Uniform Locales

# Uniform Locales

## Idea

A uniform locale is to a uniform space as a locale is to a topological space.

## Definition

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.”

### Basic notions

For any $E\in Op(X\times X)$ and any sublocale $A$ of $X$, with inclusion map $i:A\to X$, we write $E[A]$ for the image of $(i\times 1)^*(E)$ of $A\times X$ under the projection $A\times X \to X$. If $A$ is overt (such as if $A$ is an open part of $X$ and $X$ is overt, or if excluded middle holds), then $E[A]$ is an open part of $X$, and can equivalently be defined as $\bigvee \{ V\in Op(X) \mid Pos(V\cap A) \}$, where $Pos(W)$ means “$W$ is positive”. Otherwise, it is merely a sublocale.

Similarly, if $E,F\in Op(X\times X)$, we write $E\circ F$ for the image of $\pi_{23}^*(E) \cap \pi_{12}^*(F)$ under the projection $\pi_{13}:X\times X\times X\to X\times X$. If $X$ is overt, this is an open part of $X\times X$; otherwise it is merely a sublocale. We also write $E^{-1}$ for the pullback of $E$ along the twist map $X\times X \cong X\times X$.

### Covering uniformities

An [open] 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.

• $E_C = \bigcup \{ U\times U \mid U\in C \}$, and $C[A] = E_C[A]$.

• For $A \in Op(X)$, we write $C[A] = E_C[A]$ (see above definition). If $A$ is overt (such as if $X$ is overt, such as if excluded middle holds), this is equivalent to $\bigvee \{ B \in C \;|\; A \cap B \text{ is positive } \}$, where positive is in the sense of positive element.

• $C^* \coloneqq \{ C[A] \;|\; A \in C\}$.

We now define a covering uniformity on a locale $X$ to be a collection of covers, called uniform covers, such that

1. If $C$ is a uniform cover and $C \prec C'$, then $C'$ is a uniform cover.

2. The cover $\{X\}$ is a uniform cover and if $C_1, C_2$ are uniform covers, then so is $C_1 \wedge C_2$.

3. If $C$ is a uniform cover, there exists a uniform cover $C'$ such that $(C')^* \prec C$.

4. The collection of uniform covers is admissible: for any open part $A\in Op(X)$, we have

$A = \bigvee \{ B | \exists \;\text{ a uniform cover }\; C \;\text{ such that }\; C[B] \le A \}$

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.

### Entourage uniformities

An entourage (Picado and Pultr, Section XII.1.1) is an open part $E\in Op(X\times X)$ such that $\{u\in Op(X)\mid u\times u\le E\}$ is an open cover of $X$.

An entourage uniformity (Picado and Pultr, Section XII.2.3) on a locale $X$ is a collection of entourages such that:

1. If $E$ is an entourage and $E\subseteq F$, then $F$ is also an entourage.

2. The open part $X\times X$ is an entourage and if $E$ and $F$ are entourages, then so is $E\cap F$.

3. If $E$ is an entourage, then so is $E^{-1}$.

4. For any entourage $E$, there exists an entourage $F$ such that $F\circ F \subseteq E$. The sublocale $F\circ F$ is not always open, but we can still ask it to be contained in the open sublocale $E$.

5. The collection of uniform entourages is admissible: for any $U\in Op(X)$ we have $U = \bigvee \{ V\in Op(X) \mid E[V] \subseteq U \text{ for some uniform entourage } E\}$.

### From entourage uniformities to covering uniformities

Given an entourage $E$ on a locale $L$, we define an open cover $U_E$ by setting

$U_E=\{x\in L\mid x\times x\subset E\}.$

Given an entourage uniformity $\{E_i\}_i$, we construct a covering uniformity out of it by taking all open covers $V$ that are refined by some $U_{E_i}$.

### From covering uniformities to entourage uniformities

Given an open cover $U$, we construct an entourage $E_U$ by setting $$E_U=\bigvee_{x\in U}x\times x.$$

Given a covering uniformity $\{U^i\}_i$, we construct an entourage uniformity out of it by taking all entourages $E$ that contain some $E_{U^i}$.

### Equivalence between covering uniformities and entourage uniformities

Theorem XII.3.3.4 in Picado and Pultr shows that the above correspondence is bijective. Furthermore, the categories of entourage uniform locales and covering uniform locales are equivalent (Corollary XII.3.4.3 in the cited book). However, since their book uses classical logic throughout, it is not entirely clear whether the same equivalence holds constructively.

## Properties

### Uniformizability

A locale admits a uniformity if and only if it is completely regular (\cite[Theorem 2.8.2]{PP}).

### Fine uniformity

Uniformities are closed under unions, so any completely regular locale admits a largest uniformity, the fine uniformity.

The fine uniformity consists of all normal covers, where a cover $U$ is normal if there is a sequence of covers $\{U_n\}_{n\ge0}$ such that $U=U_0$ and $U_{n+1}U_{n+1}$ refines $U_n$.

### Compact regular locale

On a compact regular locale the system of all covers forms a uniformity. Maps out of compact regular locales are automatically uniform.

### Complete uniform locales

A uniform locale $L$ is complete if every dense uniform embedding $L\to X$ is an isomorphism of uniform locales.

The category of complete uniform locales is a reflective subcategory of uniform locales. The reflection functor is known as the completion functor.

Concretely, the completion of a uniform locale $L$ with a system of uniform covers $A$ can be described as follows. Its opens are downward-closed subsets $U$ of the underlying frame of $L$ that satisfy the following saturation conditions:

• If for some $a\in L$ we have $\{x\in L\mid x\lt_A a\}\subset U$, then also $a\in U$. Here $x\lt_A a$ means that there is $C\in A$ such that $C[x]\le a$.

• If for some $C\in A$ and $a\in L$ we have $a\wedge C\subset U$, then also $a\in U$.

## Criteria for completeness

Every compact regular locale is complete. The fine uniformity on a paracompact locale? is complete.

Isbell’s theorem states that a locale is paracompact if and only if it admits a complete uniformity.

A completely regular locale is compact if and only if it admits a complete uniformity that has a basis in which all covers are finite.

A detailed treatment of uniform locales can be fund in

The following paper developes covering uniformities constructively, and includes citations to several other papers that do it classically:

• Peter Johnstone, 1989; A constructive theory of uniform locales. I. Uniform covers; General topology and applications, 179–193; Lecture Notes in Pure and Applied Mathematics 134 (1991).

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.