PRODUCTS OF COMPACT SPACES AND THE AXIOM OF CHOICE Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis, and Jean E. Rubin August 22, 2001 Abstract. We study the Tychonoff Compactness Theorem for several different definitions of a compact space. 1. Introduction and Definitions In this paper we study products of compact spaces. The Tychonoff Compactness Theo- rem, the product of compact spaces is compact, is one of the oldest such theorems. In 1930, Tychonoff [ty] gave a proof of this product theorem for the closed unit interval [0, 1], but his proof was easily generalized to arbitrary compact spaces, (see [c]). In 1950, Kelley [k] proved that the Tychonoff Compactness Theorem implies the axiom of choice, AC. (The Tychonoff Compactness Theorem is P(A,A) below.) In the following definition, we define the product topology by defining its basis. Then the topology itself is the set of all unions of basis elements. Product Topology. Let {X i : i ∈ k} be a set of topological spaces and let X =  i∈k X i be their product. A basis for the product topology on X is all sets of the form  i∈k U i where U i is open in X i and for all but a finite number of coordinates, U i = X i . We shall use the forms of compactness which were defined in [dhhrs]. (The reader is referred to [dhhrs] for the reasons we chose these forms of compactness and for background information about them.) We list these definitions below. Our proofs, unless stated otherwise, will be in ZF. Definitions:. 1. A space is compact A if every open covering has a finite subcovering. (Equivalently, every filter has an accumulation point.) 2. A space is compact B if every ultrafilter has an accumulation point. 3. A space is compact C if there is a subbase for the topology such that every open covering by elements of the subbase has a finite subcovering. 1991 Mathematics Subject Classification. 03E25, 04A25, 54D30.