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.