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单词 TopologicalGroup
释义

topological group


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completely regularPlanetmathPlanetmathPlanetmath

Definitions

A topological groupMathworldPlanetmath is a group G endowed with a topologyMathworldPlanetmathsuch that the multiplication and inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath operationsMathworldPlanetmath of Gare continuousPlanetmathPlanetmath (http://planetmath.org/Continuous).That is, the map G×GG defined by (x,y)xy is continuous,where the topology on G×G is the product topology,and the map GG defined by xx-1 is also continuous.

Many authors require the topology on G to be HausdorffPlanetmathPlanetmath,which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to requiring that the trivial subgroup be a closed setPlanetmathPlanetmath.

A topology on a group G that makes G into a topology groupis called a group topology for G.

Examples

Any group becomes a topological group if it is given the discrete topology.

Any group becomes a topological group if it is given the indiscrete topology.

The real numbers with the standard topology form a topological group.More generally, an ordered group with its is a topological group.

Lie groupsMathworldPlanetmath are topological groups with additional structureMathworldPlanetmath.

Profinite groups are another important class of topological groups;they arise, for example, in infinite Galois theory.

Subgroups, quotients and products

Every subgroupMathworldPlanetmath (http://planetmath.org/Subgroup) of a topological groupeither has empty interior or is clopen.In particular, all proper subgroupsMathworldPlanetmath of a connected topological grouphave empty interior.The closureMathworldPlanetmathPlanetmath of any subgroup is also a subgroup,and the closure of a normal subgroupMathworldPlanetmath is normal(for proofs, see the entry“closure of sets closed under a finitary operation (http://planetmath.org/ClosureOfSetsClosedUnderAFinitaryOperation)”).A subgroup of a topological group is itself a topological group,with the subspace topology.

If G is a topological group and N is a normal subgroup of G,then the quotient groupMathworldPlanetmath G/N is also a topological group,with the quotient topology.This quotient G/N is Hausdorff if and only if N is a closed subset of G.

If (Gi)iI is a family of topological groups,then the unrestricted direct product iIGiis also a topological group, with the product topology.

Morphisms

Let G and H be topological groups, and let f:GH be a function.

The function f is said to be a homomorphism of topological groupsif it is a group homomorphismMathworldPlanetmath and is also continuous.It is said to be an isomorphism of topological groupsif it is both a group isomorphism and a homeomorphism.

Note that it is possible for f to be a continuous group isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath(that is, a bijectiveMathworldPlanetmath homomorphism of topological groups)and yet not be an isomorphism of topological groups.This occurs, for example, if G is with the discrete topology,and H is with its usual topology,and f is the identity map on .

Topological properties

While every group can be made into a topological group,the same cannot be said of every topological space.In this sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath we mention some of the propertiesthat the underlying topological space must have.

Every topological group is bihomogeneousand completely regular (http://planetmath.org/Tychonoff).Note that our earlier claim that a topological groupis Hausdorff if and only if its trivial subgroup is closedfollows from this:if the trivial subgroup is closed,then homogeneity ensures that all singletons are closed,and so the space is T1 (http://planetmath.org/T1Space),and being completely regular is therefore Hausdorff.A topological group is not necessarily http://planetmath.org/node/1530normal, however,a counterexample being the unrestricted direct productof uncountably many copies of the discrete group .

Every topological group is obviously an H-spaceMathworldPlanetmath.Consequently, the fundamental groupMathworldPlanetmathPlanetmath of a topological group is abelianMathworldPlanetmath.Note that because topological groups are homogeneousPlanetmathPlanetmathPlanetmath,the fundamental group does not depend (up to isomorphism)on the choice of basepoint.

Every locally compact topological groupis http://planetmath.org/node/1530normal and strongly paracompact.

Every connected locally compact topological group is σ-compactPlanetmathPlanetmath.

Other notes

Every topological group possesses a natural uniformity,which induces the topology.See the entry about the uniformity of a topological group (http://planetmath.org/UniformStructureOfATopologicalGroup).

A locally compact topological grouppossesses a natural measure, called the Haar measure.

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更新时间:2025/5/4 17:51:48