Path (topology)

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

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 $$X$$ is often denoted $$\pi_0(X).$$

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If $$X$$ is a topological space with basepoint $$x_0,$$ then a path in $$X$$ is one whose initial point is $$x_0$$. Likewise, a loop in $$X$$ is one that is based at $$x_0$$.

Definition
A curve in a topological space $$X$$ is a continuous function $$f : J \to X$$ from a non-empty and non-degenerate interval $$J \subseteq \R.$$ A ' in $$X$$ is a curve $$f : [a, b] \to X$$ whose domain $$[a, b]$$ is a compact non-degenerate interval (meaning $$a < b$$ are real numbers), where $$f(a)$$ is called the ' of the path and $$f(b)$$ is called its . A  is a path whose initial point is $$x$$ and whose terminal point is $$y.$$ Every non-degenerate compact interval $$[a, b]$$ is homeomorphic to $$[0, 1],$$ which is why a  is sometimes, especially in homotopy theory, defined to be a continuous function $$f : [0, 1] \to X$$ from the closed unit interval $$I := [0, 1]$$ into $$X.$$

An  or $C$0 in $$X$$ is a path in $$X$$ that is also a topological embedding.

Importantly, a path is not just a subset of $$X$$ that "looks like" a curve, it also includes a parameterization. For example, the maps $$f(x) = x$$ and $$g(x) = x^2$$ represent two different paths from 0 to 1 on the real line.

A loop in a space $$X$$ based at $$x \in X$$ is a path from $$x$$ to $$x.$$ A loop may be equally well regarded as a map $$f : [0, 1] \to X$$ with $$f(0) = f(1)$$ or as a continuous map from the unit circle $$S^1$$ to $$X$$
 * $$f : S^1 \to X.$$

This is because $$S^1$$ is the quotient space of $$I = [0, 1]$$ when $$0$$ is identified with $$1.$$ The set of all loops in $$X$$ forms a space called the loop space of $$X.$$

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 $$X$$ is a family of paths $$f_t : [0, 1] \to X$$ indexed by $$I = [0, 1]$$ such that The paths $$f_0$$ and $$f_1$$ connected by a homotopy are said to be homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.
 * $$f_t(0) = x_0$$ and $$f_t(1) = x_1$$ are fixed.
 * the map $$F : [0, 1] \times [0, 1] \to X$$ given by $$F(s, t) = f_t(s)$$ is continuous.

The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path $$f$$ under this relation is called the homotopy class of $$f,$$ often denoted $$[f].$$

Path composition
One can compose paths in a topological space in the following manner. Suppose $$f$$ is a path from $$x$$ to $$y$$ and $$g$$ is a path from $$y$$ to $$z$$. The path $$fg$$ is defined as the path obtained by first traversing $$f$$ and then traversing $$g$$:
 * $$fg(s) = \begin{cases}f(2s) & 0 \leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}$$

Clearly path composition is only defined when the terminal point of $$f$$ coincides with the initial point of $$g.$$ If one considers all loops based at a point $$x_0,$$ then path composition is a binary operation.

Path composition, whenever defined, is not associative due to the difference in parametrization. However it associative up to path-homotopy. That is, $$[(fg)h] = [f(gh)].$$ Path composition defines a group structure on the set of homotopy classes of loops based at a point $$x_0$$ in $$X.$$ The resultant group is called the fundamental group of $$X$$ based at $$x_0,$$ usually denoted $$\pi_1\left(X, x_0\right).$$

In situations calling for associativity of path composition "on the nose," a path in $$X$$ may instead be defined as a continuous map from an interval $$[0, a]$$ to $$X$$ for any real $$a \geq 0.$$ (Such a path is called a Moore path.) A path $$f$$ of this kind has a length $$|f|$$ defined as $$a.$$ Path composition is then defined as before with the following modification:
 * $$fg(s) = \begin{cases}f(s) & 0 \leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f| + |g|\end{cases}$$

Whereas with the previous definition, $$f,$$ $$g$$, and $$fg$$ all have length $$1$$ (the length of the domain of the map), this definition makes $$|fg| = |f| + |g|.$$ What made associativity fail for the previous definition is that although $$(fg)h$$ and $$f(gh)$$have the same length, namely $$1,$$ the midpoint of $$(fg)h$$ occurred between $$g$$ and $$h,$$ whereas the midpoint of $$f(gh)$$ occurred between $$f$$ and $$g$$. With this modified definition $$(fg)h$$ and $$f(gh)$$ have the same length, namely $$|f| + |g| + |h|,$$ and the same midpoint, found at $$\left(|f| + |g| + |h|\right)/2$$ in both $$(fg)h$$ and $$f(gh)$$; more generally they have the same parametrization throughout.

Fundamental groupoid
There is a categorical picture of paths which is sometimes useful. Any topological space $$X$$ gives rise to a category where the objects are the points of $$X$$ and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of $$X.$$ Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point $$x_0$$ in $$X$$ is just the fundamental group based at $$x_0$$. More generally, one can define the fundamental groupoid on any subset $$A$$ of $$X,$$ using homotopy classes of paths joining points of $$A.$$ This is convenient for Van Kampen's Theorem.