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Compact element of a complete partial order

In the context of a complete partial order, a compact element is an element that is "always smaller than a limit."

Compact elements are often used as notions of finite approximants of the elements of an order. However, they are not always "finite" the usual mathematical sense.

Definition

Let (D, \sqsubseteq) be a (directed) complete partial order.

An element c \in D is compact just if for any directed set S \subseteq D we have

  c \sqsubseteq \bigsqcup S
    \quad\Longrightarrow\quad
  \exists s \in S.\ c \sqsubseteq s

In other words, if c is below a limit \bigsqcup S of a set D, then c must be below one of the elements of S.

This definition readily adapts to complete lattices, and to other notions of completeness.

See also

References