A uniform locale is to a uniform space as a locale is to a topological space.
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.”
A cover of a locale $X$ is a collection $C \subseteq Op(X)$ of open parts of $X$ whose join is $X$. For covers $C_i$, we define:
$C_1$ refines $C_2$, written $C_1 \prec C_2$, if every element of $C_1$ is $\le$ in some element of $C_2$.
$C_1 \wedge C_2 \coloneqq \{ A \wedge B \;|\; A \in C_1, B \in C_2 \}$; this is also a cover.
For $A \in Op(X)$, $C[A] \coloneqq \bigcup \{ B \in C \;|\; A \cap B$ is positive }
.
$}C^* \coloneqq \{ C[A] \;|\; A \in C\}$.
We now define a covering uniformity on a locale $X$ 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 $\{X\}$ is a uniform cover.
If $C$ is a uniform cover, there exists a uniform cover $C'$ such that $(C')^* \prec C$.
If $C_1, C_2$ are uniform covers, so is some cover that refines $C_1 \wedge C_2$. In light of axiom (4), it follows that $C_1 \wedge C_2$ is a uniform cover.
If $C$ is a uniform cover and $C \prec C'$, then $C'$ is a uniform cover.
For any open part $A\in Op(X)$, we have
The last condition is the one saying that “the induced topology is again the topology of $X$.”; the other conditions correspond precisely to the uniform-cover definition of a uniform topological space.
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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 $C[A] = \bigcup \{ B \in C \;|\; A \cap B \neq \bot \}$. 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.