Definition. Scott open [math-7H66]

2023-12-09 · Lîm Tsú-thuàn

Named after Dana S. Scott.

Let $D$ be a cpo. A subset $U$ of $D$ is said to be Scott open if

whenever $x \in U$ and $x \sqsubseteq y$ then $y \in U$ ; and
whenever $M \subseteq D$ is directed and $\bigsqcup M \in U$ , then $M \cap U \ne \varnothing$ .

Definition Scott topology and continuous [math-8KRP]

2023-12-09

Scott open subsets of $D$ form a topology, this is the Scott topology on $D$ . A continuous function $f : D \to E$ will preserve directed least element $f(\sqcup_D X) = \sqcup_E f(X)$ for all directed $X \subseteq D$ .