TITLE: Grothendieck Topologies versus Pretopologies
QUESTION [12 upvotes]: The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that distinct pretopologies can give rise to the same topology on a category. My (almost certainly trivial) questions are the following:
1) What are the key differences between Grothendieck topologies and Grothendieck pretopologies, and
2) What are some nice examples of distinct pretopologies that give rise to the same topology?

REPLY [10 votes]: I think of pretopologies (and coverages and other variations of this idea) as convenient ways of presenting a topology, just as groups (and other algebraic structures) can be conveniently presented by generators and relations, or (since I'm a set theorist) a complete Boolean algebra can be conveniently presented by giving a dense subposet (called a notion of forcing).  Usually (though not always), the important thing is the bigger, more abstract object --- the Grothendieck topology, the group, or the complete Boolean algebra --- on which the interesting constructions can be based.  Then the choice of a particular presentation is, in principle, irrelevant, but it can be very useful for working with the object and understanding it, especially because the presentation is often much smaller than the abstract object.