Is there a metric space of an infinite set such that every closed set is finite except the whole space.

Question

Let $X$ be an infinite set.

Then, is it possible to construct a metric space $(X,d)$ such that every closed set except the whole space $X$ is finite?

If possible, what would be the example of such $X$ and $d$? If not possible, why? Can you give me a proof?

Answer 1

The whole space $X$ is closed, so the answer is no.

Now let us change the problem to ask whether it is possible for every closed set apart from $X$ to be finite.

Let $p$ and $q$ be distinct points. Let $\epsilon=d(p,q)/4$. Let $B_p$ be the open ball with centre $p$ and radius $\epsilon$, and define $B_q$ similarly.

If every closed set apart from $X$ is finite, then the complement of $B_p$ and the complement of $B_q$ are both finite. But the union of these complements is $X$, contradicting the fact that $X$ is infinite.

Answer 2

If $N_r(x)=\{y\in X:d(x,y)<r\}$ is the neighborhood of $x$ of radius $r$, then given $x\neq y$, let $r=\frac{d(x,y)}{2}$. Then $N_r(x)\cap N_r(y)=\emptyset$.

But $N_r(x)$ and $N_r(y)$ are non-empty and open, so they must be the complements of closed sets. If only finite sets are closed sets, can $N_r(x)^c\cup N_r(y)^c=X$?