Semilattices of totally bounded quasi-uniformities

We prove that in general the set of all compatible totally bounded quasi-uniformities (ordered by set-theoretic inclusion) on a topological space does not have good lattice theoretic properties. For instance, we show that even if it is a lattice it need not be modular. We also establish that for every nonzero cardinal κ there exists a topological space X such that X admits exactly κ totally bounded quasi-uniformities. Various further results concerning the number of compatible totally bounded quasi-uniformities on topological spaces are obtained.