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Hatcher on pg. 3 of his book "Algebraic Topology" says that two spaces are homotopy equivalent if they are both deformation retractions from the same space.

Two spaces are homotopy equivalent only if they are homeomorphic.

I don't understand why Hatcher's statement is true. For example, the open set $[0,1]\times [0,1]$ can have a deformation retraction to both $[0,1]$ and $\{0\}$. Are we to say that $[0,1]$ and $\{0\}$ are homotopy equivalent?

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  • $\begingroup$ Yes, they are homotopy equivalent. Even stronger: $\{0\}$ is a deformation retract of $[0,1]$. Btw, homeomorphic spaces are homotopy equivalent, but the converse of this is not true in general. $\endgroup$ – drhab Feb 5 '15 at 13:52
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"Two spaces are homotopy equivalent only if they are homeomorphic" This is not true. Homotopy equivalence is much weaker than being homeomorphic.

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  • $\begingroup$ That may be, but my main doubt is in the last couple of lines. $\endgroup$ – algebraically_speaking Feb 5 '15 at 10:46
  • $\begingroup$ Ah, sorry for not answering your question. Yes, they are homotopy equivalent. $\endgroup$ – Callus Feb 5 '15 at 14:12

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