# 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:**