The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set.
Suppose there are two sets $S$ and $T$ such that $S\cap T$ has a non-empty interior. Prove that there exists a constant $r>0$, such that every set $S'$ whose Hausdorff distance from $S$ is less than $r$ also intersects $T$.
Here is my attempt:
Let $P$ be a point in the interior of $S\cap T$, and let $r>0$ be the largest radius of a ball contained in $S\cap T$ whose center is $P$. If $d_H(S,S')<r$, then, because $P\in S$, there must be a point $P'\in S'$ such that $d(P,P')<r$. Therefore $P'$ is contained in the ball of radius $r$ contained in $S\cap T$, and thus $P'\in S'\cap T$.
A. Is this proof correct?
B. I am trying to slightly generalize this lemma to: there exists a constant $r>0$, such that every set $S'$ whose Hausdorff distance from $S$ is less than $r$ intersects every set $T'$ whose Hausdorff distance from $T$ is less than $r$. Is this generalization correct?
A. Your proof is correct.
B. No, this generalization is false. In $\mathbb R^n$, given two nonempty compact sets $S$ and $T$ and a number $r>0$, we can find two nonempty compact sets $S'$ and $T'$ such that $d_H(S,S')\le r$, $d_H(T,T')\le r$, and $S'\cap T'=\varnothing$.
Indeed, since $S$ is compact, for any $\epsilon>0$ we can find a finite $r/2$-net $N_S\subset S$, i.e., a set such that for every $x\in S$ there is $y\in N_S$ such that $|x-y|\le r/2$. Observe that $d_H(S,N_S)\le r/2$.
Similarly, let $N_T$ be a $r/2$-net in $T$. If $N_S$ and $N_T$ are disjoint, we are done. Otherwise, move each points of $N_T$ that belongs to $N_S$ in some direction by distance $\le r/2$, so that the new point is not in $N_S$. (Since $N_S$ is finite, this is easy to do). This perturbation creates a set $\tilde N_T$ which is disjoint from $N_S$ and satisfies $d_H(T,\tilde N_T)\le r$.