Path space fibration

Remarks

The map $p$ is a Hurewicz fibration, and its fiber $p^{-1}(x_0)$ is the loop space $\Omega X$. The space $PX$ is contractible: $h_s(\gamma)(t) = \gamma((1 - s) t)$ deforms every path to the constant path at $x_0$. So the long exact sequence of homotopy groups gives $\pi_n(\Omega X) \cong \pi_{n+1}(X, x_0)$ for $n \ge 0$.

Example: for $X = \R$ and $x_0 = 0$, the fiber over $a \in \R$ is the space of paths from $0$ to $a$, and $p$ is surjective, since the path $t \mapsto t a$ ends at $a$.

Non-example: if $X$ is not path-connected, $p$ is not surjective. For $X = \set{0, 1}$ with the discrete topology and $x_0 = 0$, every path from $0$ is constant, so $PX$ is a point and $p^{-1}(1)$ is empty.

The fiber of $p$ over $y \in X$ is the homotopy fiber over $y$ of the inclusion $\set{x_0} \to X$. So $p$ replaces this inclusion by a fibration with a contractible total space.