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In algebraic geometry, the importance of non-trivial Grothendieck topologies is very well-known. One starts out with the Zariski topology on $\mathsf{Sch}$, but concludes that it is 'too coarse' for cohomological techniques to work, and so one develops the more refined étale topology as a remedy. The validity of étale descent is central to why the topology works in the first place. Ever since then, one often finds algebraic geometers playing with the various topologies, sometimes even developing new ones that get as close as possible to their needs.

Contrast this to 'ordinary' topology, where one has the archetypical Grothendieck topology on $\mathsf{Top}$ in which coverings are simply declared to be the open coverings in the classical sense. I do not recall having ever seen any other Grothendieck topology, and certainly no interesting ones. One could declare coverings to be jointly surjective, but that topology would fail to be subcanonical. What about covering spaces? Perhaps we may declare coverings to be one-element sets $\{Y \to X\}$ in which the map $Y \to X$ is a covering space.

Do you know of any interesting Grothendieck topologies on $\mathsf{Top}$? Are they as varied as they are on $\mathsf{Sch}$, and are there any applications to their existence?

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    I would guess that the site consisting of covering spaces of $X$'s would have cohomology that computes group cohomology relative to the fundamental group. – Alex Youcis Sep 19 '19 at 15:04

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Yes. For instance, the numerable topology on Top controls bundles that are classified by maps to a classifying space.

Open covers in the numerable topology are those open covers that admit a subordinate partition of unity.

If G is a topological group, it has a classifying space BG, as well as a classifying stack G-Bun of principal G-bundles, defined as the stackification of the prestack B(Top(−,G)).

Concretely, for a topological space T the groupoid G-Bun(T) is the groupoid of numerable principal G-bundles over T, i.e., principal bundles that are trivializable over a numerable open cover.

We have a canonical isomorphism π_0(G-Bun(T))→[T,BG], i.e., isomorphism classes of numerable principal G-bundles over T are classified by homotopy classes of maps into BG.

This is false if we do not use the numerable topology.

The numerable site is essentially due to Albrecht Dold (Partitions of unity in the theory of fibrations, Annals of Mathematics (2) 78 (1963), 223–255).

Dmitri P.
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  • Interesting! I recall skimming through a paper on topological stacks by B. Noohi, and I don't remember anything like a numerable topology being needed there though. –  Sep 23 '19 at 12:19
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    @II: Noohi restricts to paracompact spaces, for which the two topologies coincide. This restriction is unnecessary if one works with the numerable topology. – Dmitri P. Sep 23 '19 at 16:09