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.”
For any and any sublocale of , with inclusion map , we write for the image of of under the projection . If is overt (such as if is an open part of and is overt, or if excluded middle holds), then is an open part of , and can equivalently be defined as , where means “ is positive”. Otherwise, it is merely a sublocale.
Similarly, if , we write for the image of under the projection . If is overt, this is an open part of ; otherwise it is merely a sublocale. We also write for the pullback of along the twist map .
refines , written , if every element of is in some element of .
; this is also a cover.
, and .
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.
An entourage uniformity on a locale consists of a collection of open parts , called [open] entourages, such that:
There exists an entourage (in light of axiom 4 below, it follows that is an entourage)
For any entourage , there exists an entourage such that , and an entourage such that . (In light of axiom 4 below, it follows that is itself an entourage.) Note that is in general only a sublocale (not necessarily open), but we can still ask it to be contained in the open sublocale .
If and are entourages, then so is some open part contained in (in light of axiom 4 below, it follows that is itself an entourage)
If is an entourage and , then is also an entourage.
For any we have .
It is not clear whether these definitions are equivalent. Most references (see below) use only covering uniformities, although Johnstone 89 promises an equivalence to entourage uniformities in a future paper.
This paper developes covering uniformities constructively, and includes citations to several other papers that do it classically:
A constructive and predicative theory in the programme of formal topology can be found 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.