Given a set X, equation image denotes the statement: “equation image has a choice set” and equation image denotes the family of all closed subsets of the topological space equation image whose definition depends on a finite subset of X. We study the interrelations between the statements equation image equation image equation image equation image and “equation imagehas a choice set”. We show: equation image iff equation image iff equation image has a choice set iff equation image. equation image iff (...) for every set X, equation image has a choice set. equation image does not imply “equation image has a choice set equation image implies equation image but equation image does not imply equation image.We also show that “For every setX, “equation imagehas a choice set” iff “for every setX, equation imagehas a choice set” iff “for every productequation imageof finite discrete spaces,equation image has a choice set”. (shrink)