  A topology is defined on the mapping class group of a compact connected
orientable surface. It is shown that a notion of "genericity" on subsets of the
mapping class group arises from this definition. Many plausible results follow
from this notion easily; for example, the set of pseudo-Anosov maps is shown to
be generic, and can be assumed to have arbitrary large stretch factor,
generically. Let M be a 3-manifold obtained from a Heegaard splitting of fixed
genus g and generic gluing map. It is shown that for such manifolds,
generically M is hyperbolic, has first Betti number zero and Heegaard genus
exactly equal to g.
