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 usual topology, form a topological group. More generally, Euclidean n-space Rn with addition and standard topology is a topological group. More generally yet, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.
The above examples are all abelian. Examples of non-abelian topological groups are given by the classical groups. For instance, the general linear group GL(n,R) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subset of Euclidean space Rn×n.
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.
Read more about this topic: Topological Group
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