Closed ball minus a point is connected Could someone help me to prove that if $B\subset\mathbb{R}^n$ ($n>1$) is a closed ball, then $B\setminus \{p\}$ is a connected set for all $p\in B$? 

HINT: Let $x$ and $y$ be any two points of $B\setminus\{p\}$. If $p$ is not on the line segment $\overline{xy}$, then that line segment is a connected set in $B\setminus\{p\}$ containing both $x$ and $y$. If $p$ does lie on the line segment $\overline{xy}$, let $z$ be any point of $B$ that is not on the line through $x$ and $y$, and show that $\overline{xz}\cup\overline{zy}$ is a connected subset of $B\setminus\{p\}$ that contains both $x$ and $y$. (This actually shows that $B\setminus\{p\}$ is not just connected, but path connected.)
