**Axiom of Countable Choice (CC)** states that for every countable family $\left\{A\_i\right\}\_{i=1}^\infty$ of nonempty sets there exists *choice function* $f \colon \mathbb{N} \to \bigcup\_{i=1}^\infty A\_i$ such that $f(i) \in A\_i$ for all $i \in \mathbb{N}$.


Consider the following axiom strongly related to the above axiom:


For every countably infinite family $\left\{A\_i\right\}\_{i=1}^\infty$ of nonempty sets there exists countably infinite subfamily $\left\{A\_{i\_k}\right\}\_{k=1}^\infty$ such that $\left\{A\_{i\_k}\right\}\_{k=1}^\infty$ has *choice function* i.e. there exists $f \colon \mathbb{N} \to \bigcup\_{k=1}^\infty A\_{i\_k}$ such that $f(k) \in A\_{i\_k}$ for all $k \in \mathbb{N}$.


Let's call it **Axiom of Countable Choice for Countable Subfamily (CCCS)**.


Of course **CC** implies **CCCS**. But I wonder if **CCCS** is strongly weaker than **CC** or equivalent to **CC**.


I can't imagine being the first person to come up with the idea of **CCCS**, so probably there is literature which consider this kind of *choice axiom* or maybe it's unnaturally to consider such an axiom. I'm amateur in that sort of set theory. What do you think? Maybe someone knows proper literature?



Countable choice for countable subfamily
They are equivalent. Given a countable family $B\_i$ of nonempty sets, let $A\_i=\prod\_{j=1}^iB\_j$. That these are nonempty is provable in ZF (we only use finite products). Now for any choice function $f$ with $f(k)\in A\_{i\_k}$, since $i\_k\geq k$, projecting $f(k)$ onto the $k$-th coordinate of $A\_{i\_k}=\prod\_{j=1}^{i\_k}B\_j$ gives a choice function for the $B\_i$.


I don't know a reference for this result, but I have seen similar arguments when establishing various equivalents of countable choice for subsets of $\mathbb R$, so I can't take full credit for the idea.


