Proof That the Components of a Totally Disconnected Space are the Singleton Sets
Theorem
The components of a totally disconnected set
are the singleton sets.
Proof
Supposeis a totally disconnected space and let
be a component of
Let
Since
is totally disconnected, there are open sets
such that
The sets
are nonempty, and
is a disconnection of
But
is a component of
so has to be connected. Hence consists of only the point
or
andis totally disconnected.
