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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

##### MSC:
 5.4e+16 Uniform structures and generalizations 5.4e+06 Proximity structures and generalizations
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##### 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
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