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Having the same homotopy type equivalence

Prove that "having the same homotopy type" is an equivalence relation on the set of topological spaces.

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Lemma: if f,f' are homotopic maps from X to Y and g,g' are homotopic
maps from Y to Z, then g o f is homotopic to g' o ...

Solution Summary

This solution is comprised of a detailed explanation to prove that "having the same homotopy type" is an equivalence relation
on the set of topological spaces.

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