Remark

Given a topological ring $A$, consider, on a left $A$-module $E$, a topology compatible with the additive group structure of $E$. By virtue of the identity

the axiom (MT) is equivalent to the conjunction of the following three axioms: (MT$_I'$). For each $x_0 \in E$, the mapping $\lambda \to \lambda x_0$ is continuous at the point $\lambda = 0$. (MT$_{II}'$). For each $\lambda_0 \in A$, the mapping $x \to \lambda_0 x$ is continuous at the point $x = 0$. (MT$_{III}'$). The mapping $(\lambda, x) \to \lambda x$ is continuous at the point $(0, 0)$.

We deduce from this a necessary and sufficient set of conditions that the filter $\mathfrak{B}$ of neighbourhoods of $0$ in an $A$*-module* $E$ must satisfy in order to define a topology on $E$ compatible with its module structure; $\mathfrak{B}$ must satisfy the axioms (GA$_I$) and (GA$_{II}$) of $§ 1$, no. 2, and in addition must satisfy the following three axioms: (MV$_I$). For each $x_0 \in E$ and $V \in \mathfrak{B}$, there is a neighbourhood $S$ of $0$ in $A$ such that $S.x_0 \subset V$. (MV$_{II}$). For each $\lambda_0 \in A$ and $V \in \mathfrak{B}$, there exists $W \in \mathfrak{B}$ such that $\lambda_0 W \subset V$. (MV$_{III}$). For each $V \in \mathfrak{B}$ there exists $U \in \mathfrak{B}$ and a neighbourhood $T$ of $0$ in $A$ such that $T.U \subset V$.

Every commutative topological group is a topological $\mathbf{Z}$-module when the ring $\mathbf{Z}$ is given the discrete topology.

If $M$ is a submodule of a topological $A$-module $E$, it is clear that the topology induced on $M$ by the topology of $E$ is compatible with the module structure of $M$. Moreover, on the quotient $A$-module $E/M$, the topology which is the quotient by $M$ of the topology of $E$ is compatible with the $A$-module structure. To see this it is enough to show that the mapping $(\lambda, x) \to \lambda x$ of $A \times (E/M)$ onto $E/M$ is continuous (where $x \to x'$ denotes the canonical mapping of $E$ onto $E/M$). Now, since we may identify the additive topological groups $A \times (E/M)$ and $(A \times E)/(\{0\} \times M)$ (§ 2, no. 9, Corollary to Proposition 26), it is enough to show that the mapping $(\lambda, x) \to \lambda x'$ of $A \times E$ into $E/M$ is continuous; and this is immediate, since the mapping in question is the composition of $x \to x'$ and $(\lambda, x) \to \lambda x$.

Let $(E_i)_{i \in I}$ be an arbitrary family of topological $A$-modules, and let $E = \prod_{i \in I} E_i$ be the $A$-module which is the product of the $E_i$. Then the product topology on $E$ is compatible with the $A$-module structure of $E$. To prove this it is enough to show that the mapping $(\lambda, x) \to (\lambda \cdot \mathrm{pr}_i x)_{i \in I}$ of $A \times E$ into $E$ is continuous, or (by Proposition 1 of Chapter I, § 4, no. 1) that for each index $x \in I$ the mapping $(\lambda, x) \to \lambda \cdot \mathrm{pr}_x x$ is a continuous mapping of $A \times E$ into $E_x$; but this mapping is the composition of $(\lambda, x_x) \to \lambda x_x$ and $(\lambda, x) \to (\lambda, \mathrm{pr}_x x)$, both of which are continuous.

Let $A$ be a Hausdorff topological ring and $E$ a Hausdorff topological $A$-module. Let $\hat{E}$ be the additive group which is the completion of the commutative topological group $E$ (§ 3, no. 5, Theorem 2). The $\mathbf{Z}$-bilinear mapping $(\lambda, x) \to \lambda x$ of the product $A \times E$ of the additive groups $A, E$ into the additive group $E$ can be extended by continuity to a $\mathbf{Z}$-bilinear mapping of $\hat{A} \times \hat{E}$ into $\hat{E}$ (no. 5, Theorem 1), and this mapping we continue to denote by $(\lambda, x) \to \lambda x$. By virtue of the principle of extension of identities, we have $\lambda (\mu x) = (\lambda \mu) x$ for $\lambda \in \hat{A}, \mu \in \hat{A}$ and $x \in \hat{E}$, and $1 \cdot x = x$ for all $x \in \hat{E}$; the external law $(\lambda, x) \to \lambda x$ therefore defines an $\hat{A}$-module structure on $\hat{E}$ compatible with its topology. The topological $\hat{A}$-module $\hat{E}$ thus defined is called the completion of the topological $A$-module $E$.

Let $E$ be a topological module over a topological ring $A$, where neither $A$ nor $E$ is necessarily Hausdorff. Let $N$ (resp. $F$) be the closure of $\{0\}$ in $A$ (resp. $E$). $N$ is a two-sided ideal of $A$ (no. 4, Proposition 5) and $F$ is a sub-$A$-module of $E$ (no. 1, Proposition 1); furthermore, by continuity we have $\lambda x \in F$ whenever $\lambda \in N$ or $x \in F$. We can therefore define, by passing to the quotients, a mapping $(\dot{\lambda}, \dot{x}) \to \dot{\lambda} \dot{x}$ of $(A/N) \times (E/F)$ into $E/F$; it is easily verified (by use of the Corollary to Proposition 26 of § 2, no. 9) that this mapping is continuous, and therefore defines a structure of a topological $(A/N)$-module on $E/F$. If we put $B = A/N$ and $L = E/F$, then the $B$-module $L$ is called the Hausdorff module associated with $E$; its completion $\hat{L}$ is a topological module over the Hausdorff completion $\hat{A}$ (equal by definition to $\hat{B}$) of $A$ (no. 5), and this module $\hat{L}$ is called the Hausdorff completion of $E$ and is denoted by $\hat{E}$. We see as in , no. 4, Proposition 8 that every continuous homomorphism $u : E \to G$ of $E$ into a complete Hausdorff $\hat{A}$-module $G$ factorizes uniquely into $u = v \circ \varphi$, where $v$ is a continuous homomorphism of $\hat{E}$ into $G$ and $\varphi$ is the canonical mapping of $E$ into $\hat{E}$. We conclude that if $E, E'$ are two topological $A$-modules and $u : E \to E'$ is a continuous homomorphism, then there is a unique continuous homomorphism $\hat{u} : \hat{E} \to \hat{E}'$ such that the diagram

is commutative, $\varphi$ and $\varphi'$ being the canonical mappings.

The definitions and results of this section apply equally well to pseudo-modules over an arbitrary topological ring, by deleting all mention of the identity element of the ring.