Proof That the Components of a Totally Disconnected Space are the Singleton Sets


The components of a totally disconnected setare the singleton sets.


Supposeis a totally disconnected space and letbe a component of


Sinceis totally disconnected, there are open setssuch that

The sets are nonempty, andis a disconnection of

Butis a component ofso has to be connected. Hence consists of only the pointorandis totally disconnected.