Is there a metric on the reals $\Bbb R$ so that a subset of $\Bbb R$ is open iff its complement is finite?

Question

I wanted to know if there is a distance function $d$ on $\Bbb R$ so that a nonempty subset $U$ of $\Bbb R$ is open with respect to $d$ if and only if its complement $\Bbb R$ \ U is finite ?

Answer 1

Perhaps overkill, but here is another proof of the impossibility of such a metric:

You can prove that in any metric space $X$ and for all $x \in X$, $$\bigcap_{n=1}^\infty B_\frac{1}{n}(x)= \{x\}$$ (where $B_\epsilon(x)$ is the open ball of radius $\epsilon$ centered at $x$).

Suppose that you had such a metric on $\Bbb R$ and fix an $x \in \Bbb R$. By assumption, the complement of every open ball centered at $x$ is finite. Thus

$$\Bbb R = \{x\} \cup \bigcup_{n=1}^\infty B^{\,\prime}_\frac{1}{n}(x)$$ is a countable union of finite sets, hence countable. But this contradicts the uncountability of $\Bbb R$.

More generally, the cofinite topology on an uncountable space isn't first-countable hence it isn't metrizable since all metric spaces are first-countable.

Answer 2

No, this is not possible. The resulting topology is not Hausdorff, and any metric topology must be.

Answer 3

Let $d$ be such a metric and $a,b$ any distinct points. Then the open balls around $a$ and $b$ of radius $\frac 12 d(a,b)$ are disjoint proper non-empty open sets, hence at most one of them can be co-finite.