Let $A$ be a subset in a metric space $(X,d)$ with non-empty interior. Let $x$ be a point in the boundary of $A$. Does $B_{\delta}(x)$ intersect the interior of $A$ for all $\delta$?
This is a step I need when doing another problem. Geometrically it feels right?
No, not necessarily. Take the subset $[0, 1] \cup \{ 2\}$ of $\Bbb{R}$. The set has non-empty interior $(0, 1)$, and is closed, so the boundary is $$([0, 1] \cup \{2\}) \setminus (0, 1) = \{0, 1, 2\}.$$ The ball $B_{\frac{1}{2}}(2)$ fails to intersect $(0, 1)$.