Vol. 65, No. 2, 1976

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Separation axioms and metric-like functions

Howard Anton and William J. Pervin

Vol. 65 (1976), No. 2, 299–306

Boltjanskii has constructed classes of semifield quasi-pseudo-metrics which are adequate to metrize topological spaces with various separation properties. In this paper we show that his condition given as adequate for T-0 spaces actually is satisfied by every semifield metric inducing the topology. On the other hand, we show that the condition he introduced for T-1 and T-2 spaces is never satisfied by a certain natural semifield quasi-pseudo-metric related to the usual (or Pervin) quasiuniformity. In this paper we completely characterize the classes of semifield quasi-pseudo-metrics which are not only adequate to metrize T-1 and T-2 spaces but actually contain all such metrics inducing such topologies. Characterizations of R-0 and R-1 inducing metrics will also be obtained. Applications to quasi-uniform and quasi-gauge spaces will be made.

Mathematical Subject Classification 2000
Primary: 54E99
Received: 18 February 1976
Published: 1 August 1976
Howard Anton
William J. Pervin