By a result of [KL1], each topological space that admits more than one quasi-uniformity admits at least 22\aleph0 nontransitive quasi-uniformities. Furthermore it was shown in [KPP] that each Hausdorff space that has a discrete subspace of infinite cardinality \kappa admits at least 22\kappa transitive totally bounded quasi-uniformities.
On the other hand, it seems fairly demanding to construct nontransitive totally bounded quasi-uniformities even on nice topological spaces. To see this, recall for instance that each compact Hausdorff space admits a unique uniformity, which is transitive if and only if the space is (strongly) zero-dimensional. So even the problem whether each infinite compact zero-dimensional Hausdorff space admits a nontransitive totally bounded quasi-uniformity is nontrivial and its positive solution evidently needs a truely nonsymmetric approach. (Note that in this case the nontransitive quasi-uniformity constructed will necessarily be finer than the compatible uniformity, since for a compact Hausdorff space X the compatible uniformity is the coarsest quasi-uniformity that X admits [FL, Proposition 1.47].)
If we consider larger classes of spaces, then the situation becomes even more difficult. The method given in this paper shows that, indeed, each infinite completely regular Hausdorff space admits a nontransitive totally bounded quasi-uniformity.
On the other hand, it is known that there are infinite T1-spaces that do not admit a nontransitive totally bounded quasi-uniformity. For instance, the cofinite topology on a countably infinite set admits a unique totally bounded quasi-uniformity, which, of course, is transitive, although the space admits 22\alpeh0 nontransitive quasi-uniformities. Moreover, in [KS] even a T1-space having infinitely many isolated points is defined that admits only one, necessarily transitive, totally bounded quasi-uniformity. Also, in [KL2] for each nonzero cardinal \kappa a T0-space X\kappa is constructed that admits exactly \kappa totally bounded quasi-uniformities all of which are transitive.
The authors now conjecture that each infinite Hausdorff space admits a nontransitive totally bounded quasi-uniformity, but have been unable so far to settle this problem.
They would also like to encourage readers, puzzled by various (seemingly ad-hoc) constructions given in this paper, to look for more canonical (possibly, categorical) methods that would yield similar results or to produce proofs that such techniques cannot exist.
For basic results on topological and quasi-uniform spaces we refer the reader to [E, FL]. In particular let us mention (compare [P]) that each topological space X admits the transitive and totally bounded Pervin quasi-uniformity generated by the entourages [G x G] \cup [(X \ G) x X] where G is an open subset of X.
We recall that a quasi-uniformity on a set X is called transitive if it possesses a base consisting of transitive entourages and it is said to be totally bounded provided that for each entourage V the cover { (V \cap V-1)(x) : x in X } has a finite subcover.
Recall also that a base B of a topological space X is called an l-base (i.e. a lattice base) if it is closed under finite unions and finite intersections and the empty set and X are in B. In [L] it is observed that for an arbitrary topological space there is a one-to-one correspondence between the set of compatible transitive totally bounded quasi-uniformities UB and the set of l-bases B. In fact, let us mention that UB = fil{[B x B] \cup [(X \ B) x X] : B in B} and B consists of the sets B which are strongly contained in themselves with respect to UB.
Date received: June 13, 2000.