TITLE: Homeomorphism of two compact sets
QUESTION [6 upvotes]: If I have two compact subsets, $A$ and $B$, of the plane $\mathbb{C}$, and we know that $\partial A$ and $\partial B$ are homeomorphic, can we say that $A$ and $B$ are homeomorphic?

REPLY [9 votes]: Let $A$ be a full circle and $B := ∂A$.

REPLY [6 votes]: No. Consider a closed annulus, such as $\{z \in \mathbb C \mid 1 \leq |z| \leq 2\}$, and the union of two closed discs, such as $\{z \in \mathbb C \mid \min(|z-2|,|z+2|) \leq 1\}$.