Hi everyone I'm struggle with the following.
Define a complete metric on $\mathbb{R}\setminus \{0,1\}$ with usual relative topology. I'd like to follow the big hint of Daniel Fischer but I have troubles because this is new:
Take a continuous function $f\colon \mathbb{R}\setminus \{0,1\}\to\mathbb{R}$, and consider $\Gamma(f) = \left\{ (t,f(t)) : t\in \mathbb{R}\setminus\{0,1\}\right\}$. The graph $\Gamma(f)$ inherits a metric from $\mathbb{R}^2$, and transporting that back, you get a metric on $\mathbb{R}\setminus\{0,1\}$ inducing the standard topology. Now you need to choose $f$ so that $\Gamma(f)$ is a closed subset of $\mathbb{R}^2$.
Suppose I have that continuous map, then $g:\mathbb{R}\setminus \{0,1\}\to \Gamma(f)$ defined by $t\to (t,f(t))$ is continuous, the $\text{pr}_1$ is its inverse which is continuous. But from here I have several questions:
1) How to define the metric on $\mathbb{R}\setminus \{0,1\}$, I don't know my guess is something like $d(x,y)=d_E(g(x),g(y))$ where $d_E$ is the euclidean metric inherited to $\Gamma(f)$
2) How this shows that $\mathbb{R}\setminus \{0,1\}$ is complete I know that $\Gamma(f)$ is because I'm assuming that is closed and how I know that it has the same topology as $\Gamma(f)$ because in this form the function which I defined as the metric is not the correct one, I think.
3) Some reference to study this particular kind of argument because are totally new for me and very interesting (I suppose any book in general topology but I prefer to know something more specific).
EDIT
Thanks in advance and I'm apologize if my questions are very stupid but this way of thinking is completely new for me. My classes of topology was one introductory courses in Real Analysis.
4) From here how do we know that $d$ as above using for example $x\to 1/x(x-1)$ give us the same topology as $d\restriction \mathbb{R}\setminus \{0,1\}\times \mathbb{R}\setminus \{0,1\}$
(1) That is the right guess, now you just have to choose the right $g$. You want some $ g $ that makes $ 0 $ and $ 1 $ infinitely far away, so that you can't have a cauchy sequence approaching $0 $ or $ 1 $. This should give you a hint on how the graph looks.
(2) It shows that $ \mathbb{R} \backslash \{0,1\} $ is complete because it is isometric to $ \Gamma(f) $ with euclidean metric, and $ \Gamma(f)$ is complete because closed subspaces of complete spaces are complete.
(3) I liked Munkres, Topology. But I don't remember doing any exercise like this.