  We establish Euclidean-type lower bounds for the codimension-1 Hausdorff
measure of sets that separate points in doubling and linearly locally
contractible metric manifolds. This gives a quantitative topological
isoperimetric inequality in the setting of metric manifolds, in the sense that
lower bounds for the codimension-1 measure of a set depend not on some notion
of filling or volume but rather on in-radii of complementary components. As a
consequence, we show that balls in a closed, connected, doubling, and linearly
locally contractible metric $n$-manifold $(M,d)$ with radius $0<r \leq
\text{diam}(M)$ have $n$-dimensional Hausdorff measure at least $c \cdot r^n$,
where $c>0$ depends only on $n$ and on the doubling and linear local
contractibility constants.
