= Outer boundary of a planar compact set
{title2=$\partial C_\infty(K)$}
The outer boundary of a compact planar set $K$ is the boundary of the unbounded component $C_\infty(K)$ of its complement. Its filled hull is $\mathbb C\setminus C_\infty(K)$. When the outer boundary is a <simple closed curve>, the hull is its closed interior. For a simply connected open domain $U$, containment of a compact trace and of its filled hull are equivalent. This lets outer-boundary pushforwards retain simply connected domain-containment information.
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