---- > [!definition] Definition. ([[discrete group]]) > A [[topological group]] $G$ is called **discrete** if it consists only of [[limit point|isolated points]]. ^definition > [!equivalence] > $G$ is discrete if and only if its identity element $e$ is an [[limit point|isolated point]]. Indeed, if $U \ni e$ witnesses isolation of $e$ [[open sets translate to open sets in topological groups|then]] $gU$ witnesses isolation of $g$, for any $g \in G$. ^equivalence ---- #### ---- #### 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 > ```