Amenable actions and exactness for discrete groups

It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Cech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups.Comment: Added remarks and a reference. Submitted to Comptes Rendus Acad. Sci. Pari

Independent families in Boolean algebras with some separation properties

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed

Syndetic Sets and Amenability

We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak*-closed convex hull of the two-sided translates of every bounded function on an infinite discrete semigroup contains a constant function. Our proofs use the algebraic properties of the Stone-Cech compactification

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

Reduced Coproducts of Compact Hausdorff Spaces

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the reduced coproduct , which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the ultracoproduct can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces