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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 ...
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).
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.
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.
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 ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are ...
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.
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneered the ...