component of identity of a topological group is a closed normal subgroup

Theorem - Let G be a topological groupMathworldPlanetmath and e its identity elementMathworldPlanetmath. The connected componentMathworldPlanetmathPlanetmathPlanetmath of e is a closed normal subgroupMathworldPlanetmath of G.

Proof: Let F be the connected component of e. All components of a topological spaceMathworldPlanetmath are closed, so F is closed.

Let aF. Since the multiplication and inversion functions in G are continuous, the set aF-1 is also connected, and since eaF-1 we must have aF-1F. Hence, for every bF we have ab-1F, i.e. F is a subgroupMathworldPlanetmathPlanetmath of G.

If g is an arbitrary element of G, then g-1Fg is a connected subset containing e. Hence g-1FgF for every gG, i.e. F is a normal subgroup.

Title component of identityPlanetmathPlanetmath of a topological group is a closed normal subgroup
Canonical name ComponentOfIdentityOfATopologicalGroupIsAClosedNormalSubgroup
Date of creation 2013-03-22 18:01:42
Last modified on 2013-03-22 18:01:42
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Theorem
Classification msc 22A05