----- > [!proposition] Proposition. ([[quotient map from topological group onto left cosets is open]]) > Let $G$ be a [[topological space|topological]] [[group]] that is [[metrizable]] and $H \leq G$ a [[subgroup]]. Then the [[topological quotient map|quotient map]] from $G$ onto the [[coset|left cosets]] of $H$, $\begin{align} \pi: G & \to G / H \\ g & \mapsto gH \end{align}$ is an [[open map]]. > [!proof]- Proof. ([[quotient map from topological group onto left cosets is open]]) > The general open set in $G / H$ is a collection of equivalence classes which union to an open set in $G$ ([[quotient space|see here]]). Let $A$ be open in $G$. Then $\pi(A)=\{ \{aH\} : a \in A \}$ must be open in $G / H$, since this collection of equivalence classes unions to an open set in $G$, for [[openness dominates in internal subspace products in topological groups|this result]] implies that $\bigcup_{a \in A}^{}aH=AH$ is open in $G$ because $A$ is. ^eb82e8 ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```