Quasi-uniformities, by Peter Fletcher

Topology Atlas Document # zaaa-47.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 2 (1996), 12.

Quasi-uniformities
by

Peter Fletcher

(Christiansburg, VA, U.S.A.)

A quasi-uniformity on a nonempty set X is a filter of reflexive relations on X with the property that for each U belonging to the filter there is a member V of the filter such that VoV (V composite V) is a subset of U. The study of quasi-uniformities was initiated by Leopoldo Nachbin (e.g., C. R. Acad. Sci., Paris 226 (1948), 381-382, 547, 774-775). The importance of quasi-uniformities is captured by the title of Nachbin's monograph, "Topology and Order": quasi-uniformities provide a natural way to study topology and order simultaneously. It is for this reason that quasi-uniformities have proved useful in theoretical computer science. Two extensive bibliographies are given in "Quasi-Uniform Spaces", Marcel Dekker, New York and Basel,(1982) and "Quasi-uniform spaces - Eleven years later" Top. Proc. 18 143-171(1993).

Received by the editors: December 14, 1995.