homotopic curves seen in $C(S^1,X)$ imply homotopic curves seen as closed paths $C([0,1],X)$

Question

I was solving an exercise and the following question came to me that I don't know how to attack:

It is true that two homotopic curves viewed in $C(S^1,X)$ are then also homotopic viewed as closed paths $C([0,1],X)$. Where $X$ is a topological space (Hausdorff if necessary).

Any ideas?

Answer 1

Both spaces are the same: $S^1$ arises from $[0,1]$ by identifying $0$ with $1$, so any continuous closed curve $\gamma : [0,1]\rightarrow X$ with $\gamma(0)=\gamma(1)$ becomes a continuous curve $\tilde\gamma : S^1\rightarrow X$. The converse is analogous, since "breaking up" $S^1$ at any point results again in $[0,1]$.

Thus, it doesn't really matter in which of these two spaces two curves are homotopic.