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.
Let be a (directed) complete partial order.
An element is compact just if for any directed set we have
In other words, if is below a limit of a set , then must be below one of the elements of S.
This definition readily adapts to complete lattices, and to other notions of completeness.
The importance of compact elements in the context of domain-theoretic semantics was first noticed by Scott in ...