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Entourage uniformities for frames. (English) Zbl 0736.54023
Let \((X,{\mathcal U})\) be a quasi-uniform space and let \(U\in{\mathcal U}\). An open set \(B\) is \(U\)-small provided that whenever \(A\) is open and \(A\cap B\neq\emptyset\), \(B\subseteq U(A)\). The quasi-uniformity \(\mathcal U\) is small-set symmetric provided that for each \(U\in{\mathcal U}\) the \(U\)-small sets cover \(X\), and \(\mathcal U\) is open-set symmetric provided that there is a base \(\mathcal B\), for \(\mathcal U\) such that for each pair \(A\), \(B\) of open sets and each \(U\in{\mathcal B}\), \(U(A)\cap B=\emptyset\) if, and only if \(U(B)\cap A=\emptyset\). A quasi-uniformity is a uniformity if and only if it is open-set symmetric and small-set symmetric. The two symmetry conditions given above are used to define a uniformity for a frame in terms of entourages, and it is shown that the category of entourage uniform frames is isomorphic with the category of covering uniform frames.
Reviewer: P.Fletcher

54E15 Uniform structures and generalizations
54E05 Proximity structures and generalizations
Full Text: DOI EuDML
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