3-connected planar spaces uniquely embed in the sphere

We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere -- i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if $\bar G$ is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in $G$ has a unique accumulation point in $\bar G$, then $\bar G$ has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.