----- > [!proposition] Proposition. ([[openness dominates in internal subspace products in topological groups]]) > Let $G$ be a [[topological group]]. Then for any subsets $A,B \subset G$, if $A$ is open then so are $AB$ and $BA$. > [!proof]- Proof. ([[openness dominates in internal subspace products in topological groups]]) > First, recall: > ![[open sets translate to open sets in topological groups#^945a65]] ![[open sets translate to open sets in topological groups#^0e970d]] > Then, recognize $AB = \{ ab : a \in A, b \in B \} =\bigcup_{a \in A}^{}aB$ and $BA = \{ ba: b \in B, a \in A \} = \bigcup_{b \in B}^{}bA$ to be open in $G$ as unions of (open, per the above) translates of open sets in $G$. ^a7a2f4 ----- #### ---- #### 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 > ```