We examine conditions under which a point in the stable set
of a hyperbolic invariant set for a $C^1$ surface
diffeomorphism is accessible via a path from the complement of
the stable set. Let $M$ be a surface, and let $\Lambda$ be a
compact saturated hyperbolic locally stably closed invariant
set possessing a local product structure. Denote the stable
set of $\Lambda$ by $W^s(\Lambda)$. Our main result states
that $z \in W^s(\Lambda)$ is accessible from $M \setminus
W^s(\Lambda)$ if and only if $z$ lies on the stable manifold
of a periodic point $p$, and there is a branch of a local
unstable manifold of $p$ disjoint from $W^s(\Lambda)$.