= Path space
{title2=$\Omega(p,q)$}
For <points> $p,q$ in a <topological space> $X$, the fixed-endpoint <path space> is
$$
\Omega(p,q)=\{\gamma:[0,1]\to X:\gamma\text{ continuous},\ \gamma(0)=p,\ \gamma(1)=q\},
$$
with the subspace <compact-open topology> from the continuous-map space. For a <metric space>, this agrees with uniform convergence of <paths> on $[0,1]$. If $p,q$ lie in different <path components>, this <path space> is empty.
If $p,q$ lie in the same <path component>, choose one <continuous path> $\alpha$ from $p$ to $q$. Appending the reversed $\alpha$ sends a <path> from $p$ to $q$ to a <based loop> at $p$. Conversely, append $\alpha$ to a <based loop>. The backtracking pieces contract through <paths> with fixed endpoints, so these maps are <homotopy inverses>. Hence $\Omega(p,q)$ has the <homotopy type> of the based <loop space> $\Omega_pX$. In the usual smooth <manifold> setting, smooth, piecewise smooth and Sobolev $H^1$ path models give the same <homotopy type>.
Back to article page
