Geodesic
Complete Regularity as a Separation Axiom
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 96-105

Voir la notice de l'article provenant de la source Cambridge University Press

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 T 1 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.) 
Groot, J. de; Aarts, J. M. Complete Regularity as a Separation Axiom. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 96-105. doi: 10.4153/CJM-1969-010-0
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[1] 1. Aarts, J. M., Dimension and deficiency in general topology, Thesis, Univ. Amsterdam, 1966. Google Scholar

[2] 2. Aarts, J. M. and de Groot, J., Complete regularity as a separation axiom, Communication at the International Congress of Mathematics, Moscow, 1966. Google Scholar

[3] 3. Frink, O., Compactification and semi-normal spaces, Amer. J. Math. 86 (1964), 602–607. Google Scholar

[4] 4. Gillman, L. and Jerison, M., Rings of continuous functions, The University Series in Higher Mathematics (Van Nostrand, Princeton, N.J., 1960). Google Scholar

[5] 5. Kelley, J. L., General topology (Van Nostrand, Toronto, 1955). Google Scholar

[6] 6. Smirnov, Yu. M., On the theory of completely regular spaces, Dokl. Akad. Nauk SSSR (N.S.) 62 (1948), 749–752. (Russian) Google Scholar

[7] 7. Steiner, E. F., Normal families and completely regular spaces, Duke Math. J. S3 (1966), 743–745. Google Scholar

[8] 8. Tychonoff, A., Über die topologische Erweiterung von Raümen, Math. Ann. 102 (1930), 544–561. Google Scholar

[9] 9. Wallman, H., Lattices and topological spaces, Ann. of Math. (2) 89 (1938), 112–126. Google Scholar

[10] 10. Colloquium Co-topology, 1964-1965, Mathematical Centre, Amsterdam. Notes by J. M. Aarts. (Mimeographed reports.) Google Scholar

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