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In mathematics, the space of loops or loop space of a topological space X is the topological space of continuous maps from the circle S1 to X with the compact-open topology. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...
In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ...
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. ...
 That is, a particular function space. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ...
In homotopy theory loop space commonly refers to the same construction applied to pointed spaces, i.e. continuous maps respecting base points. In this setting there is a natural "concatenation operation" by which two elements of the loop space can be combined. With this operation, the loop space can be regarded as a magma. If we consider the quotient of the loop space with respect to the equivalence relation of pointed homotopy, then we obtain a group, the well-known fundamental group. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ...
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i. ...
In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
The iterated loop spaces of X are formed by applying Ω a number of times.
The loop space construction is right adjoint to the suspension functor, and the version for pointed spaces to the reduced suspension. This accounts for much of the importance of loop spaces in stable homotopy theory. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...
In topology, the suspension SX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse both ends to two points. ...
In topology, the suspension SX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse both ends to two points. ...
In mathematics, stable homotopy theory is a branch of algebraic topology. ...
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