# Thread: discrete metric space n singleton sets

1. ## discrete metric space n singleton sets

"show that in a discrete metric space, singleton sets are the only nonempty connected sets"

A discrete metric space gives rise to the discrete topology. So, let $M$ be the metric space in common. Clearly each singleton is connected. But, given any $m,m'\in M$ clearly $\{m\}\cup\{m'\}=\{m,m'\}$ is a separation of $\{m,m'\}$ since each $\{m\},\{m'\}\subseteq M$ are open.