---- > [!definition] Definition. ([[cofinal monotone map]]) > Let $I,J$ be [[poset|(pre)posets]] and suppose $f:I \to J$ is a [[monotonic map|monotone map]]. Suppose also that $I,J$ are [[filtered poset|filtered]]. We then say $f$ is a **cofinal map** if for all $j \in J$, there exists $i \in I$ such that $f(i) \geq j$. > If $I \subset J$ and $f$ is [[inclusion map|inclusion]], then $I$ is called a **cofinal subset**. ^definition > [!basicexample] > - The set of [[prime number|prime numbers]] is cofinal in $(\mathbb{N}, \leq)$. >- $\{ \overline{B_{N}(0)} : N \in \mathbb{N}\}$ is cofinal in the collection $\mathcal{K}(\mathbb{R}^{n})$ of compact subset of $\mathbb{R}^{n}$, cf. [[Heine-Borel theorem|Heine-Borel]]. ^basic-example > [!basicproperties] > - [[computing colimits with cofinals]] ^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 > ```