Theorem

Connectedness is a topological property. Equivalently, ifandare topological spaces andis continuous, then ifis (dis)connected thenis (dis)connected and vice versa.

Proof

Supposeandare topological spaces andis continuous. Suppose is connected and Y is not connected. Then there are open setssuch that

Sinceis continuous,are open sets in

Sinceis a bijection,and

Henceis disconnected - a contradiction. Henceis connected.

The other implications are very similar.