I am currently reading Vickers' text "topology via logic" and Peter Johnstone's "stone spaces", and I understand the material in both of these texts to pertain directly to constructions in elementary topos theory (by which I do not mean 'the theory of elementary topoi). However, these things do not seem to be mentioned explicitly in these texts, at least not to great extent. Where might I avail myself of material which really 'brings home' the notion of topoi as 'generalized spaces' in the context of stone spaces and locales as alluded to in Vickers and Johnstone? I understand that Borceaux's third volume in the 'handbook of categorical algebra' is probably a good place to start…
[Math] Stone Spaces, Locales, and Topoi for the (relative) beginner
lo.logiclocalestopos-theory
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To the best of my knowledge, this has never been "officially" described in the set theoretic literature. This has been described by Blass and Scedrov in Freyd's models for the independence of the axiom of choice (Mem. Amer. Math. Soc. 79, 1989). (It is of course implicit and sometimes explicit in the topoi literature, for example Mac Lane and Moerdijk do a fair bit of the translation in Sheaves in Geometry and Logic.) There are certainly a handful of set theorists that are well aware of the generalization and its potential, but I've only seen a few instances of crossover. In my humble opinion, the lack of such crossovers is a serious problem (for both parties). To be fair, there are some important obstructions beyond the obvious linguistic differences. Foremost is the fact that classical set theory is very much a classical theory, which means that the double-negation topology on a site is, to a certain extent, the only one that makes sense for use classical set theory. On the other hand, although very important, the double-negation topology is not often a focal point in topos theory.
Thanks to the comments by Joel Hamkins, it appears that there is an even more serious obstruction. In view of the main results of Grigorieff in Intermediate submodels and generic extensions in set theory, Ann. Math. (2) 101 (1975), it looks like the forcing posets are, up to equivalence, precisely the small sites (with the double-negation topology) that preserve the axiom of choice in the generic extension.
Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) can be interesting, and are controlled by its absolute Galois group / etale fundamental group. For example, $\text{Spec } \mathbb{F}_q$, the Spec of a finite field, has the same finite covering theory as $S^1$, which reflects (and is equivalent to) the fact that its absolute Galois group is the profinite integers $\widehat{\mathbb{Z}}$. (So this suggests that one can think of $\text{Spec } \mathbb{F}_q$ itself as behaving like a "profinite circle.")
More generally, suppose you want to classify objects of some kind over $k$ (say, vector spaces, algebras, commutative algebras, Lie algebras, schemes, etc.). A standard way to do this is to instead classify the base changes of those objects to the separable closure $k_s$, then apply Galois descent. The topological picture is that $\text{Spec } k$ behaves like $BG$ where $G$ is the absolute Galois group, $\text{Spec } k_s$ behaves like a point, or if you prefer like $EG$, and the map
$$\text{Spec } k_s \to \text{Spec } k$$
behaves like the map $EG \to BG$. In the topological setting, families of objects over $BG$ are (when descent holds) the same thing as objects equipped with an action of $G$. The analogous fact in algebraic geometry is that objects over $\text{Spec } k$ are (when Galois descent holds) the same thing as objects over $\text{Spec } k_s$ equipped with homotopy fixed point data, which is a generalization of being equipped with a $G$-action which reflects the fact that $k_s$ itself has a $G$-action.
(I need to be a bit careful here about what I mean by "$G$-action" to take into account the fact that $G$ is a profinite group. For simplicity you can pretend that I am instead talking about a finite extension $k \to L$, although I'll continue to write as if I'm talking about the separable closure. Alternatively, pretend I'm talking about $k = \mathbb{R}, k_s = \mathbb{C}$.)
The classification of finite covers is the simplest place to see this: the category of finite covers of $\text{Spec } k_s$ is the category of finite sets, with the trivial $G$-action, so homotopy fixed point data is the data of an action of $G$, and we get that finite covers of $\text{Spec } k$ are classified by finite sets with $G$-action.
But Galois descent holds in much greater generality, and describes a very general sense in which objects over $k$ behave like objects over $k_s$ with a Galois action in a twisted sense.
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Many good books have already been mentioned; I like MacLane+Moerdijk as an introduction, and after that both books by Johnstone (in particular, Part C of the Elephant does a good job of connecting locale theory with topos theory). But I also wanted to mention Vickers' paper "Locales and Toposes as Spaces," which I think does a good job of connecting up the topology with the toposes and the logic in a way that isn't directly evident in many other introductions.