Path (topology)

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

An or 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 : , 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 = , 1/math> 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 : , 1\to X indexed by I = , 1/math> such that
* $f_t(0) = x_0$ and $f_t(1) = x_1$ are fixed.
* the map F : , 1\times , 1\to X given by $F(s, t) = f_t(s)$ is continuous.
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.

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

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 $g$:
:$fg(s) = \beginf(2s) & 0 \leq s \leq \frac \\ g(2s-1) & \frac \leq s \leq 1.\end$
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= (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 $$, a/math> 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:
:$fg(s) = \beginf(s) & 0 \leq s \leq , f, \\ g(s-, f, ) & , f, \leq s \leq , f, + , g, \end$

Whereas with the previous definition, $f,$ $g$, and $fg$ all have length $1$ (the length of the domain of the map), this definition makes $, fg, = , f$