  An inner-distal homeomorphism is one such that each of its proximal cells has
empty interior. In locally connected spaces, we prove these homeomorphisms have
the following properties: Every $cw$-distal homeomorphism is inner-distal but
not conversely. The inner-distal homeomorphisms are precisely those for which
the diameters of the iterates of every connected subset of the phase space
remain uniformly bounded away from zero. If, in addition, the phase space is
compact, every distal extension of an inner-distal-homeomorphism under an
inner-light map is inner-distal. We also study inner-distal homeomorphisms from
the measure-theoretical standpoint through the concept of inner-distal
measures. We prove that in Polish metric spaces, a homeomorphism is
inner-distal if and only if each Borel probability measure is. In compact
metric spaces, the existence of inner-distal measures guarantees the existence
of an invariant inner-distal one. In compact, locally connected spaces, the
minimal homeomorphisms supporting inner-distal homeomorphism are precisely the
inner-distal ones. We also classify the inner-distal homeomorphisms of the
circle or the interval. Finally, we give some dynamic consequences of
inner-distality.
