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

i have no idea how to go about this one =/

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- Apr 21st 2010, 07:31 AMcp05discrete metric space n singleton sets
"show that in a discrete metric space, singleton sets are the only nonempty connected sets"

i have no idea how to go about this one =/ - Apr 21st 2010, 03:36 PMDrexel28
A discrete metric space gives rise to the discrete topology. So, let $\displaystyle M$ be the metric space in common. Clearly each singleton is connected. But, given any $\displaystyle m,m'\in M$ clearly $\displaystyle \{m\}\cup\{m'\}=\{m,m'\}$ is a separation of $\displaystyle \{m,m'\}$ since each $\displaystyle \{m\},\{m'\}\subseteq M$ are open.

- Apr 21st 2010, 05:59 PMcp05
oh wow that was so obvious lol thanks so much!