Axiom of countable choice in the context of Countably infinite

The axiom of countable choice or axiom of denumerable choice, denoted AC_ω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ${\displaystyle A}$ with domain ${\displaystyle \mathbb {N} }$ (where ${\displaystyle \mathbb {N} }$ denotes the set of natural numbers) such that ${\displaystyle A(n)}$ is a non-empty set for every ${\displaystyle n\in \mathbb {N} }$, there exists a function ${\displaystyle f}$ with domain ${\displaystyle \mathbb {N} }$ such that ${\displaystyle f(n)\in A(n)}$ for every ${\displaystyle n\in \mathbb {N} }$.