Construct such $d$ that $(\mathbb{R} \setminus \mathbb{Z}, d)$ is complete metric space Good evening!
I had topology exam yesterday and I had a question that gave me problems.
Lets consider $\mathrm{G} = \mathbb{R} \setminus \mathbb{Z} $ , i.e. real line without the integers, where $\rho$ is natural metric on $\mathrm{G}$. Find metric $d$ equivalent to
$\rho$, where $(\mathrm{G},d)$ is complete metric space.

It is obvious that $(\mathrm{G},\rho)$ is not complete, but I don't know how to construct $d$. Can I do something like get rid of Cauchy's property only in integral points of real line and maintain in the other ones?

This is same as construct a metric on (0,1) that is compatible.
To do so, you just need to notice (0,1) is homeomorphic to the real line R. Now pull back the standard metric on R to (0,1), which gives you an "equivalent" but now complete metric. 
Such a map is not so hard to get and I bet you have already seen one in the lectures.
