Thread: Homeomorphisms and cutsets

Let be a homeomorphism.

Prove: If S is a cutset of X, then f(S) is a cutset of Y

Originally Posted by Andreamet

Let be a homeomorphism.

Prove: If S is a cutset of X, then f(S) is a cutset of Y

If is a homeomorphism, then

is also a homeomorphism, where S is a subset of X.

Suppose to the contrary that {f(S)} is not a cutset of Y. Then the domain of is not connected (since S is a cuset of X), but the codomain of is connected, which contradicts the fact that is a homeomorphism.

Thus, {f(S)} is a cutset of Y.