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

[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

 

[mighty2000]

for any help!

To prove this statement, we will first define what it means for a topological space to be disconnected. A topological space (X, T) is disconnected if there exist two non-empty open sets U and V such that U and V are disjoint and their union is X.

Now, let (X, T1) and (Y, T2) be homeomorphism topological spaces. This means that there exists a bijective function f: X → Y such that f and f^(-1) are both continuous.

Suppose (X, T1) is disconnected. By definition, there exist two non-empty open sets U and V in X such that U and V are disjoint and their union is X. We can then apply the homeomorphism function f to both sets, giving us f(U) and f(V).

Since f is continuous, f(U) and f(V) are both open sets in Y. Additionally, since f is a bijection, f(U) and f(V) are also disjoint and their union is Y. Therefore, (Y, T2) is also disconnected.

Hence, if (X, T1) is disconnected, then (Y, T2) is also disconnected. This completes the proof.