Simple closed curves contained in~$\epsilon$-boundaries of planar sets

The $\epsilon$-boundary of a set $A\subseteq {\mathbb{R}}^2$ is the set $\{p\in {\mathbb{R}}^2:\rho (p,A)=\epsilon \}$, where $\rho$ is the Euclidean distance. We prove that if $A,B\subseteq {\mathbb{R}}^2$ are nonempty, connected sets, $A$ is bounded, and $0<\epsilon <\rho (A,B)$, then the $\epsilon$-boundary of $A$ contains a simple closed curve (aka a Jordan curve) that separates $A$ and $B$. This statement follows from the theorem which says that if $\epsilon >0$ and $A\subseteq {\mathbb{R}}^2$ is a nonempty, bounded, connected set, then the boundary of each component of $\{p\in {\mathbb{R}}^2:\rho (p,A)>\epsilon \}$ is a simple closed curve. Another corollary of this theorem is that the $\epsilon$-boundary of a nonempty, bounded, connected set $A\subseteq {\mathbb{R}}^2$ contains a simple closed curve bounding the domain that contains the open $\epsilon$-neighbourhood of $A$. In all these statements the connectivity condition can be significantly weakened. We also show that, for all $\epsilon >0$, the $\epsilon$-boundary of a nonempty, bounded set $A\subseteq {\mathbb{R}}^2$ contains a simple closed curve.