Separation of plane sets by equidistant simple closed curves

We prove that if two subsets ${A}$ and ${B}$ of the plane are connected, ${A}$ is bounded, and the Euclidean distance $\rho({A},{B})$ between ${A}$ and ${B}$ is greater than zero, then for every positive $\varepsilon<\rho({A},{B})$, the sets ${A}$ and ${B}$ can be separated by a simple closed curve (also known as a Jordan curve) whose points all lie at distance $\varepsilon$ from the set ${A}$. We also prove that the $\varepsilon$-boundary of a connected bounded subset ${A}$ of the plane contains a simple closed curve bounding the domain containing the open $\varepsilon$-neighbourhood of ${A}$. It is shown that in both statements the connectivity condition can be significantly weakened. We also show that the $\varepsilon$-boundary of a nonempty bounded subset of the plane contains a simple closed curve. This result complements Morton Brown's statement that the $\varepsilon$-boundary of a nonempty compact subset of the plane is contained in the union of a finite number of simple closed curves.