Infima of quasi-uniform anti-atoms

Let ( q ( X ) , ⊆ ) denote the lattice consisting of the set q ( X ) of all quasi-uniformities on a set X, ordered by set-theoretic inclusion ⊆. We observe that a quasi-uniformity on X is the supremum of atoms of ( q ( X ) , ⊆ ) if and only if it is totally bounded and transitive. Each quasi-uniformity on X that is totally bounded or has a linearly ordered base is shown to be the infimum of anti-atoms of ( q ( X ) , ⊆ ) . Furthermore, each quasi-uniformity U on X such that the topology of the associated supremum uniformity U s is resolvable has the latter property.