I'm working with the set of all open convex bounded sets of $\ \mathbb{R}^n$ with the topology induced by the Hausdorff distance. Now this distance function is defined between two sets $\ X$ and $\ Y$ as follows: $\ d_H(X,Y)=\max \{\sup_{x \in X}\ d(x,Y),\sup_{y \in Y}\ d(y,X) \}$, where $\ d(x,Y)=\inf_{y \in Y}\ d(x,y)$ and $\ d$ is the Euclidean distance.
I know that $\ d_H(X,Y)=0$ implies $\overline{X}=\overline{Y}$, but why does $\overline{X}=\overline{Y}$ imply $\ X=Y$ when both $\ X$ and $\ Y$ are open, convex and bounded? (Would be enough to show that bd$\ X=$ bd$\ Y$)
Any help would be appreciated.