---- > [!definition] Definition. ([[topological group]]) > A **topological group** is a [[group]] $(G, \cdot)$, which is also a [[topological space]], such that the [[binary operation|group operation]]s $\begin{align} \cdot: G \times G \to G \\ \\ {\cdot^{-1}}: G \to G, \end{align}$ are [[continuous]], where $G \times G$ is endowed with the [[product topology]]. > Topological groups are objects of [[category]] $\mathsf{TopGrp}$. The homset $\text{Hom}(G,H)$ is the set of [[continuous]] [[group homomorphism|group homomorphisms]] $G \to H$. An analogous notion of [[topological ring]] exists. > [!basicproperties] > - Any topological invariant is preserved by translating [[left and right translations preserve connectedness and compactness for subspaces of topological groups|(e.g.)]] > ^properties ---- #### ---- #### 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 > ```