The question studied in the paper is to determine the number of quasi-uniformities that various kinds of topological spaces admit. Some questions posed in the basic article of \textit{A. Losonczi} [Acta Math. Hung. 79, No. 1-2, 85-116 (1998; Zbl 0906.54019)] concerning this problem are solved. The main result is that a topological space admits only transitive quasi-uniformities if and only if it admits a unique quasi-uniformity. Furthermore, the example of a topological space that admits exactly two quasi-proximities (totally bounded quasi-uniformities) is given. Finally it is shown that if the topological space \(X\) has the net(work) weight \(nw(X)\), then any compact quasi-uniformity of \(X\) has a base of cardinality \(\leq 2^{nw(X)}\) and \(X\) has \(\leq 2^{2^{nw(X)}}\) compatible quasi-uniformities.