View source for Topological Modules/Constructions

{{definition|quotient topological module|Let <math>M</math> be a topological module over the topological ring <math>R</math>, and let <math>N \le M</math> a submodule with the subspace topology. Then the module <math>M/N</math> together with the quotient topology, ie. the final topology induced by the quotient map <math>q: M \to M/N</math>, is called the '''quotient module''' of <math>M</math>.}}

{{proposition|quotient map of topological quotient is open|Let <math>M</math> be a topological module and <math>N \le M</math> a submodule. Then the map <math>q: M \to M/N</math> is open.}}

{{proof|Let <math>U \subseteq M</math> be any open set. We have
:<math>p^{-1}(p(U)) = \bigcup_{n \in \mathbb N} U + n</math>
which is open as the union of open sets.}}

{{proposition|quotient topological module is topological module|Let <math>M</math> be a topological module and <math>N \le M</math> a submodule. Then the quotient module <math>M/N</math> is a topological module with the subspace topology.}}

{{proof|}}

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