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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeJun 18th 2010

I got tired of making unmatched links to topological locale (aka spatial locale, or locale with enough points), so I wrote a stub.

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeJun 18th 2010

Also locale of opens. Very stubby.

I think that I really only wrote these so that I could put in a bunch of redirects for each, and head off anybody else writing them with too few redirects. (^_^)

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJun 18th 2010

I would be more inclined to call the page “spatial locale” – I’ve encountered that term much more often. Plus “topological” as an adjective usually means that we’ve put an extra topology on something, so a “topological locale” sounds like an internal locale in Top.

I’m also not really fond of calling it the “locale of opens” – I would talk about the frame of opens, but the locale is “of” the same things that the space was. E.g. the locale underlying the space of real numbers is the locale of real numbers, not the locale of open subsets of real numbers – the latter would be some sort of “hyperlocale” whose points are open sets.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeJun 18th 2010

Plus “topological” as an adjective usually means that we’ve put an extra topology on something, so a “topological locale” sounds like an internal locale in Top.

Well, yeah, but the same is true of ‘spatial’. The term ‘topological’ here is acting in same way as ‘localic’ in ‘localic topos’. I don’t like ‘spatial’ because a locale is a kind of space anyway! (But spatial locale redirects, of course.) The term ‘topological’, however, is well established to mean a topological space in the sense of Bourbaki, as when (for example) one says that a convergence space is ‘topological’.

I’m also not really fond of calling it the “locale of opens” – I would talk about the frame of opens, but the locale is “of” the same things that the space was.

Gosh, you’re right! I was a little unhappy with that name on the grounds that every locale is a locale of opens; this is really just the locale of opens in a topological space. But it’s worse than that; every locale is really a locale of points! (Just like a topological space is a space of points.)

So what is the correct term here? For the locale I mean, not the frame. I suppose that one might say ‘topological locale’ (or ‘spatial locale’) again here, but what is one supposed to say when given a topological space $X$ and wanting to speak of the locale whose underlying frame is the frame of open subspaces of $X$? Somehow I have not picked up that terminology.

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeJun 19th 2010

OK, I’ve reworked topological locale and locale of opens into a new version of topological locale. This does not use the terminology ‘locale of opens at all’, but instead introduces the notation $\Omega(-)$ for the concept. I have also written frame of opens (which I made out of the old locale of opens, although that name now redirects to topological locale, since that is the page that discusses the concept that I had earlier had at the badly named page locale of opens). And I am writing space of points as well.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJun 19th 2010

the same is true of ‘spatial’.

Not to me. I’ve never heard people talk about, say, a “spatial group” or a “spatial category” with the same meanings as the established terms “topological group” and “topological category.” But I can live with “topological locale.”

I don’t know of a standard term for “the locale underlying a topological space.” I suppose if we identify spatial locales with sober spaces, as one might argue is not unreasonable, then it would be the same as the soberification, but that’d be misleading here.