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We study the removability of compact sets for continuous Sobolev functions.In particular, we focus on sets with infinitely many complementary components,called "detour sets", which resemble the Sierpi\'nski gasket. The main theoremis that if $K\subset \mathbb R^n$ is a detour set and its complementarycomponents are sufficiently regular, then $K$ is $W^{1,p}$-removable for $p>n$.Several examples and constructions of sets where the theorem applies are given,including the Sierpi\'nski gasket, Apollonian gaskets, and Julia sets.

A removability theorem for Sobolev functions and detour sets