Intersection of open sublocale of a compact regular locale ?

Question

Hello !

It's well know that any sublocale of regular locale is the intersection of a familly of open sublocale. Hence if $X$ is a regular locale, the map which to a sublocal $Y \subset X$ associate $ \lbrace o \in \mathcal{O}(X), y \subset o \rbrace $ is injective. my question is : do we know it's image, at least when $X$ is compact and regular ?

If I'm not mistaken, a subset $I$ of $\mathcal{O}(X)$ correspond to a sublocale of $X$ if and only if :

• $u\in I, v \geqslant u \Rightarrow v \in I $

• if $\forall i, u_i \in I$ and $u = \bigcap u_i $ as sublocale, then $ u \in I$.

So I'm wondering if there is a way to make the last condition more explicit... (or equivalently to detect if $\cap u_i = \emptyset$ as sublocale ).

Thank you !

Answer 1

Here is some sort of characterization; it is not really satisfactory but may be useful for further improvements.

Let us talk in terms of nuclei on frames. For a frame $A$ let $\mathbf NA$ be the frame of its nuclei. Then the question is how to characterize those subsets of $A$ which are equal to $\varphi_*j:=\{\ a\in A\ |\ ja=1\ \}$ for some $j\in\mathbf NA$.

Each such $\varphi_*j$ is obviously a filter on $A$. Let further $\mathbf FA$ be the frame of all filters on $A$. Then it is straightforward to check that the map $\varphi_*:\mathbf NA\to\mathbf FA$ preserves all meets, so it has a left adjoint $\varphi^*:\mathbf FA\to\mathbf NA$.

To describe this $\varphi^*$ more explicitly, some shorthand terminology seems to be useful.

Let us say that a filter $\mathscr F$ on $A$ is $\neg\neg\textit{trivial}$ if all its elements are dense, that is, $\forall\ f\in\mathscr F\ \neg f=0$. And let us say that $\mathscr F$ is $\neg\neg\textit{trivial above}$ $a\in A$ if $\mathscr F\cap[a,1]$ is a $\neg\neg$trivial filter on the frame $[a,1]$. Explicitly this means$$ \forall\ f\in\mathscr F\ \ \ f\to a=a. $$

Now in these terms we may describe, for a filter $\mathscr F$, the nucleus $\varphi^*(\mathscr F)$ by naming the set of its fixed points, which is$$ \mathrm{Fix}(\varphi^*(\mathscr F))=\{\ a\in A\ |\ \mathscr F\textrm{ is $\neg\neg$trivial above }a\ \}. $$ This then gives the following "characterization" of filters in the image of $\varphi_*$:

A subset $\mathscr F$ of $A$ has form $\{ a\in A\ |\ ja=1\ \}$ for some nucleus $j$ if and only if it is a filter and satisfies$$ \forall\ a\notin\mathscr F\ \exists\ a\leqslant a'\notin\mathscr F\ \textrm{such that $\mathscr F$ is $\neg\neg$trivial above }a'. $$