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Relations on topological spaces: Urysohn's lemma

  • Y.-F. Lin (a1)

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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 AX, 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 RSS(SRS), 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.

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References

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[1]Dieudonné, J., ‘Sur les fonctions continues numériques définies dans un produit de deux espaces compacts’, C.R. Acad. Sci. Paris 205 (1937), 593595.
[2]Kelley, J. L., General topology, (New York, 1955).
[3]Nachbin, L., Topologia e ordem, (Chicago, 1950).
[4]Wallace, A. D., Relations on topological spaces (Proc. Symp. on General Topology and its Relations to Modern Analysis and Algebra (Prague 1961), 356360.)
[5]Wallace, A. D., Relation-theory, Lecture Notes, (Univ. of Fla. 19631964).
[6]Ward, L. E. Jr, ‘,Binary relations in topological spaces’, Anais A cad. Basil. Ci. 36 (1954), 357373.
[7]Ward, L. E. Jr, ‘Partially ordered topological spaces’, Proc. Amer. Math. Soc. 5 (1954), 144161.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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