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Complete Regularity as a Separation Axiom

Published online by Cambridge University Press:  20 November 2018

J. de Groot  and
J. M. Aarts
J. de Groot
Affiliation:
University of Florida, Gainesville, Florida
J. M. Aarts
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts Mathematisch Instituut der Universiteit van Amsterdam, Amsterdam, Holland
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Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be T1 and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening.

Two subsets A and B of a set X are said to be screened by the pair (C, D) if C ∪ D = X, A ∩ D = ∅ and C ∩ B = 0. (Consequently, A ⊂ C and B ⊂ D.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 96 - 105
Copyright
Copyright © Canadian Mathematical Society 1969

References

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