 Definition 2.4 (Finite elements of a cpo). If \(D\) is a cpo, an element \(d \in D\) is finite if, for every increasing chain \(x_0 \sqsubseteq x_1 \sqsubseteq \ldots\)  \[d \sqsubseteq \bigsqcup_n x_n  \Longrightarrow \exists x_i\ \left(d \sqsubseteq x_i \right).\]   For \(d \in D\), the notation \(\mathcal{A}(d)\) denotes the set of finite elements below \(d\); \(\mathcal{A}(D)\) is the set of finite elements of \(D\). Finite elements are also called compact.