$X,Y$ topological spaces. Let maps $f:X\to Y$ , $g,g':Y\to X$ s.t. $fg$ is homotopic to $Id_Y$ and $g'f$ is homotopic to $Id_X$. Prove that $f$ defines homotopy equivalence.
I would be happy for any help, thanks.
2026-04-28 15:04:54.1777388694
Prove that a function defines homotopy equivalence
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Consider $f$ and $h=g'fg$, $fh=f(g'f)g$ is homotopic to $fg$ since $g'f$ is homotopic to the identity, we deduce that $fh$ is homotopic to the identity.
On the other hand, $hf=g'(fg)f$ is homotopic to $g'f$.