While an ordinary uniform space is defined directly in terms of subsets, and the underlying topology then constructed secondarily, in the absence of an underlying set it seems more convenient to define a uniform locale as additional structure on a given locale, together with an additional axiom which essentially says “the underlying topology is the same as the one we started with.”
refines , written , if every element of is in some element of .
; this is also a cover.
For , is positive
We now define a covering uniformity on a locale to be a collection of covers, called uniform covers, such that
There exists a uniform cover; in light of axiom (4), it follows that the cover is a uniform cover.
If is a uniform cover, there exists a uniform cover such that .
If are uniform covers, so is some cover that refines . In light of axiom (4), it follows that is a uniform cover.
If is a uniform cover and , then is a uniform cover.
For any open part , we have
The last condition is the one saying that “the induced topology is again the topology of .”; the other conditions correspond precisely to the uniform-cover definition of a uniform topological space.
Mike Shulman: I do not know the state of a constructive version of uniform locale theory. Most of the papers on uniform locales seem to assume classical logic, in particular in writing . The above definition seems to me the obvious constructive version, but I don’t know how well it behaves. It’s conceivable one might need to restrict to overt locales, where positivity behaves better.
Apparently there is work on constructive uniform locale theory here:
And indeed, a constructive and predicative theory in the programme of formal topology here:
Giovanni Curi?; On the collection of points of a formal space; Annals of Pure and Applied Logic 137 (2006) 1–3, 126–146.
Giovanni Curi?; Constructive metrisability in point-free topology; Theoretical Computer Science 305 (2003) 1–3, 85–109.
But we have not read these yet.