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

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- Apr 8th 2009, 03:35 PMAndreametHomeomorphisms and cutsets
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 - Apr 8th 2009, 09:43 PMaliceinwonderland
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.