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.