Directed Sets

Directed Sets

Definition: A Directed Set is a pair $(X, \leq)$ where $X$ is a nonempty set and $\leq$ is a binary relation on $X$ that satisfies the following three properties:
1) $x \leq x$ for every $x \in X$ (Reflexivity).
2) If $x \leq y$ and $y \leq z$ then $x \leq z$ (Transitivity).
3) For all $x, y \in X$ there exists a $p \in X$ such that $x \leq p$ and $y \leq p$ (The Existence of an Upper Bound for Each Pair of Elements in $X$).
Equivalently, we say that the binary relation $\leq$ Directs the Set $X$.

In short, a directed set is a pair $(X, \leq)$ where $X$ is nonempty, and $\leq$ is a reflexive transitive binary relation which possesses an upper bound for each pair of elements in $X$.

Example 1

For example, the set of natural numbers $\mathbb{N}$ with the order relation $\leq$ of "less than or equals" is a total order on $\mathbb{N}$ and thus, $(\mathbb{N}, \leq )$ is a directed set.

Example 2

For a different example, let $(X, \tau)$ be a topological space. Take any point $x \in X$ and let $\mathcal B_x \subseteq \tau$ be the collection of all open neighbourhoods of $x$. Let $\subseteq$ be the order relation of set inclusion. Then $(\mathcal B_x, \subseteq)$ is a directed set.

Indeed by the properties of set inclusion:

  • 1. $A \subseteq A$ for every $A \in \mathcal B_x$.
  • 2. If $A \subseteq B$ and $B \subseteq C$ then $A \subseteq C$.

And most importantly:

  • 3. If $A, B \in \mathcal B_x$ then $A$ and $B$ are both open neighbourhoods of $x$. Since the union of two open sets in a topological space is also open, we see that $A \cup B \in \mathcal B_x$. If we let $P = A \cup B$ then clearly $A \subseteq P$ and $B \subseteq P$. So every pair of elements in $\mathcal B_x$ has an upper bound in $\mathcal B_x$.
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