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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

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