There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective on points, such that each map in the family has some property $X$ characteristic of the topology in question; the topology is generally then called the "$X$ topology". For instance, in the étale topology, each of the maps is required to be étale.

The properties $X$ in question vary widely, but here's one very basic feature which many of them have in common: very often "$X$" is of the form "$Y + Z$", where $Y$ is a "niceness" property and $Z$ is a "finiteness" property. For instance, "etale" decomposes as "formally etale" + "locally of finite presentation". My question is:

**Question 1:** Why do topologies used in algebraic geometry generally involve specific finiteness conditions?

I think the answer to this question may well be quite straightforward -- for instance, I'm not sure whether a "formally etale" topology exists, but if it does, I imagine it has rather pathological properties, and it would be interesting to know something about these pathologies.

**Question 2:** Are there counterexamples to this trend in algebraic geometry -- useful topologies which allow for "non-finite" morphisms to generate their covers?

**Question 3:** Are there counterexamples to this trend elsewhere in mathematics -- useful Grothendieck topologies which are more "infinitary" in nature?

For Question 3, I'm kind of suspecting that some of the topologies used in set-theoretical forcing might be more "infinitary" in nature, but I'm having trouble imagining less exotic settings where this sort of thing is likely to come up.

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