homotopic implies equinumerosity

[Lolyta]

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.

[daon2]

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.

[Lolyta]

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