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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a path in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
X is a continuous function from the closed unit interval , 1/math> into X. Paths play an important role in the fields of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
. 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
non-degenerate interval (meaning a < b are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s), 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 Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, an ...
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 In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topol ...
of X.


Homotopy of paths

Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/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, + , 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 In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...
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, Topology and groupoids, Booksurge PLC, (2006). * J. Peter May, A concise course in algebraic topology, University of Chicago Press, (1999). *
James Munkres James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including ''Topology'' (an undergraduate-level text), ''Analysis on Manifolds'', ''Elements of Alg ...
, Topology 2ed, Prentice Hall, (2000). {{DEFAULTSORT:Path (Topology) Topology Homotopy theory