Loop (topology) on:
[Wikipedia]
[Google]
[Amazon]
In
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a loop in a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is a continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
from the 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 analysi ...
to such that In other words, it is a path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desir ...
whose initial point is equal to its terminal point..
A loop may also be seen as a continuous map from the pointed 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 ...
into , because may be regarded as a quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of under the identification of 0 with 1.
The set of all loops in forms a space called the loop space of .
See also
* Free loop * Loop group * Loop space * Loop algebra *Fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
*Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...
References
Topology es:Grupo fundamental#Lazo {{topology-stub