Topological module

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left topological module

An Abelian topological group that is a module over a topological ring , in which the multiplication mapping , taking to , is required to be continuous. A right topological module is defined analogously. Every submodule of a topological module is a topological module. If the module is separated and is closed in , then is a separated module. A direct product of topological modules is a topological module. The completion of the module as an Abelian topological group can be given the natural structure of a topological module over the completion of the ring .

A topological -module, where is a topological group, is an Abelian topological group that is a -module, where the multiplication mapping is required to be continuous.


[1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
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