TITLE: genericness and the Zariski topology
QUESTION [7 upvotes]: What does it mean (in a mathematically rigorous way) to claim something is "generic?" How does this coincide with the Zariski topology?

REPLY [6 votes]: In general*, if something is "generic", it means it happens or is true "almost all of the time" or "almost everywhere".
In measure theory, for example, when you say "$P(x)$ is true for almost all $x$", this has the precise meaning that the set of $x$'s for which $P(x)$ does not hold has measure zero.
One can relate this to the Zariski topology via the fact that Zariski closed subsets of $\mathbb{C}^n$ have Lebesgue measure zero: https://mathoverflow.net/questions/25513/zariski-closed-sets-in-cn-are-of-measure-0
See also these MO posts:
https://mathoverflow.net/questions/19688/what-does-generic-mean-in-algebraic-geometry
https://mathoverflow.net/questions/2162/what-are-the-most-important-instances-of-the-yoga-of-generic-points
*or perhaps I should say... generically ;-)