In a recent lecture I've been presented with the concept of $\omega$-continuity, defined as "the preservation of joins in $\omega$-chains", where an $\omega$-chain is a non-empty sequence $\{a_n\}$ of elements $a_i \in \mathcal L$, where $\mathcal L$ is a complete lattice, such that $a_{i}\leq a_{i+1}$.
That is, $F : \mathcal L \to \mathcal L$ is $\omega$-continuous if for all $\omega$-chains $\{a_n\}$:
$$ \bigvee_n F(a_n) = F\Big(\bigvee_n a_n\Big) $$
This property was used, during the lecture, to describe Kleene's fixed point theorem. I didn't find much about this property online, and when I read about the theorem from other sources, a different definition of continuity was used: Scott continuity.
My first questions are:
- Is $\omega$-continuity a special case of Scott-continuity, or are they the same?
- If there is any, what would be an example of a function that is Scott-continuous but not $\omega$-continuous?
The second part of this question is more of a question of "how would I go about proving that a function is $\omega$-continuous?"
- What would be an example of a function that is not join-preserving in general but is $\omega$-continuous?
- What is the proof that the above function is $\omega$-continuous?