If all paths with the same endpoints are homotopic, then the space is simply connected.

Question

Let $X$ be a path connected space such that any two paths in $X$ having the same end points are path homotopic. Then prove that $X$ is simply connected.

I am totally stuck on this problem. Can someone help me please? Thanks for your time.

Answer 1

Take any point $x \in X$, and any loop $\sigma$ on $X$ based at $x$. By assumption, $\sigma$ is path-homotopic to the constant path at $x$ (they obviously have the same endpoints). What does this mean in terms of the class $[\sigma]$ of $\sigma$ in the fundamental group $\pi_1(X,x)$?

Answer 2

If every two paths having the same end points are path homotopic, then every loop $w: S^1 \longrightarrow X$ can be deformed into a constant path, a point. Which is the definition of simply connected, isn't it?