Are there nontrivial nonspatial frames/locales?

Question

Frames and locales are order theoretic objects used to formalize a "topology without points". See wikipedia for details. One of the novel things about locales is that in many cases one can find a topological space that induces that locale. More interestingly for me is that there are locales that have "no points" such as atomless boolean algebras. Locales without points are called "nonspatial".

The analog of a metric for locales is called a diameter. Given a locale $L$ a function $d:L\rightarrow\mathbb{R}_{+}\cup\{\infty\}$ is a diameter if

1. $d(0)=0$
2. $a\leq b\implies d(a)\leq d(b)$
3. $a\wedge b\neq 0\implies d(a\vee b)\leq d(a)+d(b)$
4. For every $\epsilon>0$, the set

$$U_{\epsilon}^{d}=\{a\mid d(a)<\epsilon\}$$

is a cover of $L$.

My question is: are there nontrivial examples of nonspatial locales that admit a diameter?

Answer 1

Let $X$ be any non-trivial atomless measure space and let $B$ be the $\sigma$-algebra of measurable subsets of $X$ modulo the ideal of subsets of null measure. $B$ is then an atomless boolean algebra, but possibly not complete. Let $\hat{B}$ be the poset of downward-closed subsets of $B$ that are closed under countable joins. Then $\hat{B}$ is a frame, and $b \mapsto \{ b' \in B : b' \le b \}$ is a monotone embedding that preserves finitary meets and countable joins. Hence, $\hat{B}$ corresponds to a locale $L$ with no points.

Consider the measure $\mu : B \to \mathbb{R}_{\ge 0} \cup \{ \infty \}$. We extend it to a monotone map $\hat{\mu} : \hat{B} \to \mathbb{R}_{\ge 0} \cup \{ \infty \}$ in the obvious way: $$\hat{\mu} (I) = \sup {\{ \mu (b) : b \in I \}}$$ It is clear that $\hat{\mu} (\bot) = 0$, $\hat{\mu}$ is monotone, and $\hat{\mu} (I \vee J) \le \hat{\mu} (I) + \hat{\mu} (J)$ – these reduce the corresponding properties of $\mu$ straightforwardly. The condition that $\{ I \in \hat{B} : \hat{\mu} (I) < \epsilon \}$ be a cover reduces to the condition that $\{ b \in B : \mu (b) < \epsilon \}$ be a cover, which is a (non-trivial) consequence of the hypothesis that $X$ is a non-trivial atomless measure space.