$\sigma$- compact clopen subgroup.

Question

I am given $G$ locally compact group, and I want to show that there exists a clopen subgroup $H$ of $G$ that is $\sigma$-compact.

So here's what I did so far:

for $e \in U$, where $U$ is a nbhd of the identity element we know that $x \in x\bar{U}$, and $x\bar{U}$ is compact in $G$, now take some finite open cover of this set,$\cup_{i=1}^{n} V_i$, again this set is open nbhd of $x$ so again its closure is compact, and its closure equals the union of $\bar{V_i}$, now I am kind of stuck, I wish I could take this set as the clopen subgroup, but I don't think that I know that it's closed under multiplication, right?

Answer 1

Hint: Take a compact symmetric neighborhood $K$ of the identity and consider the $\sigma$-compact subgroup $H = \bigcup_{n =1}^{\infty} K^n$.

1. Prove that $H$ is an open subgroup.

2. Prove that an open sugroup of a topological group is closed — its complement is a union of (open) cosets.

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As additional exercises I suggest:

1. Prove that the connected component of the identity is a clopen subgroup. (Connected components are open and open subgroups are closed)

2. Prove that a connected locally compact group $G$ is $\sigma$-compact.