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Grothendieck topologies (or: toposes as generalized spaces):

There is no topology on a general scheme which is e.g. fine enough to give back the cohomological dimensions expected from geometry, but with a more general notion of covering (or: of space) this works out.

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Grothendieck topologies (or: toposes as generalized spaces):

There is no topology on a general scheme which is e.g. fine enough to give back the cohomological dimensions expected from geometry, but with a more general notion of covering (or: of space) this works out.