  We address the question of what is the correct higher dimensional analogue of
Jordan curves from the point of view of quantitative rectifiability. More
precisely, we show that 'topologically stable' sets can be used as covering
objects in Analyst's Travelling Salesman Theorem-type theorems: if $E$ is lower
$d$-regular (in a certain suitable sense), then we show that there exists a
topologically stable surface $\Gamma$ so that $E \subset \Gamma$ and
  $$
  \mathrm{diam}(E)^d + \beta^d(E) \approx \mathcal{H}^d(\Gamma), $$
  where $\beta^d$ is a term quantifying the curvature of $E$. A corollary of
the main result of this paper and a construction by Hyde, is a higher
dimensional analogue of Peter Jones TST, valid for \textit{any} subset of
Euclidean space.
