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Math Help - Homeomorphisms and cutsets

  1. #1
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    Homeomorphisms and cutsets

    Let 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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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Let f:X \rightarrow Y be a homeomorphism.

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

    \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 \bar {f} is not connected (since S is a cuset of X), but the codomain of \bar {f} is connected, which contradicts the fact that \bar{f} is a homeomorphism.

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