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I am so confused about something simple

Ok, I know that Rn is contractible, just by a straightline homotopy sending all points to the origin. So this means Rn has the homotopy type of a point. So Rm, for a different integer m, has the homotopy type of a point. Since homotopy equivalence is an equivalence relation, this means that Rm is homotopic to Rn? But this is not possible right?

Can someone tell me where my logic is flawed?

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# Rn homotopic to Rm?

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