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A new characterization of the continuous functions on a locale. (English) Zbl 1101.06007

Let \(W\) denote the category of Archimedean lattice-ordered groups with weak order units. In the paper, a natural convergence, called the indicated uniform convergence on \(W\)-objects, is introduced. The authors show that, within \(W\), the objects of the form \(C(L)\), the set of continuous real-valued functions on a locale \(L\), are precisely those which are divisible and complete with respect to the indicated uniform convergence. Among others, the authors construct the corresponding completion of \(W\)-objects in a purely algebraic way in terms of Cauchy sequences.

MSC:

06D22 Frames, locales
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
54C30 Real-valued functions in general topology
54E15 Uniform structures and generalizations
54B30 Categorical methods in general topology
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