Positivity 10 (2006), 165–199
© 2006 Birkh
auser Verlag Basel/Switzerland
A New Characterization of the Continuous
Functions on a Locale
RICHARD N. BALL
and ANTHONY W. HAGER
Department of Mathematics, University of Denver, Denver, CO 80208, USA.
E-mail: email@example.com; http://www.math.du.edu/rball
Department of Mathematics Wesleyan University, Middletown, CT 06459, USA.
Received 6 November 2002; accepted 20 April 2005
Abstract. Within the category W 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 func-
tions 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 a W-object A purely algebraically in terms of Cauchy
sequences. This completion can be variously described as c
A, the “closed under count-
able 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.
Mathematics Subject Classiﬁcation 2000: Primary: 06F20, 06F25; Secondary 46E05, 46E25,
Key words: Archimedean lattice-ordered group, Cauchy sequence, compact Hausdorff
We introduce a natural convergence on W-objects called indicated uniform
convergence. Though not topological, it is closely related to ordinary uni-
form convergence. We then develop the corresponding completion, called
the indicated uniform completion, purely algebraically, i.e., by means of
Cauchy sequences without reference to any representation of the W-objects.
In fact, this completion is of the nicest sort, namely an essential epire-
ﬂection (Theorem 3.2.10), and since such reﬂections have been thoroughly
investigated (in Hager  and Ball and Hager , among other places),
this fact raises the question of exactly what the complete W-objects are.
We settle this question by showing that the W-objects which are divisible
and complete are precisely those closed under countable composition. This
famous class is of central importance in W, and coincides with the W-
objects of the form C(L) for a locale L, a theorem of Isbell ; see also