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Thread: homotopic implies equinumerosity

  1. #1
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    homotopic implies equinumerosity

    Hello. I'd like to proove:
    If X and Y are discrete and finite spaces, they are homotopic if and only if the are equinumerosity.

    Thanks for every suggestion.

  2. #2
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    Do you mean homotopy equivalent? Or else, what do you mean for two spaces to be homotopic?

    If two finite spaces with the discrete topology have the same cardinality, there is a bijection between them and their open sets, and so is a homeomorphism. Thus they are homeomorphic, which of course implies homotopy equivalence.
    Last edited by mmm4444bot; 04-25-2012 at 07:59 PM.

  3. #3
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    Quote Originally Posted by daon2 View Post
    Do you mean homotopy equivalent? Or else, what do you mean for two spaces to be homotopic?
    Sure, homotopy equivalent.
    I was looking for the other implication. If they're homotopy equivalent then they are equinumerosity

    Thanks

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