----- > [!proposition] Proposition. ([[open sets translate to open sets in topological groups]]) > Let $G$ be a [[topological group|topological]] [[group]] and $U \subset G$ an open subset of $G$. Then, for all $g \in G$, the sets $gU = \{ gu: u \in U \}. \text{ and } Ug = \{ ug : u \in U \}$ > are open in $G$. ^945a65 > [!proof]- Proof. ([[open sets translate to open sets in topological groups]]) > Given fixed $g_{0} \in G$, the maps $\begin{align} G \to & G & & G \to G & G \to G \\ g \mapsto & g \cdot g_{0} & & g \mapsto g_{0} \cdot g & g \mapsto g^{-1}\\ & & & & & & \end{align}$ are [[homeomorphism|homeomorphisms]] of $G$; the inverse maps are obviously given by $\begin{align} G \to & G & & G \to G & G \to G \\ g \mapsto & g \cdot g_{0}^{-1} & & g \mapsto g_{0}^{-1} \cdot g^{} & g \mapsto g^{-1}\\ & & & & & & \end{align}$ and all maps involved are clearly [[continuous]] because the [[binary operation|group operation]] and inversion map are. > In particular, each map above must be an [[open map]]. Thus, the image of $U$ under each is open. The image of $U$ under the the first map is $Ug_{0}$; the image of $U$ under the second map is $g_{0}U$. So, these sets are open in $G$. ^0e970d ----- #### ---- #### 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 > ```