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R1, Pairwise Compact, and Pairwise Complete Spaces

Published online by Cambridge University Press:  20 November 2018

G. D. Richardson
G. D. Richardson*
Affiliation:
East Carolina University, Greenville, North Carolina
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The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.

The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 109 - 113
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Davis, A. S., Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886-893.Google Scholar
2. Fletcher, P., Hoyle, H. B., and Patty, C. W., The comparison of topologies, Duke Math. J. 36 (1969), 325-331.Google Scholar
3. Kelley, J. C., Bitopological spaces, Proc. Londo. Math. Soc. 13 (1963), 71-89.Google Scholar
4. Murdeshwar, M. G. and Naimpally, S. A., R1topological spaces, Canad. Math. Bull. 9 (1966), 521-523.Google Scholar
5. Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform topological spaces, Noordhoff, Groningen, 1966.Google Scholar