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](http://ncatlab.org/nlab/show/superextensive+site).

If you are willing to weaken the concept of pretopology to that of [coverage](http://ncatlab.org/nlab/show/coverage) then the coverage of [smooth good open covers](http://ncatlab.org/nlab/show/good+open+cover) on manifolds also generates the same topology.