In mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set ${\displaystyle A}$ such that for any ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle A}$ there exists ${\displaystyle c}$ in ${\displaystyle A}$ with ${\displaystyle a\leq c}$ and ${\displaystyle b\leq c}$. A directed set's preorder is called a direction.

The notion defined above is sometimes called an upward directed set. A downward directed set is defined symmetrically, meaning that every finite subset has a lower bound. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward.

Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.

In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

The set of natural numbers ${\displaystyle \mathbb {N} }$ with the ordinary order ${\displaystyle \,\leq \,}$ is one of the most important examples of a directed set. Every totally ordered set is a directed set, including ${\displaystyle (\mathbb {N} ,\leq ),}$ ${\displaystyle (\mathbb {N} ,\geq ),}$ ${\displaystyle (\mathbb {R} ,\leq ),}$ and ${\displaystyle (\mathbb {R} ,\geq ).}$

A (trivial) example of a partially ordered set that is not directed is the set ${\displaystyle \{a,b\},}$ in which the only order relations are ${\displaystyle a\leq a}$ and ${\displaystyle b\leq b.}$ A less trivial example is like the following example of the "reals directed towards ${\displaystyle x_{0}}$" but in which the ordering rule only applies to pairs of elements on the same side of ${\displaystyle x_{0}}$ (that is, if one takes an element ${\displaystyle a}$ to the left of ${\displaystyle x_{0},}$ and ${\displaystyle b}$ to its right, then ${\displaystyle a}$ and ${\displaystyle b}$ are not comparable, and the subset ${\displaystyle \{a,b\}}$ has no upper bound).

Let ${\displaystyle \mathbb {D} _{1}}$ and ${\displaystyle \mathbb {D} _{2}}$ be directed sets. Then the Cartesian product set ${\displaystyle \mathbb {D} _{1}\times \mathbb {D} _{2}}$ can be made into a directed set by defining ${\displaystyle \left(n_{1},n_{2}\right)\leq \left(m_{1},m_{2}\right)}$ if and only if ${\displaystyle n_{1}\leq m_{1}}$ and ${\displaystyle n_{2}\leq m_{2}.}$ In analogy to the product order this is the product direction on the Cartesian product. For example, the set ${\displaystyle \mathbb {N} \times \mathbb {N} }$ of pairs of natural numbers can be made into a directed set by defining ${\displaystyle \left(n_{0},n_{1}\right)\leq \left(m_{0},m_{1}\right)}$ if and only if ${\displaystyle n_{0}\leq m_{0}}$ and ${\displaystyle n_{1}\leq m_{1}.}$

If ${\displaystyle x_{0}}$ is a real number then the set ${\displaystyle I:=\mathbb {R} \backslash \lbrace x_{0}\rbrace }$ can be turned into a directed set by defining ${\displaystyle a\leq _{I}b}$ if ${\displaystyle \left|a-x_{0}\right|\geq \left|b-x_{0}\right|}$ (so "greater" elements are closer to ${\displaystyle x_{0}}$). We then say that the reals have been directed towards ${\displaystyle x_{0}.}$ This is an example of a directed set that is neither partially ordered nor totally ordered. This is because antisymmetry breaks down for every pair ${\displaystyle a}$ and ${\displaystyle b}$ equidistant from ${\displaystyle x_{0},}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are on opposite sides of ${\displaystyle x_{0}.}$ Explicitly, this happens when ${\displaystyle \{a,b\}=\left\{x_{0}-r,x_{0}+r\right\}}$ for some real ${\displaystyle r\neq 0,}$ in which case ${\displaystyle a\leq _{I}b}$ and ${\displaystyle b\leq _{I}a}$ even though ${\displaystyle a\neq b.}$ Had this preorder been defined on ${\displaystyle \mathbb {R} }$ instead of ${\displaystyle \mathbb {R} \backslash \lbrace x_{0}\rbrace }$ then it would still form a directed set but it would now have a (unique) greatest element, specifically ${\displaystyle x_{0}}$; however, it still wouldn't be partially ordered. This example can be generalized to a metric space ${\displaystyle (X,d)}$ by defining on ${\displaystyle X}$ or ${\displaystyle X\setminus \left\{x_{0}\right\}}$ the preorder ${\displaystyle a\leq b}$ if and only if ${\displaystyle d\left(a,x_{0}\right)\geq d\left(b,x_{0}\right).}$