A NEW CHARACTERIZATION OF THE CONTINUOUS FUNCTIONS ON A LOCALE

Abstract. 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 cor-responding completion of aW-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c3A, the “closed under countable composi-tion hull of A, ” as C(YcA), where YcA is the Yosida locale of A, and as the largest essential reflection of A. Content