In Definition [dt-001D] we defined cpos in terms of directed completeness. We may consider chain completeness instead, where, rather than require lubs for every directed subset of our poset A, we require lubs for every chain X \subseteq A. We know from Theorem [dt-0012] that all chains are directed, so every directed complete cpo is also chain complete. It is also known that every chain complete cpo is also directed complete, so the notions are equivalent, although the proof is non-trivial, relying on transfinite induction (see these lecture notes for details).
If we weaken our completeness requirement to require lubs only for the countable \omega -chains and not necessarily all chains, we get what is called an \omega -cpo. All \omega -chains are chains and are therefore directed, but not all chains are \omega -chains.
Working with just \omega -cpos is common in denotational semantics practice, but working with directed completeness simplifies some of the properties we will discuss later.