Date of Award
Bachelor of Science
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's Infinite Product Theorem, which people were stuck on before the axiom of choice was applied. Tychonoff's 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's Theorem. The reverse direction of this proof was first presented by Kelley in 1950; however, it was slightly awed. We will go over Kelley's 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's 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.
Jia, Ruoxuan, "The Axiom of Choice in Topology" (2017). Honors Theses. 46.