Suppose $V$ is a bounded open set in $\mathbb{R}^2$ and $F$ is a closed subset of the boundary of $V$. Suppose $F$ has the property that if $y$ is a point in $F$, then there is a positive number $r_0$(depending on $y$) such that the boundary of every open ball of center $y$ and radius $0<r<r_0$ intersects $F$: $$\partial B(y,r)\cap F\not=\emptyset, $$ for all $0<r<r_0$.

My question is: Can we constract an open set $W\subset V$ such that the part of the boundary of $W$ that intersets the boudary of $V$ is just $F$ (or a part of $F$ ): $\partial W\cap \partial V=F$ (or $\partial W\cap \partial V\subset F$), and the part of the boudary of $W$ that intersects $V$ is a portion of the boundary of some ball: $\partial W\cap V\subset \partial B$, for some ball $B$? It's easy to see this problem over a shape and in $\mathbb{R}^2$ at least it seems correct. But I am not sure if some kind of "contor set" will miss it up?


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