Topological module
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.
References
[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) |
Topological module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_module&oldid=41093