We show that for every we ordered cardinal number m the Tychonoff product 2m is a compact space without the use of any choice but in Cohen's Second Mode 2ℝ is not compact.

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)

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.