  A homeomorphism of a compact metric space is {\em tight} provided every
non-degenerate compact connected (not necessarily invariant) subset carries
positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a
closed surface factors to a tight homeomorphism of a generalized cactoid
(roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero
entropy.
