In mathematics, a path in a topological space X {\displaystyle X} is a continuous function from a closed interval 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 ( X ) . {\displaystyle \pi _{0}(X).} 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}} .
Definition A curve in a topological space X {\displaystyle X} is a continuous function f : J → X {\displaystyle f:J\to X} from a non-empty and non-degenerate interval J ⊆ R . {\displaystyle J\subseteq \mathbb {R} .} A path in X {\displaystyle X} is a curve f : [ a , b ] → X {\displaystyle f:[a,b]\to X} whose domain [ a , b ] {\displaystyle [a,b]} is a compact non-degenerate interval (meaning a < b {\displaystyle a<b} are real numbers), where f ( a ) {\displaystyle f(a)} is called the initial point of the path and f ( b ) {\displaystyle f(b)} is called its terminal point. A path from x {\displaystyle x} to y {\displaystyle y} is a path whose initial point is x {\displaystyle x} and whose terminal point is y . {\displaystyle y.} Every non-degenerate compact interval [ a , b ] {\displaystyle [a,b]} is homeomorphic to [ 0 , 1 ] , {\displaystyle [0,1],} which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} from the closed unit interval I := [ 0 , 1 ] {\displaystyle I:=[0,1]} into X . {\displaystyle X.}
An arc or C0-arc in X {\displaystyle X} is a path in X {\displaystyle X} that is also a topological embedding. Importantly, a path is not just a subset of X {\displaystyle X} that "looks like" a curve, it also includes a parameterization. For example, the maps f ( x ) = x {\displaystyle f(x)=x} and g ( x ) = x 2 {\displaystyle g(x)=x^{2}} represent two different paths from 0 to 1 on the real line. A loop in a space X {\displaystyle X} based at x ∈ X {\displaystyle x\in X} is a path from x {\displaystyle x} to x . {\displaystyle x.} A loop may be equally well regarded as a map f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} with f ( 0 ) = f ( 1 ) {\displaystyle f(0)=f(1)} or as a continuous map from the unit circle S 1 {\displaystyle S^{1}} to X {\displaystyle X}
f : S 1 → X . {\displaystyle f:S^{1}\to X.}
This is because S 1 {\displaystyle S^{1}} is the quotient space of I = [ 0 , 1 ] {\displaystyle I=[0,1]} when 0 {\displaystyle 0} is identified with 1. {\displaystyle 1.} The set of all loops in X {\displaystyle X} forms a space called the loop space of X . {\displaystyle X.}
Homotopy of paths
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