A New Characterization of the Continuous Functions on a Locale

Within the categoryW of archimedean lattice-ordered groups with weak order unit, we show that 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 a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of aW-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as cA, the “closed under countable composition hull of A,” as C(Y A), where Y A is the Yosida locale of A, and as the largest essential reflection of A.