Knowlpedia

Topological group

A group with a topology making multiplication and inversion continuous.

A topological group is a GG equipped with a such that:

  1. The multiplication map μ:G×G→G\mu: G \times G \to G, (g,h)↦gh(g, h) \mapsto gh, is .
  2. The inversion map ι:G→G\iota: G \to G, g↦g−1g \mapsto g^{-1}, is continuous.
Equivalent characterizations

Equivalently, the single map (g,h)↦gh−1(g, h) \mapsto gh^{-1} is continuous.

Properties
  • Translation maps Lg:h↦ghL_g: h \mapsto gh and Rg:h↦hgR_g: h \mapsto hg are .
  • The topology is determined by neighborhoods of the identity.
  • Connected components form a .
Examples
  • (R,+)(\mathbb{R}, +) and (Rn,+)(\mathbb{R}^n, +) with the Euclidean topology.
  • (R∗,⋅)(\mathbb{R}^*, \cdot) (nonzero reals under multiplication).
  • GLn(R)GL_n(\mathbb{R}) and GLn(C)GL_n(\mathbb{C}) with the subspace topology.
  • are smooth topological groups.
  • Any group with the discrete topology.
Relation to Lie groups

A Lie group is a topological group that is also a with smooth group operations.