1 Answers
In mathematics, a path in a topological space X {\displaystyle X} is a continuous function from the closed unit interval {\displaystyle } into X . {\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 X {\displaystyle X} is often denoted π 0 . {\displaystyle \pi _{0}.}
One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X {\displaystyle X} is a topological space with basepoint x 0 , {\displaystyle x_{0},} then a path in X {\displaystyle X} is one whose initial point is x 0 {\displaystyle x_{0}}. Likewise, a loop in X {\displaystyle X} is one that is based at x 0 {\displaystyle x_{0}}.