Scott topology

The Scott topology is a topology that can be defined on any complete partial order (cpo).

Thus, the Scott topology is important for domain theory and the semantics of recursion. However, because the resultant space is a T_0 space, there is very little geometric intuition involved.

Definition

For \omega-cpos

Let (D, \sqsubseteq) be a \omega-cpo. We define O \subseteq D to be Scott-open whenever

1. x \in O \land x \sqsubseteq y \Longrightarrow y \in O
2. If (x_n)_{n \in \omega} is an increasing sequence in D with \sqcup_{n \in \omega} x_n \in O, then for some n, x_n \in O.

For dcpos

Let (D, \sqsubseteq) be a dcpo. We define O \subseteq D to be Scott-open whenever

1. x \in O \land x \sqsubseteq y \Longrightarrow y \in O
2. If D is a directed set and \sqcup D \in O, then D \cap O \neq \emptyset.

More synthetically

The nLab gives a more synthetic definition.

Let \Sigma be the Sierpiński space, i.e. the cpo

\Sigma \triangleq \{ \bot, \top \},
\quad
\bot \sqsubseteq \top

Let (D, \sqsubseteq) be a cpo, and O \subseteq D. The characteristic function \chi_O : D \to \Sigma is given by

\chi_O(x) \triangleq
	\begin{cases}
  		\top & \text{ if $x \in O$} \\
      \bot & \text{ if $x \not\in O$}
  \end{cases}

We then define O to be Scott-open whenever \chi_O is monotone and continuous with respect to the two cpo's.

Through closed sets

A topology can be equivalently described through its closed sets. We can then describe the Scott-topology through the Scott-closed sets.

From topology to order

We can recover the order from the topology, by defining

x \sqsubseteq y
  \quad\triangleq\quad
\forall \text{open sets } O.\; x \in O \Longrightarrow y \in O

References

• The formulation for \omega-cpo's is exercise 21 in the first chapter of

@techreport{plotkin_1983,
	title = {Domains ({Pisa} {Notes})},
	url = {https://homepages.inf.ed.ac.uk/gdp/publications/Domains_a4.ps},
	author = {Plotkin, Gordon},
	year = {1983},
}

• nLab page on Scott topology.