Weak axioms of choice for metric spaces

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space $(X,d)$ has a choice function, then so does the family of all non-empty, open subsets of $X$. In addition, we establish that the converse is not provable in ZF.

We also show that the statement ``every subspace of the real line $\mathbb{R}$ with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form ``every continuum sized family of non-empty subsets of reals has a choice function".