---- > [!definition] Definition. ([[linear continuum]]) > A [[strict order relation|simply ordered set]] $L$ having more than one element is called a **linear continuum** if the following hold: > 1. $L$ has the [[supremum|least upper bound]] property > 2. If $x<y$, there exists $z$ such that $x<z<y$. > > The familiar example is $\mathbb{R}$. > [!basicexample] > Another example of a [[linear continuum]] is the **ordered square**. See Munkres. ---- #### ---- #### 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 > ```