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Two semigroups of continuous relations

Published online by Cambridge University Press:  09 April 2009

A. R. Bednarek  and
Eugene M. Norris
A. R. Bednarek
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32601, U.S.A.
Eugene M. Norris
Affiliation:
Department of Mathematics and Computer Science, University of South Carolina, Columbia, South Carolina 29208, U.S.A.
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Synopsis

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In this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 1 , February 1977 , pp. 46 - 58
Copyright
Copyright © Australian Mathematical Society 1977

References

Bednarek, A. R. and Magill, K. D. Jr, (1973), ‘Semigroups of clopen relations’, (unpublished MS).Google Scholar
Day, J. M. and Franklin, S. P. (1967), ‘Spaces of continuous relations’, Math. Ann. 169, 289–293.CrossRefGoogle Scholar
Dugundji, J. (1965), Topology (Allyn and Bacon, Boston).Google Scholar
Magill, K. D. Jr, (1967), ‘Another S-admissible class of spaces’, Proc. Amer. Math. Soc. 18, 295–298.Google Scholar
Michael, E. (1951), ‘Topologies on spaces of subsets’, Trans. Amer. Math. Soc. 71, 152–182.CrossRefGoogle Scholar
Vitanza, M. (1966), ‘Mappings of semigroups associated with ordered pairs’, Amer. Math. Monthly 73, 1078–1082.CrossRefGoogle Scholar