In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
A topological vector space is a topological module over a topological field.
An abelian topological group can be considered as a topological module over where is the ring of integers with the discrete topology.
A topological ring is a topological module over each of its subrings.
A more complicated example is the -adic topology on a ring and its modules. Let be an ideal of a ring The sets of the form for all and all positive integers form a base for a topology on that makes into a topological ring. Then for any left -module the sets of the form for all and all positive integers form a base for a topology on that makes into a topological module over the topological ring