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I am trying to prove

$$\Omega^n(X,x_0)\approx M(S^n,*; X,x_0)$$

where $\Omega^n(X,x_0)$ is the $n$-loop group and $M(S^n,*; X,x_0)$ is the set of pointed continuous maps from $(S^n,*)$ to $(X,x_0)$

My experience with the compact-open topology is very weak, so I have some misgiving about my attempted proof. I would be very grateful if anyone could look through my arguments and point out any errors (which I am sure are many).

My argument is as follows:

First note that $I^n/\partial I^n \approx S^n$ for all $n\geq 1$. Let $q$ be the quotient map between $I^n$ and $I^n /\partial I^n$ and let $h$ be the homeomorphism between $I^n/\partial I^n$ and $S^n$. Since $f$ acts constantly on the fibers of $q$, there exists a unique pointed map $\tilde{f}:(I^n/\partial I^n,*) \longrightarrow (X,x_0)$ such that $f=\tilde{f}\circ q$. Now define the map $\varphi: M(I^n/\partial I^n,*; X,x_0) \longrightarrow M(I^n,\partial I^n; X,x_0)$ by $\varphi(\tilde{f})=\tilde{f}\circ q=f$. Let $U^K$ be a subbasic subset of $M(I^n,\partial I^n; X,x_0)$. By definition, $\varphi^{-1}(U^K)$ consists of the $\tilde{g} \in M(I^n/\partial I^n, * ; X,x_0)$ such that $\tilde{g}\circ q(U) \subseteq K$. Note that $q(U)$ is open in $(I^n/\partial I^n,*)$, so we see that

$$\varphi^{-1}(U^K)=(q(U))^K,$$

which is a subbasic subset of $M(I^n/\partial I^n,*; X,x_0)$. Clearly $\varphi$ is surjective because for each $f\in M(I^n,\partial I^n; X,x_0)$, we have some $\tilde{f}$ which maps to it. The uniqueness of $\tilde{f}$ implies the injectivity of $\varphi$. An inverse for $\varphi$ is given by the map $\psi: M(I^n,\partial I^n; X,x_0)\longrightarrow M(I^n/\partial I^n,*; X,x_0)$ defined by $\psi(f)=\tilde{f}$. Let $U^K$ be a subbasic subset of $I^n/\partial I^n$. Note that we have $U=q(V)$ for some open subset $V$ of $I^n$. By definition

$$\psi^{-1}=\{f \in M(I^n,\partial I^n ; X,x_0) \; | \; \psi(f)=\tilde{f}\in U^K\}$$.

Since $\tilde{f}(U)=\tilde{f}(q(V))\subseteq K$, we see that $f(V) \subseteq K$ and so

$$\psi^{-1}(U^K)=V^,$$

showing that $\psi^{-1}(U^K)$ is a subbasic subset of $M(I^n,\partial I^n; X,x_0)$, hence is open. Thus $\psi$ is continuous. Composing with $h$ show that $\Omega^n(X,x_0)\approx M(S^n,*; X,x_0)$.

What most concerns me is my conclusion that the inverse image of the subbasic subsets are subbasic, where here I mean subbasic to say they are generators for the subbasis of the compact open topology. Have I made a mistake, or are my conclusion justified.

Note: This is not a homework assignment.

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