# Basis or subbasis for Scott topology

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.

• I suppose the "(as closed sets)" remark was copied from the previous question; here it should be removed, as you define the open sets :) – Wojowu Feb 8 '19 at 20:17
• @Wojowu Thanks, that was embarrassing. – geodude Feb 8 '19 at 20:23

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.