Infinite metric space has a infinite open set whose complement is also infinite

Question

let $d$ be a metric on an infinite set $M$. prove that there is an open set $U$ in $M$ such that both $U$ and its complement are infinite.

Hint is given that either $(M,d)$ is discrete or it is not.

case of discrete is done. but what about the case when $(M,d)$ is not discrete.it mean it has at least one subset which is not open. how from this can i find a infinite open set ,whose complement is also infinite.

any hint.thank in advanced

Answer 1

If $M$ is not discrete, there exists a sequence $(x_n)$ of distinct points that converges to some limit $L$. Let $A=\{x_{2n}:n\ge1\}\cup\{L\}$ and $U=M\setminus A$.