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A topological group, G, is a topological space that is also a group such that the group operation (in this case product): ⋅ : G × G → G, (x, y) ↦ xy. and the inversion map: −1 : G → G, x ↦ x−1. are continuous. [note 1] Here G × G is viewed as a topological space with the product topology. Such a topology is said to be compatible ...
In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group . That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative . The theory of topological groups applies also to TAGs, but more can be done with TAGs.
Compact group. The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication. In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group).
Category. : Topological groups. Wikimedia Commons has media related to Topological groups. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps.
A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. [3] In this context, an inverse system consists of a directed set (,), an indexed family of finite groups {:}, each having the discrete topology, and a family of homomorphisms {:,,} such that is the identity map on and the collection satisfies the composition ...
Locally compact group. In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure.
Conversely, let G be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group ′ that is locally isomorphic to G. Then, by a version of the closed subgroup theorem , G ′ {\displaystyle G'} is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a ...
Covering group. In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which H ...