---- > [!definition] Definition. ([[discrete subgroup of Euclidean space]]) > A **discrete subgroup of $(\mathbb{R}^{n}, +)$** is one which contains no [[limit point|limit points]]. > > [!basicexample] > The discrete subgroups of $(\mathbb{R}^{2}, +)$ look like one of > - $L=\{ 0 \}$; > - A 'discrete line' $L=\{ m \v v : m \in \mathbb{Z} \} \cong \mathbb{Z}$; > - A 'lattice' $L=\{ m\v v_{1} + n \v v_{2}, m, n \in \mathbb{Z} \} \cong \mathbb{Z}^{2}$. ---- #### ---- #### 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 > ```