Exercise 8, § 1

Let $G$ be a Hausdorff topological group, and let $s \neq e$ be a point of $G$. Show that there is a symmetric neighbourhood $V$ of $e$ such that $s \notin V^2$, and that $G$ can be covered by at most 17 sets of the form $a_k V b_k$ ($1 \leq k \leq 17$). (Consider a maximal element in the set of symmetric neighbourhoods $V$ of $e$ such that $s \notin V^2$. We are thus led to consider the set $S$ of $x \in G$ such that $x^2 = s$; observe that if $x, y$ are two elements of $S$ such that $xy^{-1} \in S$, then we must have $x^2 = e$, and if $z$ is a third element of $S$ such that $xz^{-1} \in S$ and $yz^{-1} \in S$, then $z$ must commute with $xy^{-1}$, and $xy^{-1}z^{-1} \notin S$.) If $G$ is commutative, the number 17 may be replaced by 5.