Outer boundary of a planar compact set ($\partial C_{\infty }(K)$)

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.