Example. Compact elements with infinitely many approximations [dt-003X]

Liam O'Connor

Compact elements may still have an infinite number of approximations. Consider the cpo $\mathbb {N} \cup \{ \infty , \infty + 1 \}$ , where we have tacked on an additional top element $\infty + 1$ to our normal cpo of natural numbers extended with infinity. Then, while $\infty$ is not compact, $\infty + 1$ is compact — it is not the lub of an interesting directed set. If $\infty + 1$ is the lub of a set $X$ then $\infty + 1$ must be in the set $X$ . Nonetheless, there are an infinite number of approximations to $\infty + 1$ , i.e., elements $x$ such that $x \sqsubseteq \infty + 1$ .

As an aside, requiring that our compact elements have a truly finite number of approximations is the basis for the theory of Berry domains and stable functions.

Example. Compact elements with infinitely many approximations [dt-003X]

Definition. Boring and interesting sets [dt-003W]

Definition. Compactness [dt-003U]

Example. Compact elements [dt-003V]

Example. Compact elements with infinitely many approximations [dt-003X]

Theorem. Submonoids as a cpo [dt-003Y]

Exercise. Both infinite and compact [dt-003Z]

Solution.