Exercise 16, § 6

Show that on a ring $A$ (resp. a division ring $K$), the least upper bound $\mathfrak{T}$ of a family $(\mathfrak{T}_i)$ of topologies compatible with the ring structure of $A$ (resp. the division ring structure of $K$) is compatible with this structure. A set $M$ which is left (resp. right) bounded for each of the topologies $\mathfrak{T}_i$ is also left (resp. right) bounded for $\mathfrak{T}$.