  We study an invariant of compact metric spaces which combines the notion of
curvature sets introduced by Gromov in the 1980s together with the notion of
Vietoris-Rips persistent homology. For given integers $k\geq 0$ and $n\geq 1$
these invariants arise by considering the degree $k$ Vietoris-Rips persistence
diagrams of all subsets of a given metric space with cardinality at most $n$.
We call these invariants \emph{persistence sets} and denote them as
$D_{n,k}^\mathrm{VR}$. We argue that computing these invariants could be
significantly easier than computing the usual Vietoris-Rips persistence
diagrams. We establish stability results as for these invariants and we also
precisely characterize some of them in the case of spheres with geodesic and
Euclidean distances. We identify a rich family of metric graphs for which
$D_{4,1}^{\mathrm{VR}}$ fully recovers their homotopy type. Along the way we
prove some useful properties of Vietoris-Rips persistence diagrams.
