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.

In this paper it is shown that a sufficient condition for a continuum X to have the cone = hyperspace property is that there exists a selection for C(X)\{X} which, for some Whitney map for C(X), maps each nondegenerate Whitney level homeomorphically onto X. Also, we construct an example of a one-dimensional, nonchainable, noncircle-like continuum which has the cone = hyperspace property. The continuum is described by means of inverse limits using only one bonding map. Each factor space in the inverse limit sequence is the quotient space resulting from an upper semi-continuous decomposition of a disjoint union of simple triods. The bonding map is an adaptation of the bonding map defined by W. T. Ingram in his construction of an atriodic, tree-like continuum which is not chainable [4].

Definitions, notation, and terminology. By continuum we mean a nonempty, compact, connected metric space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1030 - 1048
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Bing, R. H., A simple closed curve is the only homogenous hounded plane continuum that contains an arc, Can. J. Math. 12 (1960). 209–230.Google Scholar
2. Burgess, C. E., Chainable continua and indecomposability Pac. J. Math. 9 (1959), 653–659.Google Scholar
3. Dilks, A. M. and Rogers, J. T. Jr., Whitney stability and contractible hyperspaces, Proc. Amer. Math. Soc. 83 (1981), 633–640.Google Scholar
4. Ingram, W. T., An atriodic, tree-like continuum with positive span. Fund. Math. 77 (1972), 99–107.Google Scholar
5. Kelley, J. L., Hyperspaces of a continuum. Trans. Amer. Math. Soc. 52 (1942), 22–36.Google Scholar
6. Mardesic, S. and Segal, J., e-mappings onto polxhedra, Trans. Amer. Math. Soc. 109 (1963), 146–164.Google Scholar
7. Michael, E., Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152–158.Google Scholar
8. Mioduszewski, J., Mappings of inverse limits. Coll. Math. 10 (1963), 39–44.Google Scholar
9. Moebes, T. A., An atriodic, tree-like continuum that is not weakly chainable, Doctoral Dissertation, University of Houston, Houston, Tx. (1980).Google Scholar
10. Moore, R. L., Foundations of point set theory. Amer. Math. Soc. Colloquium Publications 13 (1962).Google Scholar
11. Nadler, S. B. Jr., Continua whose cone and hyperspace are homeomorphic, Trans. Amer. Math. Soc. 230 (1977), 321–345.Google Scholar
12. Nadler, S. B. Jr., Hyperspaces of sets (Marcel Dekker, New York, 1978).Google Scholar
13. Nadler, S. B. Jr., Some basic connectivity properties of Whitney map inverses in C(X), in Studies in topology (Academic Press, New York, 1975), 393–410.Google Scholar
14. Petrus, A., Whitney maps and Whitney properties of C(X), Topology Proceedings, (Proceedings of 1976 Topology Conference, Auburn University), 1 (1976), 147–172.Google Scholar
15. Rogers, J. T. Jr., The cone = hyperspace property, Can. J. Math. 24 (1972), 279–285.Google Scholar
16. Rogers, J. T. Jr., Continua with cones homeomorphic to hyperspaces, Gen. Top. and Its Applications 3 (1973), 283–289.Google Scholar
17. Rogers, J. T. Jr., Embedding the hyperspaces of circle-like plane continua, Proc. Amer. Math. Soc. 29 (1971), 165–168.Google Scholar
18. Rogers, J. T. Jr., Whitney continua in the hyperspace C(X), Pac. J. Math. 58 (1975), 569–584.Google Scholar
19. Segal, J., Hyperspaces of the inverse limit space, Proc. Amer. Math. Soc. 10 (1959), 706–709.Google Scholar