In this paper we will generalize the following well-known result. Suppose that I is an arc in the complex sphere ℂ* and h : I ℂ* is an embedding. Then there exists an orientation-preserving homeomorphism H : ℂ* → ℂ* such that H I = h. It follows that h is isotopic to the identity. Suppose X → ℂ* is an arbitrary, in particular not necessarily locally connected, continuum. In this paper we give necessary and sufficient conditions on an embedding h : X → ℂ* to be extendable to an orientation-preserving homeomorphism of the entire sphere. It follows that in this case h is isotopic to the identity. The proof will make use of partitions of complementary domains U of X, into hyperbolically convex subsets, which have limited distortion under the conformal map φU : D → U on the unit disk. © 2010 American Mathematical Society.