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Math Help - Hoeomorphic topological spaces

  1. #1
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    Hoeomorphic topological spaces

    Consider [0,1], [0,1), and (0,1) as subspaces of R with the standard topology. Prove that none of these spaces are homeomorphic.

    Hint: try removing points and see what happens with cutsets.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Consider [0,1], [0,1), and (0,1) as subspaces of R with the standard topology. Prove that none of these spaces are homeomorphic.

    Hint: try removing points and see what happens with cutsets.
    Based on your hint, suppose we have a homeomorphism between [0,1] and [0,1) such that

    h:[0,1] \rightarrow [0,1).

    If h is homeomorphism, then the restriction of h removing 1 from the domain of h and h(1) from the codomain should be homeomorphism such that

    \bar{h}:[0,1]\setminus\{1\} \rightarrow [0,1)\setminus\{h(1)\}.

    We see that \bar{h} is not a homeomorphism because the domain of \bar{h} is connected but the codomain of \bar{h} is not connected. Contradiction.

    Thus, h is not a homeomorphism either.

    The remaining cases are similar to the above one.
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