Remark

In analysis one fairly often meets rings which satisfy axioms (AT_I), (AT_{II}) and (AT_{III a}), but not (AT_{III b}). An example is the ring of measures on a compact group, where multiplication is convolution and the topology is the vague topology.

Example 3). Let $\mathfrak{B}$ be a filter base on a ring $A$, consisting of two-sided ideals. $\mathfrak{B}$ is a fundamental system of neighbourhoods of $0$ for a topology compatible with the additive group structure of $A$, and it follows immediately from (AV_I) and (AV_{II}) that this topology is compatible with the ring structure of $A$.

Let $X$ be a topological space, and let $f$ and $g$ be two mappings of $X$ into a topological ring $A$. If $f$ and $g$ are continuous at a point $x_0 \in X$, then $f + g$, $-f$ and $fg$ are continuous at this point. It follows that the continuous mappings of $X$ into $A$ form a subring of the ring $A^X$ of all mappings of $X$ into $A$. We see also that, if $A$ is commutative, then every polynomial in n variables, with coefficients in $A$ and defined on $A^n$, is continuous on $A^n$. Again, let $f$ and $g$ be two mappings of a set $X$, filtered by a filter $\mathfrak{F}$, into a Hausdorff topological ring $A$; if $\lim_{\mathfrak{F}} f$ and $\lim_{\mathfrak{F}} g$ exist, then so do $\lim_{\mathfrak{F}} (f + g)$, $\lim_{\mathfrak{F}} (-f)$ and $\lim_{\mathfrak{F}} (fg)$, and we have (Chapter I, § 7, no. 4, Proposition 9, Corollary 1, and § 8, no. 1, Proposition 1)

(1)

(2) (3)