  The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice
$\mathsf{AC}_\omega$ are two weak forms of the Axiom of Choice that can be
stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary
relation on $X$ has an infinite chain, while $\mathsf{AC}_\omega (X)$ asserts
that any countable collection of nonempty subsets of $X$ has a choice function.
It is well-known that $\mathsf{DC} \Rightarrow \mathsf{AC}_\omega$. We study
for which sets and under which hypotheses $\mathsf{DC}(X) \Rightarrow
\mathsf{AC}_\omega (X)$, and then we show it is consistent with $\mathsf{ZF}$
that there is a set $A \subseteq \mathbb{R}$ for which $\mathsf{DC} (A)$ holds,
but $\mathsf{AC}_\omega (A)$ fails.
