connectedness of this set Suppose $A$ is a subset of $\mathbb{R}^{n}$, $n>1$, and let 
$$B=\lbrace x:dist(x,A)\leq r\rbrace,$$
where $r$ is positive. If $A$ is connected, is $B$ also connected?
 A: First note that we may assume that $A$ is closed (why?). We write $b_1 \sim b_2$ to denote that $b_1,b_2 \in B$ lie in the same connected component of $B$. We will show that $b_1 \sim b_2$ for any $b_1,b_2 \in B$, i.e. $B$ is connected.
Since $A$ is closed, there exist $a_1,a_2 \in A$ with
$$\vert b_1 - a_1 \vert, \vert b_2 - a_2 \vert \leq r.$$
It follows that the line segment joining $b_1$ and $a_1$, respectively for $b_2$ and $a_2$, lies in $B$, thus $b_1 \sim a_1$ and $a_2 \sim b_2$. Since $A$ is connected, we have $a_1 \sim a_2$. By transitivity we have $b_1 \sim b_2$.
A: Notation: $d$ denotes distance.  $\;\bar B(p,r)=\{q:d(q,p)\leq r\}.$
(1).  Let $C,D$ be open subsets of $\mathbb R^n$ such that $(C\cap B)\cap (D\cap B)=\phi$ and $(C\cap B)\cup (D\cap B)=B.$
Then ($C\cap A)\cap (D\cap A)=\phi$ and $(C\cap A)\cup(D\cap A)=A,$ and $A$ is connected, so one of the sets $(C\cap A),\;(D\cap A)$ is equal to $A$  and the other is empty... So WLOG $\;C\cap A=A.$ 
(2). Observe that $B=\cup_{p\in A}\{\bar B(p,r)\}.$ 
(3).  Now if $p\in A$ the set $\bar B(p,r)$ is connected, and the sets $C\cap \bar B(p,r),\; D\cap \bar B(p,r)$ are disjoint...[ because by (2), $\;C\cap \bar B(p,r)\subset C\cap B$ and $D\cap \bar B(p,r)\subset D\cap B$],  and their union is $\bar B(p,r)\;$...[ because  $C\cup D\supset B$, and $B\supset\bar B(p,r)$ by (2)].
So one of them is $\bar B(p,r)$ and the other is empty. But $p\in C$ by (1) so $\phi\ne C\cap \bar B(p,r)$.
So  $D\cap \bar B(p,r)=\phi$.  This holds for every $p\in A$ so by (2) we have $ D \cap B=\phi.$ So $B$ is connected.
