$X$ path connected space simple connected if only if map $S^1 \rightarrow X$ extends to $D^2 \rightarrow X$

Question

I do not know how to approach this proof. Any hints about how to prove this?

$X$ path connected space simple connected if only if map $S^1 \rightarrow X$ extends to $D^2 \rightarrow X$

Answer 1

hint: Let $C(S^1):=(S^1 \times I)/\sim$ where $(x,1) \sim (y,1)$. Show that $C(S^1)$ is homeomorphic to the disk.

Let $F:S^1 \times I \to X$ be a map such that $F \mid_{S^1 \times \{1\}}$ is constant. Then it factors through the cone by the universal property of quotients.