Unit Interval is Simply Connected

Question

Given the definition of simply connected space to be a topological space $X$ whose every connected covering is homeomorphic to $X$, i want to show that $[0,1]$ is simply connected.

Answer 1

Let $I$ be the unit interval. Let $p:X\to I$ be a path-connected covering space. It suffices to show that the number of sheets (which is constant since $X$ is connected) equals $1$ because an injective covering map is a homeomorphism. Choose a point $x_0$ in $p^{-1}(0)$. The path $0\rightsquigarrow 1$ which is Id$_I$ itself has a unique lift starting at $x_0$, which ends at a point $x_1$ in $p^{-1}(1)$.
Now choose another point $x_2$ in $p^{-1}(1)$. There is a path $h$ from $x_0$ to $x_2$ which maps to a path $p(h)$ from $0$ to $1$. This $p(h)$ is path-homotopic to Id$_I$ and this homotopy has a unique lift $F:I\times I \to X$. But $F$ is again a path-homotopy, which implies that $x_1$ is the same point as $x_2$. Since $x_2$ was choses arbitrarily, the fiber $p^{-1}(1)$ has only one element.