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Quasi-Uniform Spaces and Topological Homeomorphism Groups(1)

Published online by Cambridge University Press:  20 November 2018

Massood Seyedin
Massood Seyedin*
Affiliation:
The National University of Iran, Eveen, Tehran, Iran
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Let X be a topological space and G a subgroup of the homeomorphism group H(X) with the topology of point-wise convergence. It is well-known that if X is uniformizable and G is equicontinuous with respect to a compatible uniformity then G is a topological group. In this paper we show that essentially this same result applies when X is only an R0-space (and hence in particular if X is T1 or regular). A corresponding result for regular spaces has been proved [2].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 97 - 98
Copyright
Copyright © Canadian Mathematical Society 1974

Footnotes

(1)

This paper is based upon the author’s dissertation at Virginia Polytechnic Institute and State University, Blacksburg, Virginia, under the direction of P. Fletcher. The author wishes to thank the referee for his valuable suggestions.

References

1. Davis, A. S., Indexed systems of neighborhoods for general topological spaces, Amer. Math Monthly, 68 (1961), 886-893.Google Scholar
2. Fuller, R. V., Semiuniform spaces and topological homeomorphism groups, Proc. Amer. Math. Soc, 26 (1970), 365-368.Google Scholar
3. Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform topological spaces, Noordhoff (1966).Google Scholar