# A question about homotopy equivalence.

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?

• 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. – drhab Feb 5 '15 at 13:52