genericness and the Zariski topology What does it mean (in a mathematically rigorous way) to claim something is "generic?" How does this coincide with the Zariski topology?

Generic has different meanings in different branches of geometry.  In algebraic geometry,
it usually means that the property in question holds on a Zariski dense open set.  In other geometric contexts, it could also that the property holds on a dense open set (in whatever is the natural topology under consideration), but can also mean that it holds on a countable intersection of such sets.  (The reason for considering such intersections as being big is
motivated by the Baire category theorem, which says that in reasonable contexts the complement of such a set will be very "thin", in some sense.)
