Scott continuity

Remarks

A Scott-continuous map is monotone: if $x \le y$, apply the definition to $D = \set{x, y}$, whose supremum is $y$, to get $f(x) \le f(y)$. A map $f$ is Scott-continuous if and only if it is continuous for the Scott topologies on $P$ and $Q$. Example: on the power set of $\N$, the map $A \mapsto A \cup \set{0}$; the image of the union of a directed family $D$ is $\bigcup_{A \in D} (A \cup \set{0})$. Non-example: on the power set of $\N$, let $f(A) = \N$ if $A$ is infinite and $f(A) = \varnothing$ if $A$ is finite. This map is monotone but not Scott-continuous. The finite sets $\set{0, \ldots, n}$ form a directed family with union $\N$, and $f(\N) = \N \ne \varnothing = \bigvee_n f(\set{0, \ldots, n})$.