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    BANGNAM
    2023-03-21 11:06:35
    5 min read
    · 5:30

    Discrete topological group

    topological group

    \PMlinkescapephrase

    completely regular

    Discrete topological group

    Show
    • topological group
    • Definitions
    • Subgroups, quotients and products
    • Topological properties
    • Other notes
    • Video liên quan

    Definitions

    A topological group

    Discrete topological group
    is a group G endowed with a topology
    Discrete topological group
    such that the multiplication and inverse
    Discrete topological group
    operations
    Discrete topological group
    of G are continuous(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 Hausdorff, which is equivalent

    Discrete topological group
    to requiring that the trivial subgroup be a closed set.

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

    Examples

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

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    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

    Discrete topological group
    are topological groups with additional structure
    Discrete topological group
    .

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

    Subgroups, quotients and products

    Every subgroup

    Discrete topological group
    (http://planetmath.org/Subgroup) of a topological group either has empty interior or is clopen. In particular, all proper subgroups
    Discrete topological group
    of a connected topological group have empty interior. The closure
    Discrete topological group
    of any subgroup is also a subgroup, and the closure of a normal subgroup
    Discrete topological group
    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

    Discrete topological 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 (Gi)iI is a family of topological groups, then the unrestricted direct product iIGi is 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 groups if it is a group homomorphism

    Discrete topological group
    and is also continuous. It is said to be an isomorphism of topological groups if it is both a group isomorphism and a homeomorphism.

    Note that it is possible for f to be a continuous group isomorphism

    Discrete topological group
    (that is, a bijective
    Discrete topological group
    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 sectionwe mention some of the properties that the underlying topological space must have.

    Every topological group is bihomogeneous and completely regular (http://planetmath.org/Tychonoff). Note that our earlier claim that a topological group is Hausdorff if and only if its trivial subgroup is closed follows 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.orgode/1530normal, however, a counterexample being the unrestricted direct product of uncountably many copies of the discrete group .

    Every topological group is obviously an H-space

    Discrete topological group
    . Consequently, the fundamental group
    Discrete topological group
    of a topological group is abelian
    Discrete topological group
    . 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 group is http://planetmath.orgode/1530normal and strongly paracompact.

    Every connected locally compact topological group is σ-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 group possesses a natural measure, called the Haar measure.

    Titletopological group
    Canonical nameTopologicalGroup
    Date of creation2013-03-22 15:47:09
    Last modified on2013-03-22 15:47:09
    Owneryark (2760)
    Last modified byyark (2760)
    Numerical id29
    Authoryark (2760)
    Entry typeDefinition
    Classificationmsc 22A05
    Related topicGroup
    Related topicTopologicalSpace
    Defineshomomorphism of topological groups
    Definesisomorphism of topological groups

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