# discrete metric space n singleton sets

• Apr 21st 2010, 07:31 AM
cp05
discrete metric space n singleton sets
"show that in a discrete metric space, singleton sets are the only nonempty connected sets"

• Apr 21st 2010, 03:36 PM
Drexel28
Quote:

Originally Posted by cp05
"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.