Atoms, anti-atoms and complements in the lattice of quasi-uniformities

We describe the atoms of the complete lattice ( q ( X ) , ⊆ ) of all quasi-uniformities on a given (nonempty) set X. We also characterize those anti-atoms of ( q ( X ) , ⊆ ) that do not belong to the quasi-proximity class of the discrete uniformity on X. After presenting some further results on the adjacency relation in ( q ( X ) , ⊆ ) , we note that ( q ( X ) , ⊆ ) is not complemented for infinite X and show how ideas about resolvability of (bi)topological spaces can be used to construct complements for some elements of ( q ( X ) , ⊆ ) .