connected locally compact topological groups are $\sigma $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 $\sigma $compact^{} (http://planetmath.org/SigmaCompact) subgroup $H$.
Corollary 1  Every locally compact topological group is the topological disjoint union of $\sigma $compact spaces.
Corollary 2  Every connected locally compact topological group is $\sigma $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 $\sigma $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 $\sigma $compact subset of $G$, and $G$ is the topological disjoint union $\bigcup _{g\in G}}gH$.
Of course, if $G$ is connected then $H$ must be all of $G$. Hence, $G$ is $\sigma $compact. $\mathrm{\square}$
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Proof (Theorem) : Let us fix some notation first. If $A$ is a subset of $G$ we use the notation ${A}^{1}:=\{{a}^{1}:a\in A\}$, ${A}^{n}:=\{{a}_{1}\mathrm{\dots}{a}_{n}:{a}_{1},\mathrm{\dots},{a}_{n}\in A\}$ and $\overline{A}$ denotes the closure^{} of $A$.
Pick a neighborhood $W$ of $e$ (the identity element of $G$) with compact closure. Then $V:=W\cap {W}^{1}$ is a neighborhood of $e$ with compact closure such that $V={V}^{1}$.
Let $H:={\bigcup}_{n=1}^{\mathrm{\infty}}{V}^{n}$. $H$ is clearly a subgroup of $G$. We now only have to prove that $H$ is open and $\sigma $compact.
We have that (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups)  3, 4 and 5)

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${V}^{n}$ is open

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${\overline{V}}^{n}$ is compact

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${\overline{V}}^{n}\subset {V}^{2n}$
So $H$ is open and also $H={\bigcup}_{n=1}^{\mathrm{\infty}}{\overline{V}}^{n}$, which implies that $H$ is $\sigma $compact. $\mathrm{\square}$
Title  connected locally compact topological groups are $\sigma $compact 

Canonical name  ConnectedLocallyCompactTopologicalGroupsAresigmacompact 
Date of creation  20130322 17:37:12 
Last modified on  20130322 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 