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