Topological module

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left 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.


[1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) Zbl 0145.19302
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) Zbl 0279.13001
How to Cite This Entry:
Topological module. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article