In mathematics, a directed set is a set A together with a binary relation <= having the following properties:
- a ≤ a for all a in A (reflexivity)
- if a ≤ b and b ≤ c, then a ≤ c (transitivity)
- for any two a and b in A, there exists a c in A with a ≤ c and b ≤ c (directedness)
Examples of directed sets include:
- The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).
- If x0 is a real number, we can turn the set R - {x0} into a directed set by writing a <= b if and only if |a - x0| ≥ |b - x0|. We then say that the reals have been directed towards x0. This is not a partial order.
- If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed set by writing U <= V if and only if U contains V.
- In a poset P, every subset of the form {a| a in P, a<=x}, where x is a fixed element from P, is directed.
- A is not the empty set,
- for any two a and b in A, there exists a c in A with a <= c and b <= c (directedness),
Directed subsets are most commonly used in domain theory, where one studies orders for which these sets are required to have a least upper bound. Thus, directed subsets provide a generalization of (converging) sequences in the setting of partial orders as well.