Path

Paths play an important role in the fields of Topology and Mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space X is often denoted π₀(X).

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x₀, then a path in X is one whose initial point is x₀. Likewise, a loop in X is one that is based at x₀.