> [!definitionb] Definition > > ![[directed_set.png|400]] > > Let $A$ be a [[Set|set]] equipped with a [[Relation|relation]] $\lesssim$. $A$ is a **directed set** if: > - $a \lesssim a$ for all $a \in A$. > - $a \lesssim b$ and $b \lesssim c$ implies that $a \lesssim c$. > - For any $a, b \in A$, there exists $c \in A$ such that $a \lesssim c$, $b \lesssim c$. > [!definition] > > Let $A$ and $B$ be two directed sets. Then their product $A \times B$ is a directed set with > $ > (a, b) \lesssim (c, d) \Leftrightarrow a \lesssim b, c \lesssim d > $ > *Proof*. $a \lesssim a, b \lesssim b$ implies that $(a, b) \lesssim (a, b)$. > > If $(a, b) \lesssim (c, d)$ and $(c, d) \lesssim (e, f)$, then $a \lesssim e$ and $b \lesssim f$, so $(a, b) \lesssim (e, f)$. > > Lastly, for any $(a, b)$ and $(c, d)$ we can find $(e, f)$ such that $a, c \lesssim e$ and $b, d \lesssim f$, so $(a, b), (c, d) \lesssim (e, f)$. # Examples > [!example] [[Sequence|Sequential]] Convergence > > The natural numbers $\nat$ is a directed set with the relation $\le$. > [!example] $\varepsilon$-$\delta$ Convergence > > The real numbers $\real$ with $x \lesssim y \Leftrightarrow |x - a| \ge |y - a|$ for some $a \in \real$ is a directed set. > [!example] [[Riemann Integral]] > > Partitions of a set with $P \lesssim P' \Leftrightarrow P \supset P'$ is a directed set. An "upper bound" for two partitions is their common refinement. > [!example] Topological Convergence > > Let $(X, \topo)$ be a [[Topological Space|topological space]] and $\cn$ a [[Neighbourhood Base|neighbourhood base]] at $x \in X$, then $\cn$ is a directed set with $U \lesssim V \Leftrightarrow U \supset V$.