X is a topological space. Show that two statements are equivalent:
(1) $\pi_1(X,x_0)$ is trivial for all $x_0 \in X$
(2) All continous functions $f:S^1\to X$ can be continuously extended to $D^2$
Question: is my proof correct?
Proof: (1)->(2) By asumption there is a homotopy $h:S^1\times [0,1]\to X$ between any $f:S^1\to X$, $f(1)=x_0$ and constant map $c_{x_0}$, so that $h(s,0)=f(s)$, $h(s,1)=c_{x_0}(s)$. I claim that $F:D^2 \to X,$ $ F(z)=h(e^{iArg(z)},1-|z|)$ is a continuous extension of f. Indeed $F(s)|_{S^1}=h(s,0)=f(s)$.
(2)->(1) Let's fix one $x_0\in X$ and pick any path $f:S^1\to X$ so that $f(1)=x_0$ and let F be a continuous extension of f. I claim $h:S^1\times[0,1]\to X$, $h(s,t)=F((1-t)s+t)$ is a homotopy between f and $c_{x_0}$ relative $x_0$. Indeed $h(s,0)=F(s)=f(s)$, $h(s,1)=F(1)=f(1)=x_0$ and $h(1,t)=F(1)=f(1)=x_0$. Therefore h is a homotopy iff F is continuous. Now because I picked $x_0$ without any restrictions, it follows $\pi_1(X,x_0)$ is trivial for all $x_0 \in X$.