Post #1888490
2026-04-27 04:11 UTC
@jameshanson Thanks! So I guess this means that going from a locale to its topological space of points is not compatible with joins/unions of sublocales? Because otherwise we would get R as a union of the two half rays pointwise? It should be compatible with joins of open sublocales though.
Replies (1)
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@jameshanson@mathstodon.xyz 2026-04-27 14:33
@jdw That's right. One way to think about it is that constructively it's consistent for R to 'not have enough points'. When this happens, if you take the join of the one-point sublocales of [0,∞), you get a proper sublocale of [0,∞). That said it's also not always going to be compatible with infinitary joins of open sublocales. If you have a singular cover of R, then there's a open proper sublocale of R that contains all of the points of R.