The countable-choice principle used in the foliation pair

The countable choice principle used throughout this pair, written ${\mathrm{A}\mathrm{C}}_{\omega }$, is the assertion:

For every sequence $(A_n{)}_{n\in \mathbb{N}}$ of nonempty sets there is a function $c$ with domain $\mathbb{N}$ such that $c(n)\in A_n$ for every $n\in \mathbb{N}$.

Here "sequence" means a function on $\mathbb{N}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain). We call the displayed selector $c$ an indexed choice function for the sequence. Its domain is the index set $\mathbb{N}$, whereas a choice function in Choice function has as domain the family of sets themselves. These notions must be distinguished when factors repeat. A family choice function $g$ on $\{A_n:n\in \mathbb{N}\}$ gives an indexed selector by $c(n)=g(A_n)$; the equivalence of the two existence assertions is explained in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

This is the sequence formulation of countable choice. The equivalent nonempty-product formulation, that ${\prod }_{n\in \mathbb{N}}{A}_n$ is nonempty for every sequence $(A_n{)}_{n\in \mathbb{N}}$ of nonempty sets, is verified in Countable choice is equivalent to nonempty countable products ↗; that lemma is recorded as the well-definedness certificate of the present definition.

In this pair ${\mathrm{A}\mathrm{C}}_{\omega }$ is a stated hypothesis of the foliation theorems and of the items whose proof selects countably many plaque data. It is not assumed where a proof does not use it, and the items that consume it state the hypothesis explicitly.

This pair-local carrier is retained deliberately: it restates the published The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) in exactly the sequence form consumed by the foliation items, and the published definition supplies the same principle. The retention and its cross-batch consequences are recorded in the Step-3 report of this pair (finding F4).