# Difference between revisions of "Topological module"

An Abelian topological group $A$ that is a module over a topological ring $R$, in which the multiplication mapping $R \times A \to A$, taking $(r,a)$ to $ra$, is required to be continuous. A right topological module is defined analogously. Every submodule $B$ of a topological module $A$ is a topological module. If the module $A$ is separated and $B$ is closed in $A$, then $A/B$ is a separated module. A direct product of topological modules is a topological module. The completion $\hat A$ of the module $A$ as an Abelian topological group can be given the natural structure of a topological module over the completion $\hat R$ of the ring $R$.
A topological $G$-module, where $G$ is a topological group, is an Abelian topological group $A$ that is a $G$-module, where the multiplication mapping $G \times A \to A$ is required to be continuous.