Guided by topology I'm expecting this to be true, but I cannot find a proof for the following statement:
Every regular locale is spatial.
2026-02-23 05:29:01.1771824541
Regular locale is spatial
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The answer is no. A counterexample is $\Omega(\mathbb{Q})\times \Omega(\mathbb{Q})$, the product (in the category of locales) of (the locale corresponding to) the space of rational numbers with itself.
Johnstone gives two proofs in Stone Spaces that this locale is not spatial: Proposition II.2.14 on p.61 is a direct proof, and it also follows from Proposition III.1.4 on p. 84. This locale is regular since any product of regular locales is regular (Proposition III.1.6 on p. 85).
On the other hand, it's a fundamental theorem that every compact regular locale is spatial (Proposition III.1.10 on p. 90).
I'm curious how you were guided by topology to expect regular locales to be spatial. Your intuition from topology can be useful in generalizing from spatial locales to all locales. But I'm not sure how it could be useful in determining which locales are spatial.