Definition 3.7.6 (Locally Compact Group).label A locally compact group is a topological group $G$ whose topology is locally compact and Hausdorff. Moreover, $G$ enjoys the following properties:
- (1)
There exists a clopen $\s$-compact subgroup $G_{0}\leq G$.
- (2)
$G$ is paracompact.
- (3)
$G$ is (T4).
Proof.
- (1)
Let $U$ be a symmetric compact neighborhood of $1$ then $G_{0}\define \cups{\infty}{n=1}\prods{n}{i=1}U$ is the group generated by $U$. $G_{0}$ is open, closed and $\s$-compact by Definition 3.7.1(7,6b,4) respectively.
- (2)
By Corollary 3.13.11 and Lemma 3.7.2
- (3)
$\square$
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