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I know there are many questions on this but I ask a proof verification and a clarification. I am trying to prove that given $X$ topological space,

$X$ is contractible if and only if it is a deformation retract.

The if part is clear to me (it is just a stronger requirement than asking the identity is nullhomotopic)

The only if part: by definition of contractible, $X \sim \{x_0\}$, which holds if and only if the identity map $id_X$ is homotopic to the constant map $\text{const }x_{0}$. Then I take the following retraction: $r:X \rightarrow \{x_0\}$ given by $r = \text{const }x_{0}$.

Denoting $i$ the inclusion of $\{x_0\}$ into $X$, now $r \circ i = id_{\{x_0\}}$ and $i \circ r = \text{const }x_{0} \sim id_X$ by the fact $X$ is contractible.

Nevertheless, here people show a space which is contractible but does not deformation retract to any point. What am I missing?

Crash Bandicoot
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    Hint: Does your construction keep $x_0$ fixed? – Michael Burr Mar 06 '23 at 17:07
  • My apologies: I should specify that my definition of deformation retract is that there exists a retraction from $X$ to $x_0$ and $i \circ r \sim id_X$. Hence I think I am not considering the definition of strong deformation retract, right? I guess with this definition the proof should work, correct? – Crash Bandicoot Mar 06 '23 at 17:14
  • You seem to be mixing the language. At the top, you want to prove that a space $X$ is contractible if and only if it is the deformation retract of some other space $Y$. But below, you mention a space which is contractible but does not itself deformation retract onto a point. That's not a counterexample, since it's talking about a different thing – FShrike Mar 06 '23 at 17:21
  • I always take $Y$ to be a point of the space $X$, i.e. $Y={x_0}$, with $x_0 \in X$. – Crash Bandicoot Mar 06 '23 at 17:24

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The linked question considers strong deformation retractions, i.e. deformation retractions which fix points of the subspace at any $t\in[0,1]$.

You are correct that $X$ is contractible if and only if it weakly deformation retracts onto some (any) point.

freakish
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