connected locally compact topological groups are -compact
The main result of this entry is the following theorem (whose proof is given below). The result expressed in the title then follows as a corollary.
Corollary 2 - Every connected locally compact topological group is -compact.
We first outline the proofs of the above corollaries:
Proof (Corollaries 1 and 2) : Let be a locally compact topological group. The main theorem implies that there is an open -compact subgroup .
It is known that every open subgroup of is also closed (see this entry (http://planetmath.org/ClosednessOfSubgroupsOfTopologicalGroups)). Therefore, each is a clopen -compact subset of , and is the topological disjoint union .
Of course, if is connected then must be all of . Hence, is -compact.
Proof (Theorem) : Let us fix some notation first. If is a subset of we use the notation , and denotes the closure of .
Pick a neighborhood of (the identity element of ) with compact closure. Then is a neighborhood of with compact closure such that .
Let . is clearly a subgroup of . We now only have to prove that is open and -compact.
We have that (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups) - 3, 4 and 5)
So is open and also , which implies that is -compact.
|Title||connected locally compact topological groups are -compact|
|Date of creation||2013-03-22 17:37:12|
|Last modified on||2013-03-22 17:37:12|
|Last modified by||asteroid (17536)|