Based path space and the path-loop fibration ($\Omega X⟶P_{x_0}X\stackrel{{\mathrm{ev}}_1}{\to }X$)

The based path space $P_{x_0}X$ consists of continuous paths beginning at $x_0$, with variable endpoint and the compact-open topology. Endpoint evaluation is a Serre fibration with fiber the based loop space. Its total space is contractible by $\gamma (t)\mapsto \gamma (st)$ as $s$ decreases to zero. This differs from a space of paths with both endpoints fixed, which need not be contractible. The long exact sequence of homotopy groups of a fibration gives ${\pi }_i(\Omega X)\cong {\pi }_{i+1}(X)$ for connected $X$ in the usual based ranges.