Example 2

We shall see in Chapter IV that the topology of the rational line Q (resp. the real line R) is compatible with the ring structure of Q (resp. R).*

In a topological ring every left homothety $x \to ax$ (resp. every right homothety $x \to xa$) is continuous (and is a homeomorphism if $a$ is a unit of $A$).

Since we can write identically

the axiom (AT_{III}) [in view of (AT_I) and (AT_{II})] is equivalent to the conjunction of the following two axioms: (AT_{III a}). Given any $x_0 \in A$, the mappings $x \to x_0 x$ and $x \to xx_0$ are continuous at the point $x = 0$. (AT_{III b}). The mapping $(x, y) \to xy$ of $A \times A$ into $A$ is continuous at the point $(0, 0)$.

From this we can deduce a necessary and sufficient set of conditions which the filter $\mathfrak{B}$ of neighbourhoods of $0$ in a ring $A$ must satisfy in order to define a topology on $A$ compatible with its ring structure: $\mathfrak{B}$ must satisfy axioms (GA_I) and (GA_{II}) of § 1, and also the following two axioms: (AV_I). For all $x_0 \in A$ and all $V \in \mathfrak{B}$, there exists $W \in \mathfrak{B}$ such that $x_0 W \subset V$ and $W x_0 \subset V$. (AV_{II}). For all $V \in \mathfrak{B}$, there exists $W \in \mathfrak{B}$ such that $WW \subset V$.