  We show that a homeomorphism of a semi-locally connected compact metric space
is equicontinuous if and only if the distance between the iterates of a given
point and a given subcontinuum (not containing that point) is bounded away from
zero. This is false for general compact metric spaces. Moreover, homeomorphisms
for which the conclusion of this result holds satisfy that the set of
automorphic points contains those points where the space is not semi-locally
connected.
