Topological group

In mathematics , a topological group is a group that has a topology that is “compatible” with the group structure . The topological structure allows, for example, to consider limit values in this group and to speak of continuous homomorphisms .

definition

A group is called a topological group if it is provided with a topology such that: ${\ displaystyle G}$

1. The group link is continuous. In this case, with the product topology provided.${\ displaystyle G \ times G \ to G}$${\ displaystyle G \ times G}$
2. The inverse mapping is continuous.${\ displaystyle G \ to G}$

Examples

The real numbers with addition and the ordinary topology form a topological group. More generally, the -dimensional Euclidean space with vector addition and standard topology is a topological group. Every Banach space and Hilbert space is also a topological group with regard to addition. ${\ displaystyle \ mathbb {R}}$${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

The above examples are all Abelian . An important example of a nonabelian topological group is the group of all invertible real matrices. The topology is created by understanding this group as a subset of the Euclidean vector space . ${\ displaystyle \ operatorname {GL} (n, \ mathbb {R})}$${\ displaystyle n \ times n}$${\ displaystyle \ mathbb {R} ^ {n ^ {2}}}$

${\ displaystyle \ mathbb {R} ^ {n}}$is just like a Lie group , i.e. a topological group where the topological structure is that of a manifold . ${\ displaystyle \ operatorname {GL} (n, \ mathbb {R})}$

An example of a topological group that is not a Lie group is the additive group of rational numbers (it is a countable set that is not provided with the discrete topology ). A non-Abelian example is the subgroup of the rotation group of which is generated by two rotations around irrational multiples of (the circle number Pi) around different axes. ${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ mathbb {R} ^ {3}}$${\ displaystyle \ pi}$

In every unitary Banach algebra , the set of invertible elements with the multiplication forms a topological group.

properties

The algebraic and topological structure for a topological group are closely related. For example, in any topological group, the connected component of the neutral element is a closed normal subgroup of . ${\ displaystyle G}$ ${\ displaystyle G}$

If an element is a topological group , then the left multiplication and the right multiplication with homeomorphisms are from to , as is the inverse mapping. ${\ displaystyle a}$${\ displaystyle G}$${\ displaystyle a}$ ${\ displaystyle G}$${\ displaystyle G}$

Each topological group can be understood as a uniform space . Two elementary uniform structures that result from the group structure are the left and the right uniform structure . The left uniform structure makes the left multiplication uniformly continuous , the right uniform structure makes the right multiplication uniformly continuous. For non-Abelian groups, these two uniform structures are generally different. The uniform structures make it possible in particular to define terms such as completeness, uniform continuity and uniform convergence.

Like any topology created by a uniform space, the topology of a topological group is completely regular . In particular, a topological group which satisfies (i.e., which is a Kolmogoroff space) is even a Hausdorff space . ${\ displaystyle T_ {0}}$

The most natural notion of homomorphism between topological groups is that of continuous group homomorphism . The topological groups together with the continuous group homomorphisms form a category .

Each subgroup of a topological group is in turn a topological group with the subspace topology . For a subgroup of , the left and right secondary classes together with the quotient topology form a topological space. ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle G / H}$

If is a normal divisor of , then becomes a topological group. It should be noted, however, that if is not closed in the topology of , the resulting topology is not Hausdorffian. It is therefore natural, if one restricts oneself to the category of Hausdorff topological groups, to examine only closed normal subdivisions. ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle G / H}$${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle G / H}$

If is a subgroup of , then the closed envelope of is in turn a subgroup. Likewise, the conclusion of a normal divider is normal again. ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle H}$

literature

• Lev Pontryagin : Topological Groups. 2 volumes. Teubner, Leipzig 1957–1958.
• Guido Mislin (Ed.): The Hilton symposium 1993. Topics in Topology and Group Theory (= CRM Proceedings & Lecture Notes. Vol. 6). American Mathematical Society, Providence RI 1994, ISBN 0-8218-0273-9 .
• Terence Tao : Hilbert's fifth problem and related topics (= Graduate Studies in Mathematics. Vol. 153). American Mathematical Society, Providence RI 2014, ISBN 978-1-4704-1564-8 online .