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

A subjective spin on roulette wheels.

We provide a behavioral foundation to the notion of ‘mixture’ of acts, which is used to great advantage in he decision setting introduced by Anscombe and Aumann. Our construction allows one to formulate mixture-space axioms even in a fully sub-jective setting, without assuming the existence of randomizing devices. This simplifies the task of developing axiomatic models which only use behavioral data. Moreover, it is immune from the difficulty that agents may ‘distort’ the probabilities associated with randomizing devices. For illustration, we present simple subjective axiomatizations of some models of choice under uncertainty, including the maxmin expected utility model of Gilboa and Schmeidler, and Bewley’s model of choice with incomplete preferences.

Risk, ambiguity, and the separation of utility and beliefs.

We introduce a general model of static choice under uncertainty, arguably the weakest model achieving a separation of cardinal utility and a unique representation of beliefs. Most of the non-expected utility models existing in the literature are special cases of it. Such separation is motivated by the view that tastes are constant, whereas beliefs change with new information. The model has a simple and natural axiomatization. Elsewhere (forthcoming) we show that it can be very helpful in the characterization of a notion of ambiguity aversion, as separating utility and beliefs allows to identify and remove aspects of risk attitude from the decision maker’s behavior. Here we show that the model allows to generalize several results on the characterization of risk aversion in betting behavior. These generalizations are of independent interest, as they show that some traditional results for subjective expected utility preferences can be formulated only in terms of binary acts.

Nash Equilibrium and Axiom of Choice Are Equivalent

In this paper, I prove that existence of pure-strategy Nash equilibrium in games with infinitely many players is equivalent to the axiom of choice.Comment: 4 page

Limits in Function Spaces and Compact Groups

If B is an infinite subset of omega and X is a topological group, let C^X_B be the set of all x in X such that converges to 1. If F is a filter of infinite sets, let D^X_F be the union of all the C^X_B for B in F. The C^X_B and D^X_F are subgroups of X when X is abelian. In the circle group T, it is known that C^X_B always has measure 0. We show that there is a filter F such that D^T_F has measure 0 but is not contained in any C^X_B. There is another filter G such that D^X_G = T. We also describe the relationship between D^T_F and the D^X_F for arbitrary compact groups X.Comment: 16 page