= Based path space and the path-loop fibration
{title2=$\Omega X\longrightarrow P_{x_0}X\xrightarrow{\mathrm{ev}_1}X$}
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.
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