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# Loop (topology)

A loop in mathematics, in a topological space X is a continuous function f from the unit interval I. 

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Free loop

In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop.

## Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

## Loop algebra

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.

## Loop group

In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise.

## Loop space

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, maps from the circle S1 to X, equipped with the compact-open topology.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Path (topology)

In mathematics, a path in a topological space X is a continuous function f from the unit interval I.

## Pointed space

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint.

## Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

## Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible.

## Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Unit circle

In mathematics, a unit circle is a circle with a radius of one.

## 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.

## References

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