On some cardinal functions related to quasi-uniformities (Q2725482)

Estimations for the cardinality of certain classes of quasi-uniformities are given. Some results: 2.1: A topological space induced by at least two quasi-uniformities is induced by at least \(2^{2^\omega}\) nontransitive quasi-uniformities.NEWLINENEWLINENEWLINE3.2: A quasi-proximity space \(X\) is induced by at least \(2^{2^\kappa}\) quasi-uniformities provided it is induced by a quasiuniformity \(\mathcal W\) having the property: there exist \(A\subset X\), \(W\in\mathcal W\) such that no subsystem of cardinality less than \(\kappa\) of \(\{W[x]\), \(x\in A\}\) or of \(\{W^{-1}[x]\), \(x\in A\}\) covers \(A\).NEWLINENEWLINENEWLINE3.6: If a quasi-proximity space \(X\) is induced by at least two quasi-uniformities and the coarsest quasi-uniformity inducing \(X\) is transitive, then \(X\) is induced by at least \(2^{2^\omega}\) transitive quasi-uniformities.