compact spaces with group structure

PropositionPlanetmathPlanetmath. Assume that (G,M) is a group (with multiplication M:G×GG) and G is also a topological spaceMathworldPlanetmath. If G is compactPlanetmathPlanetmath HausdorffPlanetmathPlanetmath and M:G×GG is continuousMathworldPlanetmathPlanetmath, then (G,M) is a topological groupMathworldPlanetmath.

Proof. Indeed, all we need to show is that function f:GG given by f(g)=g-1 is continuous. Note, that the following holds for the graph of f:


where e denotes the neutral element in G. It follows (from continuity of M) that Γ(f) is closed in G×G. It is well known (see the parent object for details) that this implies that f is continuous, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title compact spaces with group structure
Canonical name CompactSpacesWithGroupStructure
Date of creation 2013-03-22 19:15:13
Last modified on 2013-03-22 19:15:13
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Corollary
Classification msc 26A15
Classification msc 54C05