Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T17:32:35.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Open mappings on spheres

Published online by Cambridge University Press:  09 April 2009

Edwin Duda
Edwin Duda
Affiliation:
University of Miami Coral Gables, Fla.
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 we study open mappings of the sphere, Sn, onto itself. In particular, sufficient conditions are given that such a mapping be a homeomorphism. For the cases n≦ 2 many of the results could be obtained from the work of G. T. Whyburn [7], [8], and [10]. For the cases n ≦ 3 the useful results of A. V. Cernavskii, [1], [4], proved to be sufficient. An application is made concerning a finite to one open mapping of one n cell onto itself. It is interesting to note that for n ≦ 2 that we could use similar proofs to show that certain quasi-monotone mappings of Sn onto Sn are necessarily monotone mappings.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 3-4 , November 1969 , pp. 330 - 334
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Černavskii, a. V., ‘Finite to one open mappings on manifolds’, Mat. Sb. (NS) 65 (107) (1964), 357–369.Google Scholar
[2]Church, P. T. and Hemmingsen, E., ‘Light open maps on n-manifolds’, Duke Math. J. 27 (1960), 527–536.CrossRefGoogle Scholar
[3]Moore, R. L., ‘Concerning upper semi-continuous collections of continua’, Trans. Amer. Math. Soc. 27 (1925), 416–428.CrossRefGoogle Scholar
[4]Väisälä, Jussi, ‘Discrete open mappings on manifolds’, Ann. Acad. Sci. Fenn. A I 392 (1966), 1–10.Google Scholar
[5]Wallace, A. D., ‘Quasi monotone transformations’, Duke Math. J. 7 (1940), 136–145.CrossRefGoogle Scholar
[6]Whyburn, G. T., ‘Quasi-open mappings’, Rev. Math. Pure Appl. 2 (1957), 47–52.Google Scholar
[7]Whyburn, G. T., ‘Topological analysis’, Princeton Math. Series-Princeton University Press, Princeton, N. J. 1958.Google Scholar
[8]Whyburn, G. T., ‘Analytic topology’, Amer. Math. Soc. Colloquim Publication, 28, (1942).Google Scholar
[9]Whyburn, G. T., ‘Open and closed mappings’, Duke Math. J. 17 (1950), 69–74.Google Scholar
[10]Whyburn, G. T., ‘Concerning plane closed point sets which are accessible from certain subsets of their complements’, Proc. Nat. Acad. Sci. 14 (1928), 657–666.Google Scholar
[11]Summary of Lectures and Seminars — Revised Edition University of Wisconsin Summer Institute on Set Theoretic Topology.Google Scholar