Mananifold

Sourced from these notes on locally compact groups by Linus Kramer. Let $G$ be a Hausdorff topological group. The following are equivalent. The topology on $G$ is metrisable by a left-invariant metric. The topology on $G$ is metrisable. The identity element has a countable neighbourhood basis, i.e., the group is first countable. Proof. We will first prove the following Lemma. Let $G$ be a topological group. Suppose that $\left(K_{n}\right)_{n\in\mathbb{Z}}$ is a family of symmetric identity neighbourhoods with the property that $K_{n}K_{n}K_{n}\subseteq K_{n+1}$ holds for all $n\in\mathbb{Z}$ , and with $\langle\bigcup_{n\in\mathbb{Z}}K_{n}\rangle=G$ . For $g\in G$ we put $$\ell(g)=\inf\mathinner{\lbrace t\geq0:\text{there is some }k\geq1\text{ and }n_1,\dots,n_{k}\in\mathbb{Z}\text{ with }t=2^{n_1}+\cdots+2^{n_{k}}\text{ and }g\in K_{n_1}\cdots K_{n_{k}}\rbrace}.$$ Then $\ell$ is a continuous length function. Moreover, $\{g\in G:\ell(g)\le 2^{n}\}\subseteq K_n$ and the...