Vol. 43, No. 3, 1972

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Quasi-uniformities with a transitive base

Peter Fletcher and William Lindgren

Vol. 43 (1972), No. 3, 619–631

Every topological space admits at least one compatible transitive quasi-uniformity, and among the compatible transitive quasi-uniformities there is always a finest one. It is shown that a topological space is compact if and only if its finest transitive quasi-uniformity is precompact. A TI space X whose fine quasi-uniformity is transitive is quasi-metrizable if and only if X has a σ Q-base. The upper semi-continuous quasi-uniformity of any topological space is transitive and countably precompact and a space X is almost realcompact if and only if it has a compatible almost complete countably precompact transitive quasi-uniformity. Consequently a space is almost realcompact if and only if its upper semi-continuous quasi-uniformity is almost complete. The latter result is the natural analogue of the result of Shirota that a space X is realcompact if and only if it is complete in the structure C(X).

Mathematical Subject Classification 2000
Primary: 54E15
Received: 11 August 1971
Revised: 22 December 1971
Published: 1 December 1972
Peter Fletcher
William Lindgren