Which of the following is true?
$ (0,1)$ is complete w.r.t a metric which induces the usual topology on $\Bbb R$.
$ (0,1)$ is compact w.r.t a metric which induces the usual topology on $\Bbb R$.
- True. Since $(0,1)$ and $\Bbb R$ have the same cardinality hence there exists a bijective function $f:(0,1)\to \Bbb R$. Let's define a metric $d_1$ on $(0,1)$ by $d_1(x,y)=\left|f(x)-f(y)\right|$ which defines a isometry.
Now since $\Bbb R$ is complete so will be $(0,1)$.
- False since in order to be compact $(0,1)$ must be closed in the usual topology which will never be.
Please check my answers and give your feedback.