Classical point-set topology using Grothendieck topologies

Question

Its well known that the category of opens $O(X) $of a topological space $X$ can be endowed with a Grothendieck topology making it into a site. I am looking for references which take the reader through the classical point-set topology using this point of view.

I am really trying to see the advantage of this point of view. My main guesses are that there is a lot of categorical machinery lying around that can be used to justify classical (and more modern) statements in point-set topology. And of course the blatant generality that this point of view provides.

Most algebraic geometry books I have read will define the Zariski topology on a spectrum of a ring in point-set terms. I have yet to see one that does so using the language of sites.

Answer 1

• Point-set topology is used to formalize the intuition of continuity and of convergence. It finds its ideal applications for example in Analysis.

• The notion of Grothendieck topology is designed to formalize the intuition of patching things together. More technically, this means dealing with descent theory (including sheaf theory of various flavors, e.g. stacks). It finds its applications mainly in Algebraic Geometry, Algebraic Topology, and Logic.

Let's make an analogy. Classical topology is not a direct generalization of metric spaces, rather it's a generalization of some aspects of metric space theory, namely those that pertain to continuity, adherence, etc. To each metric space you can canonically associate a topological space, and not all topological spaces arise in this way (so it's somehow a generalization); but in doing so you lose information since several distinct metrics give rise to the same topology (so it's not really a generalization, rather an abstraction).

In the same way, "Grothendieck topology" is not a generalization of classical point-set topology per se (for example you cannot recover the notion of convergence of a sequence to a point from the G. topology alone, simply because the notion of point is missing), rather it's a generalization of some aspects of point-set topology, namely those that allow you to do descent theory.

Non-homeomorphic topological spaces may induce equivalent Grothendieck topologies: this happens when the corresponding toposes of sheaves are equivalent (talking about categories equipped with a Grothendieck topology is essentially the same as talking about certain categories that are called (Grothendieck) toposes), so in the passage you lose information. Also, there are toposes that do not come from any topological space.

But the situation is slightly better behaved than the passage from metric spaces to topological spaces. Indeed, in topos theory there is the notion of point (which is of course expressed purely in terms of categorical abstract nonsense), and it turns out that if a topos has "enough points" [edit: and is generated under colimits by subobjects of the terminal object, see comment by Denis Nardin] it does indeed come from a topological space in the way you described. Even better, if this is the case, then there is somehow a canonical choice among all those topological spaces that induce the same topos: you take the unique sober one.

This said, I have the impression that the notion of point in the topos context is still quite useless, let alone the notion of convergence of a sequence/net. But I don't know enough about all this to be sure.

Answer 2

I'm not quite sure what you're asking, but under one way to interpret the question, an answer is that the theory of locales is a well-developed alternative to the classical theory of topological spaces. A locale is essentially the algebraic structure of the opens of a topological space (which is more or less equivalent to its structure as a site), considered in isolation from any "points" that the space might have. Locale theory (or "pointless topology") looks a little different from classical point-set topology, but can be used for many of the same purposes, and has certain advantages (e.g. it tends to work much better when using constructive logic, such as the internal logic of a topos). Some books on locale theory include Johnstone's Stone spaces, Picado and Pultr's Frames and Locales, Vickers' Topology via logic, and Part C of Johnstone's Sketches of an Elephant.

Answer 3

(This should probably be a comment but I can't post it as such).

My main guesses are that there is a lot of categorical machinery lying around that can be used to justify classical (and more modern) statements in point-set topology.

Here is a couple of observations in this spirit, but probably rather from what you actually want: a number of elementary properties such as separation axioms, compactness, can be reformulated in terms of orthogonality of morphisms, and the definition of a topological space can be understood as defining a simplicial object of some category (see my question for details and references).

Though, I do not know of an exposition of point-set topology using these observations.