Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T15:33:22.596Z Has data issue: false hasContentIssue false
  • English
  • Français

Pairwise complete regularity as a separation axiom

Part of: Spaces with richer structures Fairly general properties

Published online by Cambridge University Press:  09 April 2009

J. M. Aarts  and
M. Mršević
J. M. Aarts
Affiliation:
Delft University of Technology DelftThe Netherlands
M. Mršević
Affiliation:
University of BelgradeBelgrade, Yugoslavia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Focussing on complete regularity, we discuss the separation properties of bitopological spaces. The unifying concept is that of separation by a pair of bases (B1, B2) for the closed sets of a bitopological space (S, J1, J2). For various separation properties a characterization is presented in terms of separation by a pair of closed bases. This is extended to results concerning pairs of subbases. Here the notion of screening by pairs of subbases plays a central role and the characterization of complete regularity in a natural way fits in between those of regularity and normality. In the key lemma the relation with quasi-proximities is exhibited.

Keywords

(sub)base for the closed setsseparation axiomscomplete regularity.

MSC classification

Secondary: 54E55: Bitopologies 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 2 , April 1990 , pp. 235 - 245
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Aarts, J. M., Dimension and deficiency in general topology, (Thesis, Amsterdam, 1966).Google Scholar
[2]Brandenburg, H. and Mysior, A., ‘Short proof of an internal characterization of complete regularity,’ Canad. Math. Bull. 27 (1984), 461–462.CrossRefGoogle Scholar
[3]Frink, O., ‘Compactification and semi-normal spaces,’ Amer. J. Math. 86 (1964), 602–607.CrossRefGoogle Scholar
[4]de Groot, J. and Aarts, J. M., ‘Complete regularity as a separation axiom,’ Canad. J. Math. 21 (1969), 96–105.CrossRefGoogle Scholar
[5]Hamburger, P., ‘Complete regularity as a separation axiom,’ Period. Math. Hungar. 4 (1973), 39–44.CrossRefGoogle Scholar
[6]Kelley, J. L., General topology, (Princeton, N. J., 1955).Google Scholar
[7]Kelly, J. C., ‘Bitopological spaces,’ Proc. London Math. Soc. 13 (1963), 71–89.CrossRefGoogle Scholar
[8]Kerstan, J., ‘Eine Charakterisierung der vollständig regulären Räume,’ Math. Nachr. 17 (1958), 27–46.CrossRefGoogle Scholar
[9]Koh, Hyung-Joo, ‘Separation axioms in bitopological spaces,’ Bull. Korean Math. Soc. 16 (1979), 11–14.Google Scholar
[10]Lane, E. P., ‘Bitopological spaces and quasi-uniform spaces,’ Proc. London Math. Soc. 17 (1967), 241–256.CrossRefGoogle Scholar
[11]Lane, E. P. ‘Quasi-proximities and bitopological spaces,’ Portugal. Math. 28 (1969), 151–159.Google Scholar
[12]van Mill, J., ‘The superextension of the closed unit interval is homeomorphic to the Hilbert cube,’ Fund. Math. 103 (1979), 151–175.CrossRefGoogle Scholar
[13]Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform topological spaces, (Noordhoff, Groningen, 1966).Google Scholar
[14]Reilly, I. L., ‘On bitopological separation properties,’ Nanta Math. 5 (1972), 14–25.Google Scholar
[15]Reilly, I. L., ‘On essentially pairwise Hausdorff spaces,’ Rend. Circ. Mat. Palermo (2) 25 (1976), 47–52.CrossRefGoogle Scholar
[16]Seagrove, M. J., ‘Pairwise complete regularity and compactification in bitopological spaces,’ J. London Math. Soc. 7 (1973), 286–290.CrossRefGoogle Scholar
[17]Smirnov, Yu. M., ‘On the theory of completely regular spaces,’ Dokl. Akad. Nauk SSSR (N.S.) 62 (1948), 749–752.Google Scholar
[18]Steiner, E. F., ‘Normal families and completely regular spaces,’ Duke Math. J. 33 (1966), 743–745.CrossRefGoogle Scholar
[19]Verbeek, A., Superextensions of topological spaces, Mathematical Centre Tracts, No. 41 (Mathematisch Centrum, Amsterdam 1973).Google Scholar