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Don’t recall ever seeing it. Probably should be renamed “spatial locale”.
Hang on, I think there was a reason for calling it “topological locale” even if that term doesn’t appear elsewhere in the literature. The point is that according to a broad perspective, a locale is itself already a kind of space. The term “spatial locale” assumes that “space” means “topological space”, but that’s often an unwarranted assumption.
Are you saying it should be titled “topological locale” after all?
As far as I know, there is never any question that “spatial locale” means “topological spatial locale” – that’s how everyone uses the term – so how could it be an unwarranted assumption?
We already have a whole section on terminology in the article.
“Topological X” very consistently refers to an X-object in the category of topological spaces. Topological group, topological ring, topological vector space. Frames are also algebraic structures.
Also, “topological locale” suggests that these particular locales have more to do with topology or topological spaces than the other locales, which is not true.
I agree that “spatial locale” is bad terminology and something like “pointful locale” or “sober locale” (but definitely not “topological locale”) would be much better.
However, I would very strongly argue that nLab should stick to established terminology where it exists and not use some made-up terms, since the latter make it very difficult to follow up with a literature search after reading some nLab article.
The term “locale with enough points” would match the terminology and intent used in the entry and is used in standard texts as well as in current discussion (according to a quick Google search)
Re #9: Yes, and this term is already mentioned in the article. It is introduced by Johnstone in “Stone spaces”.
Why do “locale of opens” and “locale of open subspaces” redirect here?!
See the old discussion I linked in #7.
I like “locale with enough points” as a name for the page (with “spatial locale” mentioned prominently as a standard terminology at the top of the page, and of course as a redirect).
A problem with the terminology “topological locale” is that it clashes with Isbell’s notion of “topological topology” from his paper “function spaces and adjoints” (https://www.jstor.org/stable/24491137). According to this, a topological frame would be a frame equipped with a topology making the frame operations continuous.
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