In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational.
Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected.