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.
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.
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 08:59 PM.
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