# Locally compact connected topological group is paracompact

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

• Try to show $G$ is $\sigma$-compact and hence Lindelöf regular, so strongly paracompact. – Henno Brandsma Nov 13 '19 at 5:51
• I can finish the proof if I suppose $U$ is subgroup of $G$. However, I am struggling to prove the assumption... – HooMun Nov 13 '19 at 6:24
• 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 $G$.. – Henno Brandsma Nov 13 '19 at 17:31
• Oh Thanks a lot! – HooMun Nov 15 '19 at 0:41