I am trying to construct a choice function on the open subset of $\mathbb{R}^2$ (with the topology induced by the euclidiean distance)
I cannot use maximum and minimum because if the set is open I might not have these. My first try is to find a choice function on the open sets of $\mathbb{R}$ to take the product of this choice function to have one on $\mathbb{R}^2$.
I know that the open sets of $\mathbb{R}$ are the union of open interval so if $I=]a,b[$ then $i=\frac{sup(I)+inf(I)}{2} \in I$ so I have a choice function on any non-empty interval but I do not know how to generalize to an explicit choice function on the non-connected open set. Is it possible to construct one ? (I know that it is not possible to construct explicitly a choice function on the borelian).