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

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

My attempt:-

1.$(\mathbb N,d)$, where $d$ is a discrete metric space is complete

2. $(\mathbb N,d)$, where $d(x,y)=|x-y|$ is not a complete

3. I am not able to judge

4. $(\mathbb N,d)$, where $d$ is a discrete metric space is not compact.

Answer 1

All four options are correct. Consider the spaces $X_1=\big\{\frac{1}{n}:n\in \Bbb N\big\}$ and $X_2=\{0\}\cup\big\{\frac{1}{n}:n\in \Bbb N\big\}$ with usual distance metric $|\cdot|$ on $\Bbb R$.

Now consider a bijection between $X$ with these spaces, say $f:X\to Y$ be a bijection, where $Y$ is either $X_1$ or $X_2$. then, $d(a,b)=|f(a)-f(b)|$ for all $a,b\in X$ is a metric on $X$.

$X_1$ gives you options 2. and 4. and $X_2$ gives you options 1. and 3.