# Topological Modules/Constructions

Definition (quotient topological module):

Let ${\displaystyle M}$ be a topological module over the topological ring ${\displaystyle R}$, and let ${\displaystyle N\leq M}$ a submodule with the subspace topology. Then the module ${\displaystyle M/N}$ together with the quotient topology, ie. the final topology induced by the quotient map ${\displaystyle q:M\to M/N}$, is called the quotient module of ${\displaystyle M}$.

Proposition (quotient map of topological quotient is open):

Let ${\displaystyle M}$ be a topological module and ${\displaystyle N\leq M}$ a submodule. Then the map ${\displaystyle q:M\to M/N}$ is open.

Proof: Let ${\displaystyle U\subseteq M}$ be any open set. We have

${\displaystyle p^{-1}(p(U))=\bigcup _{n\in \mathbb {N} }U+n}$

which is open as the union of open sets. ${\displaystyle \Box }$

Proposition (quotient topological module is topological module):

Let ${\displaystyle M}$ be a topological module and ${\displaystyle N\leq M}$ a submodule. Then the quotient module ${\displaystyle M/N}$ is a topological module with the subspace topology.

Proof: ${\displaystyle \Box }$