Interior is Union of Way Above Closures

Theorem

Let $\left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology.

Let $X \subseteq S$.

Then $X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$

where

$X^\circ$ denotes the interior of $X$,

$x^\gg$ denotes the way above closure of $x$.

We have:

$\left\{ {G \in \left\{ {g^\gg: g \in S}\right\}: G \subseteq X}\right\} = \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$

$\left\{ {x^\gg: x \in S}\right\}$ is basis of $\left({S, \tau}\right)$.

$X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$

$\blacksquare$