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Relations on topological spaces: Urysohn's lemma

Published online by Cambridge University Press:  09 April 2009

Y.-F. Lin
Y.-F. Lin
Affiliation:
The University of South Florida Tampa, Florida
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Let X be a topological space equipped with a binary relation R; that is, R is a subset of the Cartesian square X×X. Following Wallace [5], we write Deviating from [7], we shall follow Wallace [4] to call the relation R continuous if RA*⊂(RA)* for each A⊂X, where * designates the topological closure. Borrowing the language from the Ordered System, though our relation R need not be any kind of order relation, we say that a subset S of X is R-decreasing (R-increasing) if RS ⊂ S(SR ⊂ S), and that S is Rmonotone if S is either R-decreasing or R-increasing. Two R-monotone subsets are of the same type if they are either both R-decresaing or both Rincreasing.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 1 , February 1968 , pp. 37 - 42
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Dieudonné, J., ‘Sur les fonctions continues numériques définies dans un produit de deux espaces compacts’, C.R. Acad. Sci. Paris 205 (1937), 593–595.Google Scholar
[2]Kelley, J. L., General topology, (New York, 1955).Google Scholar
[3]Nachbin, L., Topologia e ordem, (Chicago, 1950).Google Scholar
[4]Wallace, A. D., Relations on topological spaces (Proc. Symp. on General Topology and its Relations to Modern Analysis and Algebra (Prague 1961), 356–360.)Google Scholar
[5]Wallace, A. D., Relation-theory, Lecture Notes, (Univ. of Fla. 1963– 1964).Google Scholar
[6]Ward, L. E. Jr, ‘,Binary relations in topological spaces’, Anais A cad. Basil. Ci. 36 (1954), 357–373.Google Scholar
[7]Ward, L. E. Jr, ‘Partially ordered topological spaces’, Proc. Amer. Math. Soc. 5 (1954), 144–161.CrossRefGoogle Scholar