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 $;\; href="/html/ALL/s/,\_1.html"\; ;"title=",\; 1">,\; 1$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 $\backslash pi\_0(X).$
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 in atopological 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 ...

$f\; :\; J\; \backslash to\; X$ from a non-empty and non-degenerate interval $J\; \backslash subseteq\; \backslash R.$
A in a $X$ is a curve $f\; :\; [a,\; b]\; \backslash to\; X$ whose domain $[a,\; b]$ is a Compact space, 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
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 ...

$f\; :\; [0,\; 1]\; \backslash to\; X$ from the closed unit interval $I\; :=;\; href="/html/ALL/s/,\_1.html"\; ;"title=",\; 1">,\; 1$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 $f\_t\; :\; [0,\; 1]\; \backslash to\; X$ indexed by $I\; =;\; href="/html/ALL/s/,\_1.html"\; ;"title=",\; 1">,\; 1$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 $g$: :$fg(s)\; =\; \backslash beginf(2s)\; \&\; 0\; \backslash leq\; s\; \backslash leq\; \backslash frac\; \backslash \backslash \; g(2s-1)\; \&\; \backslash frac\; \backslash leq\; s\; \backslash leq\; 1.\backslash end$ 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, $[(fg)h]\; =\; [f(gh)].$ 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 $\backslash pi\_1\backslash left(X,\; x\_0\backslash right).$ 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 $a\; \backslash geq\; 0.$ A path $f$ of this kind has a length $,\; f,$ defined as $a.$ Path composition is then defined as before with the following modification: :$fg(s)\; =\; \backslash beginf(s)\; \&\; 0\; \backslash leq\; s\; \backslash leq\; ,\; f,\; \backslash \backslash \; g(s-,\; f,\; )\; \&\; ,\; f,\; \backslash leq\; s\; \backslash leq\; ,\; f,\; +\; ,\; g,\; \backslash end$ 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 $(fg)h$ 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 $\backslash left(,\; f,\; +\; ,\; g,\; +\; ,\; h,\; \backslash right)/2$ in both $(fg)h$ and $f(gh)$; 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.See also

* * Path space (disambiguation) *References

* Ronald Brown (mathematician), Ronald Brown, Topology and groupoids, Booksurge PLC, (2006). * J. Peter May, A concise course in algebraic topology, University of Chicago Press, (1999). * James Munkres, Topology 2ed, Prentice Hall, (2000). {{DEFAULTSORT:Path (Topology) Topology Homotopy theory