Hostname: page-component-7bb8b95d7b-cx56b Total loading time: 0 Render date: 2024-09-13T14:19:44.760Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Open Extension of Maps

Published online by Cambridge University Press:  20 November 2018

J. K. Kohli
J. K. Kohli*
Affiliation:
Indian Institute of Technology Kanpur, Kanpur, India; Hindu College, University of Delhi, Delhi, 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.

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains (see Whyburn [10], Baur [3], Krolevec [8], Dickman [5], Franklin and Kohli [6]). It was shown in [6] that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. In [6] and [7] we also related the topological properties of the domain and range of the mapping with the new domain; also these results were then used to obtain analogues and improvements of recent theorems of Arhangelskii, Čoban, Hodel, Keesling, Nagami, Okuyama, and Proizvolov. In this note we give a method of unifying the domain and range of a mapping so as to yield a meaningful open extension.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1139 - 1144
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Arhangelskiĭ, A. V., Existence criterion of a bicompact element in a continuous decomposition. A theorem on the invariance of weight for open-closed finitely multiple mappings, Dokl. Akad. Nauk SSSR 166 (1966), 1263–1266 (Russian) ; translated as Soviet Math. Dokl. 7 (1966), 249–253.Google Scholar
3. Arhangelskiĭ, A. V., A theorem on the metrizability of the inverse image of a metric space under an openclosed finite-to-one mapping. Example and unsolved problems, Dokl. Akad. Nauk SSSR 170 (1966), 759–762 (Russian); translated as Soviet Math. Dokl. 7 (1966), 1258–1262.Google Scholar
3. Baur, H., Konservative Abbildungen local kompater Raume, Math. Ann. 138 (1959), 398–427.Google Scholar
4. Čoban, M. M., Open finite-to-one mappings, Soviet Math. Dokl. 8 (1967), 603–605.Google Scholar
5. Dickman, R. F., Jr., On closed extensions of functions, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 326–332.Google Scholar
6. Franklin, S. P. and Kohli, J. K., On open extensions of maps, Can. J. Math. 22 (1970), 691–696.Google Scholar
7. Kohli, J. K., On open extensions of maps. II, Notices Amer. Math. Soc. 17 (1970), 684; Technical Report No. 15, Indian Institute of Technology Kanpur (Nov. 1971).Google Scholar
8. Krolevec, N., Locally perfect mappings, Dokl. Akad. Nauk SSSR 175 (1967), 1008–1011.Google Scholar
9. Proizvolov, V. V., On finite-to-one open mappings, Dokl. Akad. Nauk SSSR 166 (1966), 38–40 (Russian); translated as Soviet Math. Dokl. 7 (1966), 35–38.Google Scholar
10. Whyburn, G. T., A unified space for mappings, Trans. Amer. Math. Soc. 74 (1953), 344–350.Google Scholar