topological group

\\PMlinkescapephrase

completely regular

Definitions

A topological group is a group $G$ endowed with a topologysuch that the multiplication and inverse operations of $G$ are continuous (http://planetmath.org/Continuous).That is, the map $G\\times G\\to G$ defined by $(x,y)\\mapsto xy$ is continuous,where the topology on $G\\times G$ is the product topology,and the map $G\\to G$ defined by $x\\mapsto x^{-1}$ is also continuous.

Many authors require the topology on $G$ to be Hausdorff,which is equivalent to requiring that the trivial subgroup be a closed set.

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 groups are topological groups with additional structure.

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

Subgroups, quotients and products

Every subgroup (http://planetmath.org/Subgroup) of a topological groupeither has empty interior or is clopen.In particular, all proper subgroups of a connected topological grouphave empty interior.The closure of any subgroup is also a subgroup,and the closure of a normal subgroup 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 group $G/N$ 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 $(G_{i})_{i\\in I}$ is a family of topological groups,then the unrestricted direct product $\\prod_{i\\in I}G_{i}$ is also a topological group, with the product topology.

Morphisms

Let $G$ and $H$ be topological groups, and let $f\\colon G\\to H$ be a function.

The function $f$ is said to be a homomorphism of topological groupsif it is a group homomorphism 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 isomorphism(that is, a bijective homomorphism of topological groups)and yet not be an isomorphism of topological groups.This occurs, for example, if $G$ is $\\mathbb{R}$ with the discrete topology,and $H$ is $\\mathbb{R}$ with its usual topology,and $f$ is the identity map on $\\mathbb{R}$ .

Topological properties

While every group can be made into a topological group,the same cannot be said of every topological space.In this section 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 $\\hbox{T}_{1}$ (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 $\\mathbb{Z}$ .

Every topological group is obviously an H-space.Consequently, the fundamental group of a topological group is abelian.Note that because topological groups are homogeneous,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 $\\sigma$ -compact.

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.

单词	TopologicalGroup
释义	topological group \\PMlinkescapephrase completely regular Definitions A topological group is a group $G$ endowed with a topologysuch that the multiplication and inverse operations of $G$ are continuous (http://planetmath.org/Continuous).That is, the map $G\\times G\\to G$ defined by $(x,y)\\mapsto xy$ is continuous,where the topology on $G\\times G$ is the product topology,and the map $G\\to G$ defined by $x\\mapsto x^{-1}$ is also continuous. Many authors require the topology on $G$ to be Hausdorff,which is equivalent to requiring that the trivial subgroup be a closed set. 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 groups are topological groups with additional structure. Profinite groups are another important class of topological groups;they arise, for example, in infinite Galois theory. Subgroups, quotients and products Every subgroup (http://planetmath.org/Subgroup) of a topological groupeither has empty interior or is clopen.In particular, all proper subgroups of a connected topological grouphave empty interior.The closure of any subgroup is also a subgroup,and the closure of a normal subgroup 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 group $G/N$ 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 $(G_{i})_{i\\in I}$ is a family of topological groups,then the unrestricted direct product $\\prod_{i\\in I}G_{i}$ is also a topological group, with the product topology. Morphisms Let $G$ and $H$ be topological groups, and let $f\\colon G\\to H$ be a function. The function $f$ is said to be a homomorphism of topological groupsif it is a group homomorphism 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 isomorphism(that is, a bijective homomorphism of topological groups)and yet not be an isomorphism of topological groups.This occurs, for example, if $G$ is $\\mathbb{R}$ with the discrete topology,and $H$ is $\\mathbb{R}$ with its usual topology,and $f$ is the identity map on $\\mathbb{R}$ . Topological properties While every group can be made into a topological group,the same cannot be said of every topological space.In this section 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 $\\hbox{T}_{1}$ (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 $\\mathbb{Z}$ . Every topological group is obviously an H-space.Consequently, the fundamental group of a topological group is abelian.Note that because topological groups are homogeneous,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 $\\sigma$ -compact. 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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