# Prove: If (X; T1) and (Y; T2) are homeomorphic topological

1. Apr 22, 2012

### math25

Hi,

Prove: If (X; T1) and (Y; T2) are homeomorphism topological spaces
and (X; T1) is disconnected then (Y; T2) is disconnected.

I think I need some help with this proof
Proof:
Let (X, T1) be disconnected and let f be a homeomorphism. If f(X,T1) is disconnected then there exist two non-empty disjoint sets A and B whose union is f(X,T1). As f^(-1) is continuous, f^(-1) A and f^(-1)B are open. As f^(-1) is a bijection, they are disjoint sets whose union is (X,T1). Therefore, (Y,T2) IS disconnected.

thanks