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A generalization of Tong's theorem and properties of pairwise perfectly normal spaces

Published online by Cambridge University Press:  09 April 2009

Manuel López-Pellicer
Affiliation:
C´tedra de Matem´ticas, E.T.S.I. Agrónomos, Universidad Politécnica, Camino de Vera s.n., 46022-Valencia, Spain
Angel Gutiérrez
Affiliation:
Departamento de Matem´ticas, E.U. del Profesorado de E.G.B., Alcalde Reig, 8 46006-Valencia, Spain
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Abstract

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In this paper we give some properties of the pairwise perfectly normal spaces defined by Lane. In particular we prove that a space (X, P, Q) is pairwise perfectly normal if and only if every P(Q)–closed set is the zero of a P(Q)–l.s.c. and Q(P)–u.s.c. function. Also we characterize the pairwise perfect normality in terms of sequences of semicontinuous functions by means of a result which contains the known Tong's characterization of perfectly normal topological spaces, whose proof we modify by using the technique of binary relations.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Arya, S. P. and Singhal, A., “A note on pairwise D1 spaces”, Glasnik Math. 14 (34) (1979), 147150.Google Scholar
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[3]Gutiérrez, A. and Romaguera, S., “Sobre espacios pairwise estratificables”, Rev. Roumaine Math. Pures Appl., to appear.Google Scholar
[4]Lane, E. P., “Bitopological spaces and quasi-uniform spaces”, Proc. London Math. Soc. (3) 17 (1967), 241256.CrossRefGoogle Scholar
[5]López, M. and Santos, J. L., “A necessary and sufficient condition for the insertion of a continuous function”, Proc. Amer. Math. Soc., to appear.Google Scholar
[6]Patty, C. W., “Bitopological spaces”, Duke Math. J. 34 (1967), 387392.CrossRefGoogle Scholar
[7]Romaguera, S., “Two characterizations of quasi-pseudometrizable bitopological spaces”, J. Austral. Math. Soc. Ser. A 35 (1983), 327333.CrossRefGoogle Scholar
[8]Tong, H., “Some characterizations of normal and perfectly normal spaces”, Duke Math. J. 19 (1952), 289292.CrossRefGoogle Scholar
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