Ball intersecting with an open set

Question

Is the following statement correct?

Let $B$ be an open ball in $\mathbb R^d$ that intersects an open set $U$. Suppose $B$ contains an exterior point of $U$. Then $B$ has to intersect $\partial U$.

My further questions are:

1. Is the connectivity (or path-connectedness) of $B$ only used in the proof?
2. If 1 is true, then it is true in a general topological space assuming $B$ is connected (or path-connected)?

Answer 1

Exactly.

Suppose $U$ and $B$ are any subsets of a topological space, and assume $B$ is disjoint from $\partial U$, but intersects both $\mathrm{int}(U)$ and $\mathrm{ext}(U)=\mathrm{int}(U^\complement)\,$. Then $$B=(B\cap\mathrm{int}(U))\,\cup\,(B\cap\mathrm{ext}(U))$$ is a disjoint union of two nonempty open subsets of $B\ $ (in the subspace topology).