# No point left behind

Every infinite set admits a metric with no isolated points.

1. Every infinite set is a disjoint union of countable sets. Each of these sets is in bijection with the rational numbers between $0$ and $1$. Glue these metrics together by declaring the distance between points in different sets to be $1$.
2. Another solution: any infinite set $X$ is in bijection with $X\times\Bbb{Q}$. Give $X$ the discrete metric and $\Bbb{Q}$ the usual metric. Then the supremum metric on $X\times\Bbb{Q}$ has no isolated point, and one may transport this to $X$.