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Stream: learning: questions

Topic: products of discrete locales

Reid Barton (Jul 22 2024 at 22:11):

Does the functor from topological spaces to locales preserve (infinite) products of discrete spaces? (Assume a classical metatheory.)
I checked that it is true for countable products, but I'm not sure if the method will extend to the case of uncountably many factors.

Graham Manuell (Jul 23 2024 at 05:41):

It does not. In Loc, $\mathbb{N}^{\omega_1}$ is not spatial. You can see this by noting that the corresponding topological space is not Lindelöf, but products of regular Lindelöf locales are always Lindelöf.

Reid Barton (Jul 24 2024 at 17:39):

Interesting, thanks!