We begin by formalising the notion of an element in a cpo representing a finite amount of information.
Following the above intuition, we might be tempted to say that the "infinite" elements of a cpo are those which are the lub of an interesting set, but this notion is too weak. Consider this cpo X:
By the above definition, the only infinite element would be \infty , but if we consider the following isomorphic cpo, ordered by subset inclusion:
Then the set x cannot be a finite set, as any finite set would be a subset of one of the finite sets in the chain \emptyset \sqsubseteq \Set { 0 } \sqsubseteq \Set {0,1} \sqsubseteq \cdots . Thus, it makes more sense for us to call x an "infinite" element as well.
In the example cpo X above, all the elements except \infty and x would be compact. Thus, compactness better captures our notion of an element representing a finite amount of information. This understanding of compact elements is a generalisation of the notion of a finite element from the theory of algebraic lattices.