Remark 2

Let Θ be a subcogebra of Θ(A). We endow K with the discrete topology and the algebra A with the coarsest topology for which the mappings $f: A\rightarrow K$ for $f$ running through Θ are continuous (Gen. Top., I, §2, No. 2, p. 175). This topology endows A with the structure of a topological K-module. A two-sided ideal $\mathfrak{a}$ of A is open if and only if it has finite codimension and its orthogonal $\mathfrak{a}'$ is contained in Θ. By Proposition 3 of VIII, p. 379, the open two-sided ideals of A form a fundamental system of neighborhoods of 0 in A. The topology on A is therefore compatible with its ring structure.

Let $\mathscr{C}$ be the hereditary set of classes of A-modules of finite dimension over K such that $\Theta = \Theta_{\mathscr{C}}$ (VIII, p. 391, Proposition 9). Let E be a left A-module of finite dimension over K. We endow it with the discrete topology. The following properties are equivalent:

(i) The A-module E is of type $\mathscr{C}$.

(ii) The annihilator of the A-module E is open in A.

(iii) The mapping $(a, x)\mapsto ax$ from $A\times E$ to E is continuous.

The last property means that E is a topological A-module.

Let $\Theta^*$ be the dual algebra of the cogebra Θ. We endow $\Theta^*$ with the coarsest topology for which the mappings $\varphi \mapsto \varphi (u)$ from $\Theta^*$ to K, for $u$ running through Θ, are continuous. The topology on the algebra $\Theta^*$ is compatible with the additive group structure on $\Theta^*$. The orthogonals in $\Theta^*$ of the sets of the form $\Theta_E$(A), where E is an A-module of type $\mathscr{C}$, form a fundamental system of neighborhoods of 0. Now, such a set is a subcogebra of Θ, so its orthogonal is an ideal of $\Theta^*$. The topology on $\Theta^*$ is therefore compatible with its ring structure. The canonical algebra homomorphism $\eta : A\rightarrow \Theta^*$ (that sends $a\in A$ to the linear form $f\mapsto f(a)$ on Θ) defines an isomorphism from the completed Hausdorff space $\widehat{A}$ of A (Gen. Top., II, §3, No. 7, p. 191) to $\Theta^*$.