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Topologizing Homeomorphism Groups
Author
Source
Journal of Function Spaces and Applications
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-14, 14 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-05-29
Country of Publication
Egypt
No. of Pages
14
Main Subjects
Abstract EN
This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms ℋ(X) of a Tychonoff space X, which yield continuity of both the group operations and at the same time provide continuity of the evaluation function or, in other words, make the evaluation function a group action of ℋ(X) on X.
By means of a compact extension procedure, beyond local compactness and in two essentially different cases of rim-compactness, we show that the complete upper-semilattice ℒH(X) of all admissible group topologies on ℋ(X) admits a least element, that can be described simply as a set-open topology and contemporaneously as a uniform topology.
But, then, carrying on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure, we show that rim-compactness is not a necessary condition for the existence of the least admissible group topology.
Finally, we give necessary and sufficient conditions for the topology of uniform convergence on the bounded sets of a local proximity space to be an admissible group topology.
Also, we cite that local compactness of X is not a necessary condition for the compact-open topology to be an admissible group topology of ℋ(X).
American Psychological Association (APA)
Di Concilio, A.. 2013. Topologizing Homeomorphism Groups. Journal of Function Spaces and Applications،Vol. 2013, no. 2013, pp.1-14.
https://search.emarefa.net/detail/BIM-506082
Modern Language Association (MLA)
Di Concilio, A.. Topologizing Homeomorphism Groups. Journal of Function Spaces and Applications No. 2013 (2013), pp.1-14.
https://search.emarefa.net/detail/BIM-506082
American Medical Association (AMA)
Di Concilio, A.. Topologizing Homeomorphism Groups. Journal of Function Spaces and Applications. 2013. Vol. 2013, no. 2013, pp.1-14.
https://search.emarefa.net/detail/BIM-506082
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-506082