On the cardinality of compatible quasi-uniformities (Q1975539)

The author starts his investigations on the number of quasi-uniformities that topological spaces admit. Among other things, he shows that on a topological space there exist either a unique compatible quasi-uniformity or at least \(2^{2^{\aleph_0}}\) transitive compatible quasi-uniformities finer than its Pervin quasi-uniformity. Furthermore he proves that each infinite \(T_2\)-space admits at least \(2^{2^{\aleph_0}}\) non-transitive quasi-uniformities. The reals with their usual topology admit exactly \(2^{2^{\aleph_0}}\) quasi-uniformities (transitive quasi-uniformities, totally bounded quasi-uniformities). These results were strengthened and extended in various directions during subsequent investigations conducted mainly by the author and the reviewer; see for instance [\textit{H. P. A. Künzi} and \textit{A. Losonczi}, On some cardinal functions related to quasi-uniformities, Houston J. Math. 26, 299-313 (2000)].