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.

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


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