Open ball in $M=\mathbb{R}\setminus\{-1,1\}$

Question

Consider $M=\mathbb{R}\setminus\{-1,1\}$ with induced metric by usual in $\mathbb{R}$. Show that closed ball $B[0,1]$ is an open subset in $M$.

My question is, how can I show that close ball with center 0 and radius 1 consides with open ball in that space?

Thanks!

Answer 1

You can use the fact that$$[-1,1]\cap M=(-1,1)\cap M.$$

Answer 2

Note that $\{x\in M\mid d(0,x)=1\}=\varnothing$. That's because the "candidates" for points that are $1$ unit away from $0$ would be $\pm 1$, but they are not in $M$.

It follows that $\{x\in M\mid d(0,x)\leq 1\} = \{x\in M\mid d(0,x)< 1\}$.