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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 , 1/math> into X. Paths play an important role in the fields of
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
path-connected
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 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 : [a, b] \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] \to X from the closed unit interval I := , 1/math> into X. An or 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 Topological curve, 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 (topology), 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 (topology), 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

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] \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 : [0, 1] \times [0, 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 [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) = \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] = [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 \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. 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, + , 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 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