Directed Set - Examples

Examples

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).
• Let D₁ and D₂ be directed sets. Then the Cartesian product set D₁ D₂ can be made into a directed set by defining (n₁, n₂) ≤ (m₁, m₂) if and only if n₁ ≤ m₁ and n₂ ≤ m₂. In analogy to the Product order this is the product direction on the Cartesian product.
• The set N N of pairs of natural numbers can be made into a directed set by defining (n₀, n₁) ≤ (m₀, m₁) if and only if n₀ ≤ m₀ and n₁ ≤ m₁.
• 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 an example of a directed set that is not ordered (neither totally nor partially).
• A (trivial) example of a partially ordered set that is not directed is the set {a, b}, in which the only order relations are a ≤ a and b ≤ b. A less trivial example is like the previous example of the "reals directed towards x₀" but in which the ordering rule only applies to pairs of elements on the same side of x₀.
• 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.
  • For every U: U ≤ U; since U contains itself.
  • For every U,V,W: if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then U contains W.
  • For every U, V: there exists the set U V such that U ≤ U V and V ≤ U V; since both U and V contain U V.
• In a poset P, every lower closure of an element, i.e. every subset of the form {a| a in P, a ≤x} where x is a fixed element from P, is directed.