Continuous path

A continuous path in $X$ is a continuous map $\gamma :[0,1]\to X$.

The fixed-endpoint path space $\Omega (p,q)$ consists of paths in a manifold with specified initial and terminal points. Smooth, piecewise smooth and Sobolev $H^1$ versions have the same homotopy type in the usual manifold setting. Joining a path to a fixed path back from $q$ to $p$ gives a homotopy equivalence with the based loop space.

A continuous path with finite arc length on every compact parameter interval. Families of such paths are used in the measurable-density definition of extremal length.

A topological embedding $\gamma :S^1\to X$ of a circle into a topological space. On an oriented surface, an essential such curve is noncontractible; on a punctured surface, nonperipheral means that it does not surround only one puncture.

A collection of locally rectifiable paths, specified by their endpoints, winding or homotopy class when useful. Its extremal length compares its shortest metric length with total metric area.