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

Theorem

The components of a totally disconnected setare the singleton sets.

Proof

Supposeis a totally disconnected space and letbe a component of

Let

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.