Scott topology

Remarks

The Scott topology makes $P$ a topological space that is $T_0$, and its specialization preorder is the order $\le$; the closure of $\set{x}$ is $\set{y : y \le x}$. It is $T_1$ only when the order is trivial. Example: in $\N \cup \set{\infty}$, the set $U = \set{n \in \N : n \ge 3} \cup \set{\infty}$ is Scott-open. It is an upper set, and a directed $D$ with $\bigvee D \in U$ either contains its supremum, or is infinite and so contains some $n \ge 3$. Non-example: in the same poset, $\set{\infty}$ is an upper set but is not Scott-open, since $D = \N$ has supremum $\infty$ and $D \cap \set{\infty} = \varnothing$. Closed sets of the Scott topology are the down-sets closed under suprema of directed subsets.