Open iff Upper and with Property (S) in Scott Topological Lattice

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \struct {S, \preceq, \tau}$ be an up-complete topological lattice.

Let $A$ be a subset of $S$.


Then $A$ is open if and only if $A$ is upper and with property (S).


Proof

Sufficient Condition

Let $A$ be open.

Thus by definition of Scott topology:

$A$ is an upper section.

Let $D$ be a directed subset of $S$ such that

$\sup D \in A$

By definition of Scott topology:

$A$ is inaccessible by directed suprema.

By definition of inaccessible by directed suprema:

$A \cap D \ne \O$

By definition of non-empty:

$\exists y: y \in A \cap D$

By definition of intersection:

$y \in A$ and $y \in D$

Thus $y \in D$.

Thus by definition of upper section:

$\forall x \in D: y \preceq x \implies x \in A$

$\Box$


Necessary Condition

Assume that

$A$ is upper and with property (S).

According to the definition of Scott topology it should be proved that

$A$ is upper and inaccessible by directed suprema.

Thus by assumption:

$A$ is upper.

Let $D$ be a directed subset of $S$ such that

$\sup D \in A$

By definition of property (S):

$\exists y \in D: \forall x \in D: y \preceq x \implies x \in A$

By definition of reflexivity:

$y \preceq y$

Then

$y \in A$

By definition of intersection:

$y \in A \cap D$

Thus by definition of non-empty:

$A \cap D \ne \O$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Open_iff_Upper_and_with_Property_(S)_in_Scott_Topological_Lattice&oldid=576792"