Here's Lemma 9.2 in Topology by James R. Munkres, 2nd edition:
Given a collection $\mathscr{B}$ of non-empty sets (not necessarily disjoint), there exists a function $$c \colon \mathscr{B} \to \bigcup_{B \in \mathscr{B}} B$$ such that $$c(B) \in B$$ for each $B \in \mathscr{B}$.
The function $c$ is called a choice function for the collection $\mathscr{B}$.
This lemma has been derived using the Axiom of Choice, which is as follows:
Given a collection $\mathscr{A}$ of disjoint non-empty sets, there exists a set $C$ consisting of exactly one element from each element of $\mathscr{A}$; that is, a set $C$ such that $C$ is contained in the union of the elements of $\mathscr{A}$, and for each $A \in \mathscr{A}$, the set $C \cap A$ contains a single element.
The set $C$ can be thought of as having been obtained by choosing one element from each of the sets in $\mathscr{A}$.
Now let $\mathscr{B}$ be the collection of all the non-empty subsets of the set $\{0,1\}^\omega$ of all the infinite sequences of $0$s and $1$s.
Can we explicitly find a choice function for this collection without using the Axiom of Choice?
I know that this is possible for each of the collections of all the non-empty subsets of $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$, respectively.
Please note that my knowledge of mathematical logic and set theory is very very limited. So I would like to have an answer to my question using only the concepts that Munkres has talked about in the first nine sections of his book.
The suggested questions are not formulated in this particular context.