Exercise 21, § 6

a) Let $\mathfrak{T}$ be a locally bounded Hausdorff topology on a division ring $K$, with respect to which $K$ is connected. Show that there are non-zero topologically nilpotent elements [Exercise 9 a)] in $K$. (If $U$ is a symmetric bounded neighbourhood of $o$, and $t$ is a non-zero element of $K$ such that $Ut + tU \subset U$, show that $K$ is the union of the sets $Ut^{-n}$, by using Proposition 6 of § 2, no. 2).

b) Let $\mathfrak{T}$ be a locally bounded Hausdorff topology on $K$, with respect to which $K$ has non-zero topologically nilpotent elements. Show that there is a neighbourhood $U$ of $o$ such that, for each neighbourhood $V$ of $o$, there is an integer $n_0$ such that $U^n \subset V$ whenever $n \geq n_0$; this implies that all the elements of $U$ are topologically nilpotent and that every point of $K$ has a countable fundamental system of neighbourhoods with respect to $\mathfrak{T}$. (Let $t \neq o$ be topologically nilpotent and let $W$ be a bounded neighbourhood of $o$; take $U$ to be a bounded neighbourhood of $o$ such that $UW \subset Wt$.) The set $T$ of topologically nilpotent elements of $K$ is therefore a neighbourhood of $o$ with respect to $\mathfrak{T}$.