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Often, topological properties of locales can be generalized naturally to toposes (or, often, to geometric morphisms). … Is there a notion of "regularity" for toposes that generalizes the topological notion of regularity for spaces and locales? …

Locales are often better-behaved than topological spaces in constructive mathematics (i.e. in the absence of the law of excluded middle). … My question is: suppose $f:R_f \to R_f$ is a continuous map of locales, such that $f(x)>0$ for all points $x$ of $R_f$ (i.e. for all actual real numbers $x$); does it follow (constructively) that $f$ factors …

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of …

Can "strict pro-sets" be identified with "pro-discrete locales"? Note that some hypothesis such as "surjective transition maps" is necessary. … For instance, the pro-set $\cdots \xrightarrow{+1} \mathbb{N} \xrightarrow{+1} \mathbb{N} \xrightarrow{+1} \mathbb{N}$ is not isomorphic to the trivial pro-set ∅, but its limit in the category of locales …