Exercise 20, § 6

a) Let $K$ be a division ring endowed with a non-discrete Hausdorff topology $\mathfrak{T}$ compatible with the ring structure of $K$. A subset $M$ of $K$ is right (resp. left) bounded if and only if, for each neighbourhood $U$ of $o$ there exists $a \in K^*$ such that $aM \subset U$ (resp. $Ma \subset U$); $M$ is then right (resp. left) bounded for every Hausdorff topology $\mathfrak{T}'$ which is coarser than $\mathfrak{T}$ and is compatible with the ring structure of $K$.

b) If $\mathfrak{T}$ is locally bounded (Exercise 12) and if $U$ is a bounded neighbourhood of $o$ with respect to $\mathfrak{T}$, then the sets $xU$ (resp. $Ux$) form a fundamental system of neighbourhoods of $o$ (with respect to $\mathfrak{T}$) as $x$ runs through $K^*$. Every point of $K$ has a countable fundamental system of neighbourhoods of $o$ with respect to $\mathfrak{T}$ if and only if there is a sequence $(a_n)$ of points of $K$ which has $o$ as a cluster point.

c) The least upper bound of any finite family of locally bounded topologies on $K$ (compatible with the ring structure of $K$) is locally bounded. Conversely, if the least upper bound $\mathfrak{T}$ of a family $(\mathfrak{T}_i)$ of locally bounded topologies on $K$ is locally bounded, then $\mathfrak{T}$ is the least upper bound of a finite subfamily of $(\mathfrak{T}_i)$. (Remark that if $U$ is a neighbourhood of $o$ which is bounded with respect to $\mathfrak{T}$, then there is a finite system of indices $(l_k)$ and for each $k$ a neighbourhood $V_k$ of $o$, bounded with respect to $\mathfrak{T}_{l_k}$, such that $\bigcap V_k \subset U$; on the other hand, there exists $a_k \in K^*$ such that $U \subset a_k V_k$.)

d) A subset $F$ of a division ring $K$ is said to be a quasi-ring if: (i) $F \neq K$, $o \in F$, $1 \in F$, $-F = F$, $FF \subset F$; (ii) there is an element $c \in F^* = F \cap K^*$ such that $c(F + F) \subset F$; and (iii) for each $x \in F$, there exists $y \in F$ such that $yF \subset Fx$ and $Fy \subset xF$. Show that if $\mathfrak{T}$ is a non-discrete Hausdorff topology which is locally bounded and compatible with the ring structure of $K$, then every symmetric bounded neighbourhood $U$ of $o$ (with respect to $\mathfrak{T}$), such that $UU \subset U$ and $1 \in U$, is a quasi-ring, and that $U^*(U^*)^{-1} = K^*$ [Exercise 19 b)]. If $V$ is any symmetric bounded neighbourhood of $o$ with respect to $\mathfrak{T}$, then the set $U$ of elements $x \in K$ such that $xV \subset V$ (or $Vx \subset V$) is a symmetric bounded neighbourhood of $o$ with respect to $\mathfrak{T}$ such that $UU \subset U$ and $1 \in U$, and hence is a quasi-ring.

e) Conversely, let $F$ be a quasi-ring in $K$ such that $F^*(F^*)^{-1} = K^*$. Show that there is a Hausdorff topology $\mathfrak{T}$ which is not discrete and is compatible with the ring structure of $K$, such that $F$ is a bounded neighbourhood of $o$ with respect to $\mathfrak{T}$. In particular we may take $F$ to be any subring $A$ of $K$ such that $K$ is the division ring of right fractions of $A$. In particular, if $K = \mathbf{Q}$ and $F = \mathbf{Z}$, the corresponding topology $\mathfrak{T}$ on $K$ is not compatible with the field structure of $K$.