For another example, the pretopologies on the category of finite dimensional smooth manifolds given by

• open covers
• maps of the form $\coprod U_i \to X$ for a given open cover $(U_i)$
• surjective local diffeomorphisms
• surjective submersions

all generate the same topology. The last three are nested, and the second is cofinal in the other two. The second and the first are equivalent because of superextensivity.

If you are willing to weaken the concept of pretopology to that of coverage then the coverage of smooth good open covers on manifolds also generates the same topology.

For another example, the pretopologies on the category of finite dimensional smooth manifolds given by

• open covers
• maps of the form $\coprod U_i \to X$ for a given open cover $(U_i)$
• surjective local diffeomorphisms
• surjective submersions

all generate the same topology. The last three are nested, and the second is cofinal in the other two. The second and the first are equivalent because of superextensivity.

If you are willing to weaken the concept of pretopology to that of coverage then the coverage of (EDIT: coproducts of) smooth good open covers on manifolds also generates the same topology.