Complete metric space problem

Question

Let $X$ be a countable set. Then which of the following are true?

1. There exists a metric $d$ on $X$ such that $(X,d)$ is complete.
2. There exists a metric $d$ on $X$ such that $(X,d)$ is not complete.
3. There exists a metric $d$ on $X$ such that $(X,d)$ is compact.
4. There exists a metric $d$ on $X$ such that $(X,d)$ is not compact.

I have no idea how to connect countability with completeness and compactness with respect to a metric. Please help me.

Answer 1

1) True. For instance, put $d(x,y)=1$ for each distinct $x,y\in X$ and $d(x,x)=0$ for each $x\in X$. Then each Cauchy sequence of $(X,d)$ is eventually constant, so it converges.

2) True. For instance, fix any bijection $f$ between $X$ and a subspace $\{1/n: n\in\Bbb N\}$ of the real line endowed with the standard metric $\rho$. For each $x,y\in X$ put $d(x,y)=\rho(f(x), f(x))$. It is easy to see that $d$ is a metric and a Cauchy sequence $\{f^{-1}(n):n\in\Bbb N\}$ does not converge.

3) True. For instance, fix any bijection $f$ between $X$ and a subspace $\{0\}\cup\{1/n: n\in\Bbb N\}$ of the real line endowed with the standard metric $\rho$. For each $x,y\in X$ put $d(x,y)=\rho(f(x), f(x))$. Then $f$ is an isometry between the space $(X,d)$ and a convergent sequence, which is compact.

4) True. Both examples for 1) and 2) fit.