I read somewhere that for, a path connected CW complex $X$, there is a homotopy equivalence of pairs between $(P_1X,\Omega X)$ and $(C\Omega X,\Omega X)$ where $P_1X$ denotes the set oh paths $\gamma$ such that $\gamma(1)=*$. I need to learn more this fact, especially its proof. Any suggestions, comments, references, ... are welcomed.
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One has $$C\Omega X=I\times \Omega X \ \ / \ \{0\} \times \Omega X.$$ Map a point $(t,\gamma)$ in the cone to the restriction of $\gamma$ to $[0,t]$ (reparametrized).
This is a map of pairs $(C\Omega X,\Omega X)\to (P_1X,\Omega X)$, which is the identity on the subspace, and a homotopy equivalence $C\Omega X\to P_1X$, as both spaces are contractible.
Since everything has the homotopy type of a CW complex, it follows from the long exact sequence of homotopy groups that one has a homotopy equivalence of pairs.
