Let $\displaystyle f:X \rightarrow Y$ be a homeomorphism.
Prove: If S is a cutset of X, then f(S) is a cutset of Y
If $\displaystyle f:X \rightarrow Y$ is a homeomorphism, then
$\displaystyle \bar{f}:X\setminus S \rightarrow Y\setminus \{f(S)\}$ 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 $\displaystyle \bar {f}$ is not connected (since S is a cuset of X), but the codomain of $\displaystyle \bar {f}$ is connected, which contradicts the fact that $\displaystyle \bar{f}$ is a homeomorphism.
Thus, {f(S)} is a cutset of Y.