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Separability axioms and metric-like functions

Published online by Cambridge University Press:  17 April 2009

Howard Anton
Howard Anton
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, USA.
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Abstract

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We call a topological space completely regular if points and closed sets can be separated by continuous real-valued functions, and we call a topological space Tychonoff or T3α if it is T1 and completely regular. It is well known that the Tychonoff spaces are precisely the gauge spaces, that is the topological spaces whose topologies are induced by separating families of pseudometrics. V.G. Boltjanskiĭ has given analogous characterizations for T0, T1, T2, regular, and regular-T1 spaces in terms of families of “metric-like” functions. The purpose of this note is to fill in a case missing in Boltjanskiĭ's paper, the completely regular spaces, and to relate Boltjanskiĭ's work to results on quasi-uniformization by William J. Pervin and results about semi-gauge spaces by J.V. Michalowicz.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 2 , October 1976 , pp. 265 - 271
Copyright
Copyright © Australian Mathematical Society 1976

References

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