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.