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I´ve a question about how you can talk about contractible. I suppose I´m wrong in something with the definition.

We say a space $X$ is contractible if it has the homotopy-type of a point, that is, there is a map $f : X \to \{p\}$ and a map $g : \{p\} \to X$ such that $f \circ g \sim Id_p$ and $g \circ f \sim Id_X$.

My doubt is related with the application $f$, that necessarily is constant, and the set $Im(g) = g(p)$ is only one point. So, how you can get an homotopy with $Id_X$?

Thanks!

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  • $\begingroup$ Why wouldn't you be able to get a homotopy with $Id_X$? $\endgroup$ – Eric Wofsey Mar 12 at 15:17
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It is better to have an example in mind. Let $X=\mathbb{R}^n$ and $p=0$, $f$ is the constant function and $g(0)=0$, $g\circ f(x)=0$ is homotopic to the identity by setting $H_t(x)=tx$, $H_1(x)=x, H_0(x)=0$.

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