# Constructing an open set when a portion of its boundary is given

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 $$0 intersects $$F$$: $$\partial B(y,r)\cap F\not=\emptyset,$$ for all $$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?