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A METHOD OF CONSTRUCTING COMPATIBLE QUASI-UNIFORMITIES FOR AN ARBITRARY FRAME

Abstract: Unlike a uniformity, a quasi-uniformity is not determined by its quasiuniform 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. Brümmer, together with various examples and applications that illustrate its remarkable usefulness. By this construction, many kinds of interiorpreserving open covers (e.g. locally finite, open spectrum, well-monotone) induce compatible quasi-uniformities on an arbitrary frame.