×

zbMATH — the first resource for mathematics

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

MSC:
54E15 Uniform structures and generalizations
54E05 Proximity structures and generalizations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Banaschewski, B., Pultr, A.: Samuel compactification and completion of uniform frames. Math. Proc. Camb. Phil. Soc.108(1), 63-78 (1990). · Zbl 0733.54020
[2] ?sászar, Á.: Extensions of quasi-uniformities. Acta. Math. (Hung.)37, 121-145 (1981). · Zbl 0431.54012
[3] Dowker, C. H.: Mappings of Proximity Structures. In: General Topology and Its Relations to Modern Analysis and Algebra. (J. Novák ed.) pp. 139-141. New York: Academic Press. 1961.
[4] Fletcher, P., Hunsaker, W.: Symmetry conditions in terms of open sets. Topology and its Appl. To appear. · Zbl 0766.54025
[5] Fletcher, P., Lindgren, W. F.: Quasi-uniform spaces. New York: Marcel Dekker. 1982. · Zbl 0501.54018
[6] Hunsaker, W., Lindgren, W.: Construction of quasi-uniformities. Math. Ann.188, 39-42 (1970). · Zbl 0187.44602
[7] Johnstone, P.: Stone Spaces. Cambridge: Univ. Press. 1982. · Zbl 0499.54001
[8] Pultr, A.: Pointless uniformities I. Complete regularity. Comm. Math. Univ. Carolinae25, 91-104 (1984). · Zbl 0543.54023
[9] Pultr, A.: Pointless uniformities II. (Dia)metrization. Comm. Math. Univ. Carolinae25, 105-120 (1984). · Zbl 0543.54023
[10] Pultr, A.: Some recent topological results in locale theory. In: General Topology and its Relations to Modern Analysis and Algebra VI. (Z. Frolik ed.) pp. 451-467. Berlin: Heldermann Verlag. 1988.
[11] Smyth, M. B.: Stable local compactifications. Department of Computing, Imperial College, London. (1988) (Preprint)
[12] Vickers, S.: Topology Via Logic. Cambridge: Univ. Press. 1989. · Zbl 0668.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.