connected locally compact topological groups are σ-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.
Theorem - Every locally compact topological group G has an open σ-compact (http://planetmath.org/SigmaCompact) subgroup H.
Corollary 1 - Every locally compact topological group is the topological disjoint union of σ-compact spaces.
Corollary 2 - Every connected locally compact topological group is σ-compact.
We first outline the proofs of the above corollaries:
Proof (Corollaries 1 and 2) : Let G be a locally compact topological group. The main theorem implies that there is an open σ-compact subgroup H.
It is known that every open subgroup of G is also closed (see this entry (http://planetmath.org/ClosednessOfSubgroupsOfTopologicalGroups)). Therefore, each gH is a clopen σ-compact subset of G, and G is the topological disjoint union ⋃g∈GgH.
Of course, if G is connected then H must be all of G. Hence, G is σ-compact. □
Proof (Theorem) : Let us fix some notation first. If A is a subset of G we use the notation A-1:=, 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)
-
•
is open
-
•
is compact
-
•
So is open and also , which implies that is -compact.
Title | connected locally compact topological groups are -compact |
---|---|
Canonical name | ConnectedLocallyCompactTopologicalGroupsAresigmacompact |
Date of creation | 2013-03-22 17:37:12 |
Last modified on | 2013-03-22 17:37:12 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 9 |
Author | asteroid (17536) |
Entry type | Theorem |
Classification | msc 22A05 |
Classification | msc 22D05 |