Proof That a Finite Product of Connected Spaces is Connected
Theorem
A finite product
of connected space
is connected.
Proof
Let
and
be connected and let
The set
is homeomorphic to
and sinceis connected, so is
Similarly,
is homeomorphic to
and is connected.
Hence
is connected because
and each ofandare connected.
Also
so
and
are in the same component of
Sinceandare arbitrary, there is only one component and
is connected.
The proof is completed by an obvious induction argument.