  We provide a measure based topology for certain unions of C2 rectifiable
submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets
remain in the limit as measures when higher dimensional sets collapse down to
them. For example a decreasing sequence of spheres may have a limit consisting
of just a point. The n-1 dimensional space of outward pointing vectors can be
used for such a measure. It represents all C2 rectifiable sets of codimension
at least one of Rn as rectifiable sets in RnXSn-1 with n-1 dimensional
Hausdorff measure. When viewed as (n-1)-rectifiable varifolds or currents in
RnXSn-1 they come equipped with compactness theorems. The projection of their
limits to Rn recovers the rectifiable sets of mixed dimensions giving the
limits for the desired topology on subsets of Rn. Both varifold and current
compactness are required as there are sequences, such as honeycombs, that
converge as varifolds but not as currents. Conversely sequences such as lifts
of polyhedral approximations converge as currents but not as varifolds.
