Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called [Scott-open](https://ncatlab.org/nlab/show/Scott+topology) 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.



Basis or subbasis for Scott topology
There is nothing to say in the general case, but in the case of a *continuous* dcpo $D$ there is a well-known construction of certain bases for the Scott topology on $D$. 


We say $d$ is way below $e$, or $d \ll e$ if for each directed set $(d\_i)\_{i \in I}$ such that $e \leq \bigvee\_{i \in I}d\_i$, there exists $i \in I$ such that $d \leq d\_i$. For example, if $\mathcal{O}(X)$ is the lattice of open sets of a topological space $X$, then if $U,V$ are open sets, and $K \subseteq X$ is a compact subspace such that $U \subseteq K \subseteq V$, then $U \ll V$ in $\mathcal{O}(X)$. 


For each $d \in D$ in a dcpo, define 
$$
\mathbf{waydown}(d) = \{ e \in D \mid e \ll d \}.
$$
This is usually written like $\downarrow d$ but with a double arrowhead, but I couldn't get this symbol on MathOverflow.


A dcpo $D$ is said to be *continuous* if for all $d \in D$ the set $\mathbf{waydown}(d)$ is directed and $d = \bigvee\mathbf{waydown}(d)$.


We can now define
$$
\mathbf{wayup}(d) = \{e \in D \mid d \ll e \}
$$
In a continuous dcpo $D$, each $\mathbf{wayup}(d)$ is Scott open, and these sets form a base for the Scott topology on $D$. 


Additionally, there is the notion of a *basis for a dcpo*. This is a set $B \subseteq D$ such that for all $d \in D$, $B \cap \mathbf{waydown}(d)$ is directed and has supremum $d$. If $B$ is a basis for $D$, then $\{\mathbf{wayup}(b)\}\_{b \in B}$ is a base for the Scott topology of $D$.


You can find more information on this sort of thing in the book *Continuous Lattices and Domains* by Gierz, Hoffman, Keimel, Lawson, Mislove and Scott, particularly section III-4.


