Directed set

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
• if a <= b and b <= c, then a <= c
• for any two a and b in A, there exists a c in A with a <= c and b <= c

Directed sets are mainly used to define nets in topology. Nets generalize sequences and unite the various notions of limit.

Note that directed sets need not be antisymmetric and therefore in general are not partial orders.

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 neighborhoods of x₀ into a directed set by writing U <= V if and only if U contains V.