The Axiom of Choice in Topology

Cantor believed that properties holding for finite sets might also hold for infinite sets. One such property involves choices; the Axiom of Choice states that we can always form a set by choosing one element from each set in a collection of pairwise disjoint non-empty sets. Since its introduction in 1904, this seemingly simple statement has been somewhat controversial because it is magically powerful in mathematics in general and topology in particular. In this paper, we will discuss some essential concepts in topology such as compactness and continuity, how special topologies such as the product topology and compactification are defined, and we will introduce machinery such as filters and ultrafilters. Most importantly, we will see how the Axiom of Choice impacts topology. Most significantly, the Axiom of choice in set theory is the foundation on which rests Tychonoff\u27s Infinite Product Theorem, which people were stuck on before the axiom of choice was applied. Tychonoff\u27s Theorem asserts that the product of any collection of compact topological spaces is compact. We will present proofs showing that the Axiom of Choice is, in fact, equivalent to Tychonoff\u27s Theorem. The reverse direction of this proof was first presented by Kelley in 1950; however, it was slightly awed. We will go over Kelley\u27s initial proof and we will give the correction to his proof. Also, we introduce the Boolean Prime Ideal Theorem (a weaker version of the Axiom of Choice), which is equivalent to Tychonoff\u27s Theorem for Hausdorff spaces. Finally, we will look at an interesting topological consequences of the Axiom of Choice: the Stone-Cech Compactification. We will see how the Stone-Cech Compactification is constructed from ultrafilters, whose existence depends on the Axiom of Choice

Countable Choice and Compactness

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F

The axiom of choice and the paradoxes of the sphere.

Thesis (M.A.)--Boston UniversityThe Axiom of Choice is stated in the following form: For every set Z whose elements are sets A, non-empty and mutually disjoint, there exists at least one set B having one and only one element from each of the sets A belonging to Z. Examples are given to show the use of the Axiom of Choice and also to show when it is not needed. Two other fundamental terms are defined, namely "congruence" and "equivalence by finite decomposition", and examples are given. Congruence is defined as follows: The sets of points A and B are congruent: A B, if there exists a function f, which transforms A into B in a one-to-one manner such that if a1 and a2 are two arbitrary points of the set A, then d(a1, a2)=d[f(a1), f(a2)]; d(a, b) is a real number called the distance between the points a and b. The following definition of equivalence by finite decomposition is given: Two sets of points, A and B are equivalent by finite decomposition, Af=B, provided sets A1 , A2, ..., An and B1, B2, ..., Bn exist with the following properties: (1) A=A1+A2+...+An B=B1+B2+...+Bn (2) Aj • Ak=Bj • Bk=0 1 ≤ j < k ≤ n (3) Aj≅Bj 1 ≤ j ≤ n An historic measure problem is discussed briefly. Two paradoxes of the sphere, the Hausdorff Paradox and the Banach and Tarski Paradox are stated and discussed in detail. The Hausdorff Paradox reads as follows: The surface K of the sphere can be decomposed into four disjoint subsets A, B, C, and Q such that (1) K=A+B+C+Q and (2) A≅B≅C, A≅B+C where Q is denumerable. A refinement of this Paradox is introduced in which the denumerable set Q is eliminated. The Banach and Tarski Paradox states that in any Euclidean space of dimension n≥3, two arbitrary sets, bounded and containing interior points, are equivalent by finite decomposition. Various refinements of this paradox are noted. It is observed that the proofs of both paradoxes require the aid of the Axiom of Choice. The controversy over the Axiom of Choice is discussed at length. A wide range of viewpoints is studied, ranging from total rejection by the intuitionists to practically complete acceptance of the axiom. Seven theorems on cardinal numbers that are equivalent to the Axiom of Choice are listed. Six examples of theorems which require the aid of the Axiom of Choice in their proof are given. Based on the results of Hausdorff, Banach and Tarski, and Robinson, three specific questions are answered as follows : with the aid of the Axiom of Choice (1) the surface of a sphere can be decomposed into subsets in such a way that a half and a third of the surface may be congruent to each other. (2) A solid sphere of fixed radius can be decomposed into a finite number of pieces and these pieces can be reassembled to form two solid spheres of the given radius. (3) The minimum number of pieces required in the above problem is five. It is concluded that the general question, "Should the Axiom of Choice be accepted or rejected" is unanswerable at the present time. It is pointed out that the problem of existence and t he paradoxes that result from the axiom are major arguments against its use. However, the axiom simplifies many parts of set theory, analysis, and topology. The fact that Godel has proved the Axiom of Choice consistent with other generally accepted axioms of set theory, provided they are consistent with one another, is a second major point in its favor. Finally, Appendix I contains some statements equivalent to the Axiom of Choice, and Appendix II contains some importru1t theoren1s of Banach and Tarski

Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

On the roles of variants of Axiom of Choice in variations of Tychonoff Theorem (Set-Theoretic and Geometric Topology, and their applications to related fields)

An updated and extended version of this paper with more details and proofs is downloadable.In this purely expository note, we examine the roles of Axiom of Choice and its weak variants in topology with emphasis on their connections with Tychonoff Theorem and its variations