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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 !

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Surely the word is "locale"? –  Yemon Choi Sep 15 '12 at 17:58
    
Well, "locale" is locally "local" . . . –  Noah S Sep 15 '12 at 21:07
    
Sorry, that was a translation problem... –  Simon Henry Sep 15 '12 at 22:34
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