In mathematics, a path in a topological space is a continuous function from the closed unit interval into 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 is often denoted One can also define paths and loops in pointed spaces, which are important in homotopy theory. If is a topological space with basepoint then a path in is one whose initial point is . Likewise, a loop in is one that is based at . Definition A curve in a topological space is a continuous function from a non-empty and non-degenerate interval A in is a curve whose domain is a compact non-degenerate interval (meaning are real numbers), where is called the of the path and is called its . A is a path whose initial point is and whose terminal point is Every non-degenerate compact interval is homeomorphic to which is why a is sometimes, especially in homotopy theory, defined to be a continuous function from the closed unit interval into An or 0 in is a path in that is also a topological embedding. Importantly, a path is not just a subset of that "looks like" a curve, it also includes a parameterization. For example, the maps and represent two different paths from 0 to 1 on the real line. A loop in a space based at is a path from to A loop may be equally well regarded as a map with or as a continuous map from the unit circle to This is because is the quotient space of when is identified with The set of all loops in forms a space called the loop space of Homotopy of paths Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed. Specifically, a homotopy of paths, or path-homotopy, in is a family of paths indexed by such that and are fixed. the map given by is continuous. The paths and connected by a homotopy are said to be homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all