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.