Hello !

It's well know that any sublocal of regular local is the intersection of a familly of open sublocal. Hence if $X$ is a regular local, 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 sublocal 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 sublocal, then $ u \in I$.

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

Thank you !

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 !