Topological group

The real numbers form a topological group under addition

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.[1] A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example in physics.

Formal definition

A topological group G is a topological space and group such that the group operations of product:

${\displaystyle G\times G\to G:(x,y)\mapsto xy}$

and taking inverses:

${\displaystyle G\to G:x\mapsto x^{-1}}$

are continuous functions. Here, G × G is viewed as a topological space by using the product topology.

Although not part of this definition, many authors[2] require that the topology on G be Hausdorff; this corresponds to the identity map ${\displaystyle *\to G}$ being a closed inclusion (hence also a cofibration). The reasons, and some equivalent conditions, are discussed below. In the end, this is not a serious restriction—any topological group can be made Hausdorff in a canonical fashion.[3]

In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions. Adding the further requirement of Hausdorff (and cofibration) corresponds to refining to a model category.

Homomorphisms

A homomorphism between two topological groups G and H is just a continuous group homomorphism G ${\displaystyle \to }$ H. An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any nondiscrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.

Topological groups, together with their homomorphisms, form a category.

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.

The real numbers R, together with addition as operation and its usual topology, form a topological group. More generally, Euclidean n-space Rn with addition and standard topology is a topological group. More generally yet, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.

The above examples are all abelian. Examples of non-abelian topological groups are given by the classical groups. 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 Rn×n.

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 every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.

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. This is because if C is the identity component, a*C is the component of G (the group) containing a. In fact, the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. Therefore, the quotient topology induced by the quotient map from G to G/C is totally disconnected.[4]

The inversion operation on a topological group G is a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism GG.

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 one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if a topological group is T0 (Kolmogorov) then it is already T2 (Hausdorff), even T (Tychonoff).

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 : GG/H continuous). One can show that the quotient map q : GG/H is always open.

Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G \ H.

If H is a normal subgroup of G, then the factor group, G/H becomes a topological group when given the quotient topology. However, if H is not closed in the topology of G, then G/H will not 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.

The isomorphism theorems known from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if f : GH is a homomorphism then G/ker(f) is isomorphic to im(f) if and only if the map f is open onto its image.

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.

A topological group G is Hausdorff if and only if the trivial one-element subgroup is closed in G. If G is not Hausdorff then one can obtain a Hausdorff group by passing to the quotient space G/K where K is the closure of the identity. This is equivalent to taking the Kolmogorov quotient of G.

The fundamental group of a topological group is always abelian. This is a special case of the fact that the fundamental group of an H-space is abelian, since topological groups are H-spaces.

Relationship to other areas of mathematics

A compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

Of particular importance in harmonic analysis are the locally compact groups, because they admit a natural notion of measure and integral, given by the Haar measure. The theory of group representations is almost identical for finite groups and for compact topological groups. In general, σ-compact Baire topological groups are locally compact.

In topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. The homeomorphism group can be given a topology, such as the compact-open topology (in the case of regular, locally compact spaces), making it into a topological group.

Generalizations

Various generalizations of topological groups can be obtained by weakening the continuity conditions:[5]

Notes

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2. Armstrong, p. 73; Bredon, p. 51; Willard, p. 91.
3. D. Ramakrishnan and R. Valenza (1999). "Fourier Analysis on Number Fields". Springer-Verlag, Graduate Texts in Mathematics. Pp. 6–7.
4. {{#invoke:citation/CS1|citation |CitationClass=citation }}
5. Arhangel'skii & Tkachenko, p12

References

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• Bourbaki, Nicolas (1966). General Topology, Part 1, chapter 3: "Topological Groups", Hermann, Paris.
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