First we will state a Teichmüller-Tukey Lemma:
Let $A$ be a set and $\phi$ be a property defined on all finite subset of $A$. Assume that $B$ is a subset of $A$ such that each finite subset of $B$ have property $\phi$. Then $B$ can be extended to a maximal subset $M$ of $A$ such that each finite subset of $M$ has a property $\phi$.
My question is :- In the statement there is condition of finite. In case we have countable why we could not show that its equivalent with the axiom of choice.
The problem is that when you try to build a maximal subset by transfinite induction, the process breaks down at limit stages. An obvious counterexample would be: Let $A$ be the set of positive real numbers. For a countable subset $X$ of $A$, define $\phi$ by: $X$ satisfies $\phi$ iff the sum of all members of $X$ is finite.