On theorems of Tychonoff, Alexander, and R. Rado

Abstract

It is well known [4], [7] that the Tychonoff theorem on products of compact spaces is equivalent to the axiom of choice (AC). It is therefore natural to consider Tychonoff's theorem for products of finite spaces as a restricted form of AC, and to ask whether there are other restricted versions of AC to which it is equivalent. In this note we shall show that, assuming the axiom of choice for families of finite sets (ACF), the following three theorems are all equivalent.