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Let $G$ be a locally compact, connected topological group. Show that $G$ is a paracompact.

I was trying to prove it with defining $U_n$ by defining

$U_1$ to be a symmetric neighborhood of $e$ having compact closure, and $U_{n+1} =\bar U_n \circ U_1$.

Now I have only step to show $U=\cup U_n$ is subgroup of $G$. However, I was stuck to show

For each $a\in U$, $a^{-1} \in U$

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  • $\begingroup$ Try to show $G$ is $\sigma$-compact and hence Lindelöf regular, so strongly paracompact. $\endgroup$ – Henno Brandsma Nov 13 '19 at 5:51
  • $\begingroup$ I can finish the proof if I suppose $U$ is subgroup of $G$. However, I am struggling to prove the assumption... $\endgroup$ – HooMun Nov 13 '19 at 6:24
  • $\begingroup$ Note that $U_1$ is symmetrical, so $a \in U_1$ implies $-a \in U_1$. Closures preserve this property. Products too... Open subgroups are closed too. The only oopen-and-closed non-empty set in a connected $G$ equals $G$.. $\endgroup$ – Henno Brandsma Nov 13 '19 at 17:31
  • $\begingroup$ Oh Thanks a lot! $\endgroup$ – HooMun Nov 15 '19 at 0:41

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