Show that the following are equivalent for a path connected space $X$.
(a) $X$ is simply connected. (with the definition that the fundamental group is trivial)
(b) If two paths $\alpha$ and $\beta: I\to X$ are such that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$, then $\alpha$ is path homotopic to $\beta$.
(c) Every continuous map $f: S^1 \to X$ is continuously extendible on $D^2$.
(d) Every continuous map $f: S^1 \to X$ is null-homotopic.
I've shown the directions $(a)\to (b)$ and $(c)\iff (d)$. However, I can't find the link between (a) or (b) to (c) or (d). How can I complete this proof? I would greatly appreciate any help.
Consider a path $\gamma: I\to X$. Since $\gamma(1) = \gamma(0)$, you may as well consider it as a function $\tilde\gamma: S^1\simeq I/(0=1)\to X$. Then you should be good. :-)