Topological Modules/Constructions

Definition (quotient topological module):

Let be a topological module over the topological ring , and let a submodule with the subspace topology. Then the module together with the quotient topology, ie. the final topology induced by the quotient map , is called the quotient module of .

Proposition (quotient map of topological quotient is open):

Let be a topological module and a submodule. Then the map is open.

Proof: Let be any open set. We have

which is open as the union of open sets.

Proposition (quotient topological module is topological module):

Let be a topological module and a submodule. Then the quotient module is a topological module with the subspace topology.

Proof: