Jorge Picado; Maria João Ferreira;
A method of constructing compatible quasi-uniformities for an arbitrary frame
Unlike a uniformity, a quasi-uniformity is not determined by its quasi-uniform covers. However, a classical construction, due to Fletcher, which assigns a transitive quasi-uniformity to each family of interior-preserving open covers, allows to describe all transitive quasi-uniformities on the topological spaces in terms of those families of covers.
In this paper we develop a pointfree generalization of this, which solves a problem posed by G.C.L. Brummer, together with various examples and applications that illustrate its remarkable usefulness. By this construction, many kinds of interior-preserving open covers (e.g. locally finite, open spectrum, well-monotone) induce compatible quasi-uniformities on an arbitrary frame.
Kyungpook Mathematical Journal