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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 ...
Fundamental group. In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group.
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 ...
The affine group of one dimension is a two-dimensional matrix Lie group, consisting of. 2 × 2 {\displaystyle 2\times 2} real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form.
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.
Extension of a topological group. In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence where and are topological groups and and are continuous homomorphisms which are also open onto their images. [1] Every extension of topological groups is therefore a ...
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.
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.