Notes on the coarsest element of \(\pi(\delta)\) (Q700344)

The main result of this paper is the following: Let a (transitive) quasi-uniformity \({\mathcal V}\) be given on \(X\), and let \({\mathcal V}_{\omega}\) be the finest totally bounded quasi-uniformity among the quasi-uniformities which are coarser than \({\mathcal V}\). Then we can get \({\mathcal V}_{\omega}\) as the (transitive) infimum of \({\mathcal V}\) and the Pervin quasi-uniformity corresponding to the topology defined by \({\mathcal V}\). As a consequence of this result, the author proves that there is a topological space \((X,\tau)\) such that the set of all quasi-uniformities which induce \(\tau\) constitutes a (complete) lattice but the distributive law does not hold.