Path (topology)

In mathematics, a path in a topological space ${\displaystyle X}$ is a continuous function from a closed interval into ${\displaystyle X.}$

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 ${\displaystyle X}$ is often denoted ${\displaystyle \pi _{0}(X).}$

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If ${\displaystyle X}$ is a topological space with basepoint ${\displaystyle x_{0},}$ then a path in ${\displaystyle X}$ is one whose initial point is ${\displaystyle x_{0}}$. Likewise, a loop in ${\displaystyle X}$ is one that is based at ${\displaystyle x_{0}}$.