Path space ($\Omega (p,q)$)

For points $p,q$ in a topological space $X$, the fixed-endpoint path space iswith 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 ${\Omega }_{p}X$. In the usual smooth manifold setting, smooth, piecewise smooth and Sobolev $H^1$ path models give the same homotopy type.