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In
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 po ... X is a
continuous function from the closed
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ... , 1
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> 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, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ... .
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 component s. 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 space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ... s, 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 mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ... '' 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 po ... X is a continuous function f : J \to X from a non-empty and non-degenerate interval
In mathematics, a (real) interval is a set (mathematics), set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers ... J \subseteq \R.
A in X is a curve f : , b
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 ... \to X whose domain , b
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> 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 British ... 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
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> is homeomorphic to , 1
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 ... which is why a is sometimes, especially in homotopy theory, defined to be a continuous function f : , 1
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 ... \to X from the closed unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ... I := , 1
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> into X.
An or 0 in X is a path in X that is also a topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ... .
Importantly, a path is not just a subset of X that "looks like" a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ... , it also includes a parameterization
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, d ... . 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
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 ... \to X with f(0) = f(1) or as a continuous map from the unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ... S^1 to X
:f : S^1 \to X.
This is because S^1 is the quotient space of I = , 1
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> 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 topolo ... of X.
Homotopy of paths
Paths and loops are central subjects of study in the branch of algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ... 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 defor ... 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
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 ... \to X indexed by I = , 1
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> such that
* f_t(0) = x_0 and f_t(1) = x_1 are fixed.
* the map F : , 1
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 ... \times , 1
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 ... \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, + , 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
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ... where the objects are the points of X and the morphism s are the homotopy classes of paths. Since any morphism in this category is an isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ... this category is a groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*'' Group'' with a partial func ... , 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 to ... of X. Loops in this category are the endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ... s (all of which are actually automorphism s). The automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ... 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
A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across ... .
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 Alge ... , Topology 2ed, Prentice Hall, (2000).
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