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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"?

Tnatsissa H Craeser
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  • What's a "locale"? Is that a neighborhood? EDIT: Okay, I googled it. Interesting, never heard of that before, will read more about it. +1 for the question! – MPW Sep 18 '15 at 21:59
  • I'm not really sure what your question is. A locale with no points is just a locale with no points; points don't have anything to do with the definition of a locale. Are you really asking what application such locales have, or something like that? – Eric Wofsey Sep 19 '15 at 03:07
  • On the surface I wanted to understand the example I'm quoting (locales of surjections from $N$ to $R$). I understand that points are superfluous in topology axioms and the definition of a "locale" looks like a right idea. On a deeper level I'm trying to get a sense of how much locales differ from topological spaces and what's interesting about this "difference". Just as an example of this train of thought: once you go from points to ideals (schemes etc.), you see lots of new useful "geometric" objects. – Tnatsissa H Craeser Sep 22 '15 at 17:40
  • Regarding the locale of surjections $\mathbb N\to\mathbb R$, look at Ingo Blechschmidt's comment on this MO question. A more general construction can be found in Peter Johnstone's Sketches of an elephant, C1.2, example 1.2.8. – Jackozee Hakkiuz Aug 16 '20 at 21:05

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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.

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Adam Bartoš
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  • Thanks for the answer! If I understand that a construction with the completely prime filters: in case the poset is a power-set ordered by inclusion, then short of specifying a point, one can specify all sets containing that point. In case you do not have a point to work with (to specify such principal filter) you'll end up with the set of all prime filters. Isn't this a strict superset of the set of principal filters? I may be missing a fact here, but prime filters aren't necessarily maximal for example. – Tnatsissa H Craeser Sep 22 '15 at 18:53
  • @TnatsissaHCraeser: Not that I was tlaking about completely prime filters, not just prime. Yes, they aren't necessarily maximal. In case of power-set they coincide. Also, they are not even superset of principal filters. If the principal filter is not generated by a completely meet irreducible element (e.g. atom), then it is not completely prime. – Adam Bartoš Sep 22 '15 at 20:10
  • @user87690: I meant to say "superset of principal prime filters", thanks for the correction. What I'm having trouble with respect to "completeness" in the context of locales, is that requiring it already goes out of the scope of open sets (their infinite intersections do not have to be open of course). So, such "points" cannot be defined in the language of locales, where one operates only within the lattice. In other words, it seems like having a complete lattice is too strong a condition. Perhaps this is the reason why there can be "not enough points" resulting from this construction? – Tnatsissa H Craeser Sep 23 '15 at 01:12
  • @TnatsissaHCraeser: Note that the locale knows nothing about intersections. The lattice of open sets is complete, every family of open sets has a meet, but the meet doesn't have to be intersection. – Adam Bartoš Sep 23 '15 at 08:38
  • @user87690: My bad, locales are complete lattices! I was distracted by the analogy with open sets. Thanks for pointing this out. I now understand that in the powerset case, the elements of the lattice do not need to be open sets. I was kind of hoping that locales would deal only with open sets, but perhaps it's an unreasonable thing to ask. – Tnatsissa H Craeser Sep 23 '15 at 21:28
  • @TnatsissaHCraeser: But it deals only with open sets. Open sets of a topological space form a complete lattice. It is just that it is not a complete sublatice of the powerset. Namely, the big meets of open sets are not intersections, but interiors of intersections. – Adam Bartoš Sep 24 '15 at 07:46
  • @user87690: I see, from wiki "The supremum is given by the union of open sets and the infimum by the interior of the intersection.". Thanks again for following through. – Tnatsissa H Craeser Sep 24 '15 at 16:51
  • The construction of surjections $\mathbb N\to\mathbb R$ can be found in Peter Johnstone's Sketches of an elephant, C1.2, example 1.2.8.

    I also added an answer with a presentation of that locale in the question you linked.

     – Jackozee Hakkiuz Aug 16 '20 at 21:02
  • The above comment should say "The construction of the locale of surjections..." etc. – Jackozee Hakkiuz Aug 17 '20 at 21:33