The Birkhoff-Kakutani Theorem

• 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 therefore $\bigcap_{n\in\mathbb{Z}}K_{n}=\mathinner{\lbrace g\in G:\ell(g)=0\rbrace.}$
• Proof of Lemma. Since $K_n$'s are symmetric and generate $G$, every element is a finite product of elements in $K_n$'s, implying that $\ell$ is well-defined. Triangle inequality, symmetry, and the fact that $\ell(e)=0$ are all easily proved, showing that $\ell$ is a length function.
• Now we prove the continuity of this length function. First note that if $g\in K_n$, then $\ell(g)\leq 2^n$. Fix $g\in G$ and $\varepsilon>0$. Choose $n\in\mathbb{Z}$ such that $2^n\leq\varepsilon$. We claim that $\lvert\ell(g)-\ell(h)\rvert\leq\varepsilon$ holds for all $h\in gK_n$. Since $K_n$ is symmetric, both $g^{-1}h$ and $h^{-1}g$ are contained in it. Applying triangle inequality twice, we obtain $\ell(g)=\ell(hh^{-1}g)\leq\ell(h)+2^n$ and $\ell(h)=\ell(gg^{-1}h)\leq\ell(g)+2^n$, thus implying $\lvert\ell(g)-\ell(h)\rvert\leq 2^n\leq\varepsilon$ for all $h\in gK_n$.
• We come to the final part of the claim. Suppose $g\in G$ is such that $\ell(g)<2^n$. We wish to show that $g\in K_n$. We already have $n_1,\dots,n_k\in\mathbb{Z}$ such that $g\in K_{n_1}\cdots K_{n_k}$ with $2^{n_1}+\cdots +2^{n_k}<2^n$. We now prove the following
• Claim. Suppose that $2^{n_1}+\cdots+2^{n_k}<2^n$. Then $K_{n_1}\cdots K_{n_k}\subseteq K_n$.
• Proof of Claim. Clearly $n_{j}, so $K_{n_1}\cdots K_{n_k}\subseteq K_{n-1}\cdots K_{n-1}$ (since $K_n\subseteq K_nK_nK_n\subseteq K_{n+1}$ for all $n\in\mathbb{Z}$, the $K_n$'s inherit the order from $\mathbb{Z}$). This proves the claim for $k=1,2,3$.
• For $k\geq 4$, we split into the following cases, inducting on $k$:
• If $2^{n_1}+\cdots +2^{n_k}<2^{n-1}$, then $K_{n_1}\cdots K_{n_{k-1}}\subseteq K_{n-1}$ by the inductive hypothesis, and $K_{n_k}\subseteq K_{n-1}$, implying $K_{n_1}\cdots K_{n_k}\subseteq K_{n-1}K_{n-1}\subseteq K_n$.
• If $2^{n-1}\leq 2^{n_1}+\cdots +2^{n_k}<2^n$, we choose the smallest $1\leq r\leq k$ such that $2^{n-1}\leq 2^{n_1}+\cdots +2^{n_r}$. Then $2^{n_1}+\cdots +2^{n_{r-1}}<2^{n-1}$, so again by our inductive hypothesis we get $K_{n_1}\cdots K_{n_{r-1}}\subseteq K_{n-1}$. On the other hand, $2^{n_{r+1}}+\cdots+2^{n_k}<2^{n-1}$ as well, implying $K_{n_{r+1}}\cdots K_{n_k}\subseteq K_{n-1}$ as well. Thus, $K_{n_1}\cdots K_{n_k}\subseteq K_{n-1}K_{n_r}K_{n-1}\subseteq K_n$.
• Proof of Theorem. The forward implications are trivial, so we show that the third statement implies the first.
• Step 1: Choosing $K_n$'s. let $(V_n)_{n\in\mathbb{N}}$ be a neighbourhood basis of the identity. Set $K_n=G$ for $n\geq 1$. For $n\leq 0$, we inductively choose symmetric identity neighbourhoods with $K_n\subseteq V_{-n}$ such that $K_{n-1}K_{n-1}K_{n-1}\subseteq K_n$. This can be done since the map $\widetilde{m}:G\times G\times G\longrightarrow G$, $(x,y,z)\mapsto xyz$ is continuous and preserves the identity. For example, once a symmetric neighbourhood $K_0$ in $V_0$ has been chosen, choose a basic open set $A\times B\times C\in\widetilde{m}^{-1}(K_0\cap V_1)$, and set $K_1=(A\cap B\cap C)\cap (A^{-1}\cap B^{-1}\cap C^{-1})$.
• Step 2: Non-degeneracy of $\ell$. the length function $\ell$ with the family $(K_n)$ above is non-degenerate, i.e., $\ell(g)=0$ iff $g=e$ since $e\in\bigcap_{n\in\mathbb{Z}}K_{n}\subseteq\bigcap_{n\geq0}V_{n}=\mathinner{\lbrace e\rbrace}$, where the last equality is a consequence of Hausdorffness of the group.
• Step 3: Definition of the Metric. Set $d(g,h)=\ell(g^{-1}h)$. This is a continuous, left-invariant metric on $G$ by the Lemma above.
• Step 4: $d$ generates the topology. For any open $U\subseteq G$ and $g\in U$, there is $n\in\mathbb{N}$ such that $gV_n\subseteq U$. Thus, $B_{2^{-n}}(g)\subseteq gK_{-n}\subseteq gV_n\subseteq U$, proving the theorem.

Here is some intuition behind the proof. Dyadic approximation of a chosen number on the real line involves choosing largest possible powers of two (still smaller than the number) and adding them to get closer to it. One may make a suboptimal approximation by overshooting. A good approximation can be found by simply taking the infimum over these dyadic sums. The functionally important detail for us is the existence of these "dyadic scales" that can be translated around from the origin. First countability, as adapted in the proof, supplies us with these fine scales to define our metric.