Consider a two-point set $M = \{a,b\}$ whose topology consists of the two sets, $M$ and the empty set.

Question

Consider a two-point set $M = \{a,b\}$ whose topology consists of the two sets, $M$ and the empty set. Why does this topology not arise from a metric on M?

May someone please clarify this question (I'm new to analysis)! Does it have anything to do with the fact that $\{a\}, \{b\}$ are each closed sets and their complements, $\{b\}$ and $\{a\},$ respectively, are also closed sets?

Answer 1

$\{a\}$ and $\{b\}$ are not closed because their complements are not open. The only closed sets are $\emptyset$ and $M$. If there is a metric on the space, the only thing of interest is $d(a,b)=k$. Then the open ball around $a$ of radius $\frac k2$ only includes $a$, so $\{a\}$ must be open but it is not. Therefore the topology does not arise from a metric.

Answer 2

Because a metric space is separable. There must open subsets $U,V, a\in U, b\in V, U\cap V$ is empty. Suppose $d$ is a metric, $d(a,b)=r>0$, $B(a,r/2)\cap B(b,r/2)$ is empty.