1. 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. 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.

3. Originally Posted by daon2
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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