# Continuous dcpos are $\omega_1$-algebraic

Recall that a poset is directed if every finite subset has an upper bound, and directed-complete (a dcpo) if every directed subset has a join (supremum). In a dcpo, $x \ll y$ is defined to mean that every directed subset with join $\ge y$ contains some element $\ge x$, and $x$ is compact if $x \ll x$. A dcpo is continuous if every element $x$ is the join of a directed set of elements $\ll x$, and algebraic if every element is the join of a directed set of compact elements.

Define also the above terms prefixed with "$\sigma$-" by replacing "finite" with "countable" throughout.

I'm looking for (a) a written-down reference for the following (easy) fact, which I'm guessing is standard in domain theory: every continuous dcpo is $\sigma$-algebraic.

(Proof: given $x \ll y$, find $x \ll x_1 \ll x_2 \ll \dotsb \ll y$; then $\bigvee_i x_i$ is $\sigma$-compact, and it is easy to check that the set of such elements is $\sigma$-directed below $y$. I think there may also be an "abstract nonsense" proof using the Urysohn-Lawson lemma; pointers appreciated.)

Furthermore, it seems (I haven't checked the details) that one can characterize the posets which are the $\sigma$-compact elements in some continuous dcpo (from which the dcpo can thus be recovered as the $\sigma$-ideals): they are the posets with joins of countable increasing sequences, which are "continuous" with respect to such joins. I'd appreciate (b) a reference which also mentions this (assuming it's true), and possibly other facts concerning such posets ("continuous" wrt countable directed joins).

I've checked Continuous Lattices and Domains (aka A Compendium of Continuous Lattices) and Stone Spaces, but neither seems to mention $\sigma$-algebraicity, etc.

There appears to be a notion of "continuous $\omega$-chain-complete poset ($\omega$-ccpo)" in the literature (e.g., here), but it doesn't seem to quite coincide with the posets mentioned in (b) above. (For example, it excludes an uncountable antichain.)