  We present a simplified treatment of stability of filtrations on finite
spaces. Interestingly, we can lift the stability result for combinatorial
filtrations from [CSEM06] to the case when two filtrations live on different
spaces without directly invoking the concept of interleaving. We then prove
that this distance is intrinsic by constructing explicit geodesics between any
pair of filtered spaces. Finally we use this construction to obtain a
strengthening of the stability result.
