TheInfoList

In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a path in a
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$X$ is a
continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
from the closed unit interval
Topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...
and Mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be . Any space may be broken up into path-connected components. The set of path-connected components of a space $X$ is often denoted $\pi_0\left(X\right).$ 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 in a
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$X$ is a
continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$f : J \to X$ from a non-empty and non-degenerate interval $J \subseteq \R.$ A in a $X$ is a curve $f : \left[a, b\right] \to X$ whose domain $\left[a, b\right]$ is a Compact space, compact non-degenerate interval (meaning $a < b$ are real numbers), where $f\left(a\right)$ is called the of the path and $f\left(b\right)$ is called its . A is a path whose initial point is $x$ and whose terminal point is $y.$ Every non-degenerate compact interval $\left[a, b\right]$ is homeomorphic to $\left[0, 1\right],$ which is why a is sometimes, especially in homotopy theory, defined to be a
continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$f : \left[0, 1\right] \to X$ from the closed unit interval

# Homotopy of paths

Image:Homotopy between two paths.svg, A homotopy between two 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 $f_t : \left[0, 1\right] \to X$ indexed by

# 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 $g$: :$fg\left(s\right) = \beginf\left(2s\right) & 0 \leq s \leq \frac \\ g\left(2s-1\right) & \frac \leq s \leq 1.\end$ 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, $\left[\left(fg\right)h\right] = \left[f\left(gh\right)\right].$ Path composition defines a Group (mathematics), 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 $\pi_1\left\left(X, x_0\right\right).$ 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 $\left[0, a\right]$ to $X$ for any real $a \geq 0.$ A path $f$ of this kind has a length $, f,$ defined as $a.$ Path composition is then defined as before with the following modification: :$fg\left(s\right) = \beginf\left(s\right) & 0 \leq s \leq , f, \\ g\left(s-, f, \right) & , f, \leq s \leq , f, + , g, \end$ Whereas with the previous definition, $f,$ $g$, and $fg$ all have length $1$ (the length of the domain of the map), this definition makes $, fg, = , f, + , g, .$ What made associativity fail for the previous definition is that although $\left(fg\right)h$ and $f\left(gh\right)$have the same length, namely $1,$ the midpoint of $\left(fg\right)h$ occurred between $g$ and $h,$ whereas the midpoint of $f\left(gh\right)$ occurred between $f$ and $g$. With this modified definition $\left(fg\right)h$ and $f\left(gh\right)$ have the same length, namely $, f, + , g, + , h, ,$ and the same midpoint, found at $\left\left(, f, + , g, + , h, \right\right)/2$ in both $\left(fg\right)h$ and $f\left(gh\right)$; more generally they have the same parametrization throughout.

# Fundamental groupoid

There is a Category theory, categorical picture of paths which is sometimes useful. Any topological space $X$ gives rise to a Category (mathematics), 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 the Van Kampen's Theorem.