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Finite Topological Spaces and Quasi-Uniform Structures

Published online by Cambridge University Press:  20 November 2018

P. Fletcher
P. Fletcher*
Affiliation:
Virginia Polytechnic Institute, Blacksburg Virginia 24061
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In [6], H. Sharp gives a matrix characterization of each topology on a finite set X = {x1, x2,…, xn}. The study of quasi-uniform spaces provides a more natural and obviously equivalent characterization of finite topological spaces. With this alternate characterization, results of quasi-uniform theory can be used to obtain simple proofs of some of the major theorems of [1], [3] and [6]. Moreover, the class of finite topological spaces has a quasi-uniform property which is of interest in its own right. All facts concerning quasi-uniform spaces which are used in this paper can be found in [4].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 6 , December 1969 , pp. 771 - 775
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bonnet, D.A. and Porter, J. R., Matrix characterizations of topological properties. Canad. Math. Bull. 11 (1968) 95–105.Google Scholar
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3. Krishnamurthy, V., On the number of topologies on a finite set. Amer. Math. Monthly 73 (1966) 154–157.Google Scholar
4. Murdeshwar, M.G. and Naimpally, S.A., Quasi-uniform topological spaces. (Noordhoff, 1966.)Google Scholar
5. Pervin, W.J., Quasi-uniformization of topological spaces. Math. Ann. 147 (1962) 316–317.Google Scholar
6. Sharp, H., Quasi-orderings and topologies on finite sets. Proc. Amer. Math. Soc. 17 (1966) 1344–1349.Google Scholar