Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-09T02:14:45.750Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Topological Properties of Maps and Open Extensions of Maps

Published online by Cambridge University Press:  20 November 2018

J. K. Kohli
J. K. Kohli*
Affiliation:
Hindu College, University of Delhi, Delhi 110007, India
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.

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 6 , 01 December 1977 , pp. 1121 - 1128
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Arhangel'skii, A. V., An addition theorem for the weight of sets lying in bicompacta, Dokl. Nauk SSSR 126 (1959), 239–241 (Russian).Google Scholar
2. Alexandroff, P. S., Uber abzahlbar-fache offene Abbildungen, Dokl. Akad. Nauk SSSR 4 (1936), 295–299.Google Scholar
3. Franklin, S. P. and Kohli, J. K., On open extensions of maps, Can. J. Math. 22 (1970), 691–696.Google Scholar
4. Kohli, J. K., A note on open extensions of maps, Can. J. Math. 24 (1972), 1139–1144.Google Scholar
5. Kohli, J. K., Finite-to-one maps and open extensions of maps, Proc. Amer. Math. Soc. 48 (1975). 464–468.Google Scholar
6. Kolmogoroff, A. N., Points of local topologicity of enumerably folden open mappings, Dokl. Adad. Nauk SSSR 30 (1941), 479–481.Google Scholar
7. Olmsted, J. M. H., Counter examples in analysis (Holden-Day San Francisco, 1964).Google Scholar
8. Pasynkov, B., On open mappings, Soviet Math. Dokl. 8 (1967), 853–856.Google Scholar
9. Proizvolov, V. V., A generalization of Kolmogoroff 's theorem on the points of local homeomorphism and its implications, Soviet Math. Dokl. 8 (1967), 192–194.Google Scholar
10. Väisälä, J., Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. AI 392 (1966), 1–10.Google Scholar
11. Vâisàlâ, J. Local topological properties of countable maps, Duke Math. J. 159 (1974), 541–546.Google Scholar
12. Whyburn, G. T., On the interiority of real functions, Bull. Amer. Math. Soc. 48 (1942), 942–946.Google Scholar