TITLE: Open set in a general metric space.
QUESTION [6 upvotes]: Let d define a metric on an infinite set $M$. Show that there exists an open set $U$ such that $U$ and its complement are infinite. (Infinite referring to cardinality in both instances)
I know this is trivial with the discrete metric and also trivial on the reals with the usual metric but I'm not sure how to proceed for a general metric.

REPLY [5 votes]: Let $S$ be the set of isolated points of $M$.
If $S$ is infinite, then, since isolated points are open, writing $S$ as the disjoint union of two infinite sets gives what's needed.
Suppose $S$ is finite. Then since $M$ is infinite, there are distinct elements  $m_1$ and $m_2$ of $M\setminus S$. Choose disjoint open nhoods  $N_1$ of $m_1$ and $N_2$ of $m_2$. Both $N_1$ and $N_2$ are infinite since $m_1$ and $m_2$ are not isolated. Taking $U=N_1$, then, gives what's needed.