Locales with no points I'm very puzzled by the concept of a locale with no points. I understand that once one switches to the language of open sets and operations on them, points become optional: an open set may or may not have points. 
More puzzling are locales which cannot have points: an example of such a thing given in nLab, considers surjections $N \rightarrow R$ from natural to real numbers. This locale has no points because there are no such surjections and that's fine: looks like an empty "something" (an empty set is an abstraction of this sort). 
However this emptiness also has a bunch of sub-locales generated by pairs $(n,x): n \rightarrow x$. None of these can exist either (or rather has no elements). Formally, these descriptors do look different because a different $n$ is NOT mapped to a different $x$, but I'm not sure when this point of view becomes useful since the reason why either of these pairs fails to define a surjection is the same. Maybe this example is too boiled down? what's the context when these logical subtleties start to "work"? 
 A: First it should be made clear what the notion of point means here. I don't understand the example with surjections – there are just no such surjections. And I'm not the only one – see Info on the locale of surjections from the Natural Numbers to the Real Numbers.
So what is a locale? It is a complete lattice satisfying the frame distributivity condition. A topological space certainly has points, and to every topological space you can associate the corresponding locale – by taking the family of all open sets. The point is that not every locale arises this way. There may be a locale for which there is no topological space inducing it. That it not every locale is spatial.
If you have a locale and you want to find a topological space inducing the locale, then you need to find the set of point for the topological space. The standard construction is that a point of a locale is a completely prime filter on that locale. This way you can always construct a topological space associated to a locale, but it may be that the locale induced by that space is not the original locale, because there are not enough points – not enough complelety prime filters on the locale.
Given the above, I would think that “locale with no points” means a locale with no completely prime filters. And there are such locales – every atomless Boolean algebra is such since every completely prime filter in a Boolean algebra is an ultrafilter closed under arbitrary meets.
