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The Semicontinuous Quasi-uniformity of a Frame
Ferreira, Maria Joao,Picado, Jorge Department of Mathematics 2006 Kyungpook mathematical journal Vol.46 No.2
The semicontinuous quasi-uniformity is known to be one of the most important examples of transitive quasi-uniformities. The aim of this paper is to show that various facts in classical topology connected with the semicontinuous quasi-uniformity and semicontinuous real functions may be easily extended to pointfree topology via a construction introduced by the authors in a previous paper. Several consequences are derived.
A Method of Constructing Compatible Frame Quasi-uniformities
Maria Joao Ferreira,Jorge Picado 경북대학교 자연과학대학 수학과 2004 Kyungpook mathematical journal Vol.44 No.3
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.