Theorem

A finite productof connected spaceis connected.

Proof

Letandbe connected and let

The setis homeomorphic toand sinceis connected, so isSimilarly,is homeomorphic toand is connected.

Henceis connected becauseand each ofandare connected.

Alsosoandare in the same component of Sinceandare arbitrary, there is only one component andis connected.

The proof is completed by an obvious induction argument.