Permutable Pairs of Quasi-Uniformities

Abstract

In this project we study permutable pairs of quasi-uniformities. Our objective is to investigate the notion of permutability of quasi-uniformities in the context of conjugates, complements, adjacent and independent quasi-uniformities, following the work of authors like H. Weber, E.P de Jager and H.P.A. K¨unzi. The notion of quasi-uniformity which is a generalization of the concept of a uniformity is discussed in terms of surroundings. Two quasi-uniformities U and V on a set X are said to be permutable provided U ◦V = V◦U. Some characterizations of permutability are presented. Among other results, we show that conjugates of permutable quasi-uniformities permute and conjugates of adjacent quasi-uniformities are adjacent. Complementary quasi-uniformities are discussed and a connection between complementary quasi-uniformities and permutability is shown. Furthermore, the notion of independent quasi-uniformities is discussed. In particular we present a well known result from ([3], corollary 4) in terms of independent quasi- uniformities. Some useful examples are also discussed.