Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called Scott-open if and only if it is:

• Upward-closed: $x\in S$ and $x\le y$ implies $y\in S$;
• Inaccessible by directed suprema: if $D\subseteq S$ is directed and $\sup D\in S$, then there exist $d\in D\cap S$.

Scott-open sets are closed under taking finite intersections and arbitary unions, and the topology they define (as closed sets) is called the Scott topology.

However, working with the class of all Scott-open sets can be hard to work with. What would be a basis or subbasis that one could use instead?

I'm mostly interested in the case where $X$ is a complete lattice.

Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called Scott-open if and only if it is:

• Upward-closed: $x\in S$ and $x\le y$ implies $y\in S$;
• Inaccessible by directed suprema: if $D\subseteq S$ is directed and $\sup D\in S$, then there exist $d\in D\cap S$.

Scott-open sets are closed under taking finite intersections and arbitary unions, and the topology they define is called the Scott topology.

However, working with the class of all Scott-open sets can be hard to work with. What would be a basis or subbasis that one could use instead?

I'm mostly interested in the case where $X$ is a complete lattice.