Topological Modules/Constructions

Definition (quotient topological module):

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

Proposition (quotient map of topological quotient is open):

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

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

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

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

Proposition (quotient topological module is topological module):

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

Proof: $\Box$