In mathematics, a topological group ''G'' is a group (mathematics) that is also a topological space such that the group multiplication ''G'' × ''G'' → ''G'' and the inverse operation ''G'' → ''G'' are continuity (topology) maps. Here, ''G'' × ''G'' is viewed as a topological space by using the product topology. Though we do not do so here, it is common to also require that the topology on ''G'' be Hausdorff space. The reasons, and some equivalent conditions, are discussed below. In fancier language, one can say that topological groups are group objects in the category of topological spaces. Almost all objects investigated in mathematical analysis are topological groups (usually with some additional structure). == 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 ordinary topology, form a topological group. More generally, Euclidean space R''n'' 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. The above examples are all abelian group. 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'' Matrix (mathematics) with real entries can be viewed as a topological group with the topology defined by viewing GL(''n'',R) as a subspace (topology) of Euclidean space R''n''×''n''. All the examples above are Lie groups (if one views the infinite-dimensional vector spaces as infinite-dimensional "flat" Lie groups). 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 space containing the identity element) is a closed set normal subgroup. 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''. 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 (topology), uniformly continuous and uniform convergence on topological groups. As a uniform space, every topological group is completely regular space. It follows that if a topological group is T0 (i.e. Kolmogorov space), then it is already T2 (i.e. Hausdorff space). 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 theory. 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 finer 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 map. 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 set 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''. If ''H'' is a subgroup of ''G'' then the closure (topology) of ''H'' is also a subgroup. Likewise, if ''H'' is a normal subgroup, the closure of ''H'' is normal. ==Relationship to other areas of mathematics== Of particular importance in harmonic analysis are the locally compact space topological groups, because they admit a natural notion of measure (mathematics) 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. ==See also== *Lie group *algebraic group *topological ring Topological groups
Is it OK to say that 'Continuous group' is the same than 'Topological group' ?? :User:Looxix 23:30 Feb 23, 2003 (UTC) Continuous group is an older term that possibly could also mean Lie group, so I would be careful. User:AxelBoldt 21:16 Mar 2, 2003 (UTC)
See other meanings of words starting from letter:
Words begining with Topological_group:
Topological_group
Topological_group
Topological_groups
Topological_groups
Topological group
In mathematics, a topological group ''G'' is a group (mathematics) that is also a topological space such that the group multiplication ''G'' × ''G'' → ''G'' and the inverse operation ''G'' → ''G'' are continuity (topology) maps. Here, ''G'' × ''G'' is viewed as a topological space by using the product topology. Though we do not do so here, it is common to also require that the topology on ''G'' be Hausdorff space. The reasons, and some equivalent conditions, are discussed below. In fancier language, one can say that topological groups are group objects in the category of topological spaces. Almost all objects investigated in mathematical analysis are topological groups (usually with some additional structure). == 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 ordinary topology, form a topological group. More generally, Euclidean space R''n'' 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. The above examples are all abelian group. 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'' Matrix (mathematics) with real entries can be viewed as a topological group with the topology defined by viewing GL(''n'',R) as a subspace (topology) of Euclidean space R''n''×''n''. All the examples above are Lie groups (if one views the infinite-dimensional vector spaces as infinite-dimensional "flat" Lie groups). 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 space containing the identity element) is a closed set normal subgroup. 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''. 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 (topology), uniformly continuous and uniform convergence on topological groups. As a uniform space, every topological group is completely regular space. It follows that if a topological group is T0 (i.e. Kolmogorov space), then it is already T2 (i.e. Hausdorff space). 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 theory. 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 finer 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 map. 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 set 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''. If ''H'' is a subgroup of ''G'' then the closure (topology) of ''H'' is also a subgroup. Likewise, if ''H'' is a normal subgroup, the closure of ''H'' is normal. ==Relationship to other areas of mathematics== Of particular importance in harmonic analysis are the locally compact space topological groups, because they admit a natural notion of measure (mathematics) 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. ==See also== *Lie group *algebraic group *topological ring Topological groups
Topological group
Is it OK to say that 'Continuous group' is the same than 'Topological group' ?? :User:Looxix 23:30 Feb 23, 2003 (UTC) Continuous group is an older term that possibly could also mean Lie group, so I would be careful. User:AxelBoldt 21:16 Mar 2, 2003 (UTC)
See other meanings of words starting from letter:
T
TA | TB | TC | TD | TE | TF | TG | TH | TI | TJ | TK | TL | TŁ | TM | TN | TO | TP | TR | TS | TU | TW | TX | TY | TZ |Words begining with Topological_group:
Topological_group
Topological_group
Topological_groups
Topological_groups
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