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I have no idea why the stronger definition was there for so long. Itâ€™s just not true in applications, most particularly the pretopology consisting of surjective submersions on the category of manifolds.
Need to fix this false statement:
Notice that a basis for the topology of X is not a Grothendieck pretopology on Op(X) (since it is in general not closed under pullback, which here is restriction) but is a coverage on Op(X).
The definition of a pretopology does not require morphisms that belong to some covering family to be closed under pullbacks. Rather, it requires that base changes of morphisms in covering families always exist.
In particular, base changes of inclusions whose domains belong to the base always exist, even though the resulting inclusion has as its domain an open subset that need not belong to the base.
As discussed in this answer
there are at least 3 different sites one can produce from a base of a topological space, and only one of them (arguably the least natural one) gives a pretopology.
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