In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the product topology. Although we do not do so here, it is common to also require that the topology on G be Hausdorff, namely that any two points in this space have disjoint neighborhoods. The reasons, and some equivalent conditions, are discussed below. In the language of category theory, one would say that topological groups are group objects in the category of topological spaces. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. ... The category Top has topological spaces as objects and continuous maps as morphisms. ...

Almost all objects investigated in analysis are topological groups (usually with some additional structure). Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...

Examples

Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ...

The real numbers R, together with addition as operation and its ordinary topology, form a topological group. More generally, Euclidean n-space Rⁿ with addition and standard topology is a topological group. More generally still, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...

The above examples are all abelian. Examples of non-abelian topological groups are given by Lie groups (topological groups that are also manifolds). For instance, the general linear group GL(n,R) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subset of Euclidean space R^n×n. All Lie groups are locally compact. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R. This is a countable space and it does not have the discrete topology. For a nonabelian example, consider the subgroup of rotations of R³ generated by two rotations by irrational multiples of 2π about different axes. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication. In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

Properties

The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the identity component (i.e. the connected component containing the identity element) is a closed normal subgroup. In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

The inversion operation on a topological group G gives a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps. If G is not abelian, then these two need not coincide. The uniform structures allow to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups. In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ...

As a uniform space, every topological group is completely regular. It follows that if a topological group is T₀ (i.e. Kolmogorov), then it is already T₂ (i.e. Hausdorff). In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

The most natural notion of homomorphism between topological groups is that of a continuous group homomorphism. Topological groups, together with continuous group homomorphisms as morphisms, form a category. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous). One can show that the quotient map q : G → G/H is always open. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... For quotient spaces in linear algebra, see quotient space (linear algebra). ... In mathematics, the possible topologies on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer (alt. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...

If H is a normal subgroup of G, then the factor group, G/H becomes a topological group, and the isomorphism theorems known from ordinary group theory remain valid in this setting. However, if H is not closed in the topology of G, then G/H won't be T₀ even if G is. It is therefore natural to restrict oneself to the category of T₀ topological groups, and restrict the definition of normal to normal and closed. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup, the closure of H is normal. In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

Relationship to other areas of mathematics

Of particular importance in harmonic analysis are the locally compact topological groups, because they admit a natural notion of measure and integral, given by the Haar measure. In many ways, the locally compact topological groups serve as a generalization of countable groups, while the compact topological groups can be seen as a generalization of finite groups. The theory of group representations is almost identical for finite groups and for compact topological groups. Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a measure is a function that assigns a number, e. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

See also

• Lie group
• algebraic group
• topological ring