A Method of Constructing Compatible Frame Quasi-uniformities
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Abstract
Unlike a uniformity, a quasi-uniformity is not determined by its quasi-uniform covers. However, a classical construction, due to P. 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 point free generalization of this, which solves a problem posed by G. C. L. Br¨ummer, together with various examples and applications that illustrateits remarkable usefulness. By this construction, many kinds of interior-preserving open covers (e.g. locally finite, point-finite, spectrum, well-monotone) induce compatible quasi-uniformities on an arbitrary frame.
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