Stylianos Negrepontis

Spaces Homeomorphic to (2 α ) α

We begin this section with an abstract study, independent of previous material, of the spaces (2 α ) α for regular infinite α. These Cantor-like spaces have a canonical structure and their characterization with an economical list of properties (Theorem 15.3) suggests the possibility of their frequent “natural” occurrence. The property of ℬ-compactness in the characterization theorem suggests that if X is a compact space satisfying certain natural cardinality conditions then the associated space X α is homeomorphic to (2 α ) α . That this is indeed the case is a simple and particularly useful result (Theorem 15.9). A special case occurs as follows: assuming that 2 α is a regular cardinal and that no uniform ultrafilter on α has a filter base of cardinality less than 2 α , there is a homeomorphism between U(α)2α and (2(2α))2α (Theorem 15.14(a)). This homeomorphism forms the crucial link relating the general results about the spaces (2 α ) α considered thus far to the study of ultrafilters. It implies that the topological properties of (2(2α))2α may be studied and applied profitably to the space U(α)2α.

Topology of Spaces of Ultrafilters

The object of this section is the discussion and organization of the properties of the topology of certain subspaces of the Stone-Cech compactification of a cardinal. Stronger topologies on the space of uniform ultrafilters and their applications will be studied in § 15, while the characterization of the Gleason space of the space of uniform ultrafilters is already given in § 12.

Basic Facts on Ultrafilters

With this introductory section we begin in earnest our study of ultrafilters. Some basic definitions already mentioned are collected here once more for emphasis. In §2, z-filters on spaces and filters on Boolean algebras were defined and considered; from now on we deal almost exclusively with filters on cardinals.

Saturation of Ultraproducts

This section studies the two questions, which are related as it turns out, of the saturation properties of ultraproducts and of the characterization of elementary equivalence in terms of ultrapowers.

The Jónsson Class of Ordered Sets

This is the principal example of a Jonsson class. Arguments such as the back-and-forth induction of Cantor and Hausdorff occur here in their initial and prototypical form (Theorem 5.3). There is also a useful analogy between the simple arguments concerning the η α -property of ordered sets and the more technically detailed arguments involving the (H α ) and (R α ) properties of Boolean algebras studied in §6 below; in this sense the material of the present section is preparatory to § 6.

The General Theory of Jónsson Classes

The theory of Jonsson classes is the setting of proper generality for the application of the classical principle of back-and-forth induction of Cantor and Hausdorff. Cantor conceived of this principle and used it to prove that the set of rational numbers is described as an ordered set by the following properties: it is countable, without a least or a greatest element, and densely ordered. Hausdorff generalized this result in his theory of η α -sets (studied in § 5 below). Erdős-Gillman-Henriksen proved similar results for the theory of real-closed fields.

Topology and Boolean Algebras

The material of this section is divided into four sub-sections dealing with Topology, the finitary properties of Boolean algebras, the duality of Stone between Boolean algebras and compact totally disconnected spaces, and the completion of a Boolean algebra and the (essentially dual) Gleason space of a compact space. Three spaces are involved: The Stone-Cech compactification of a space, the Stone space of a Boolean algebra, and the Stone space of the (complete) Boolean algebra of regular-open sets of a space. The definitions use either the topological version of an ultrafilter, namely a z-ultrafilter on a space, or the algebraic version of an ultrafilter, namely an ultrafilter of a Boolean algebra; these are (non-comparable) generalizations of the set-theoretic version of an ultrafilter, namely the ultrafilter on a cardinal number.

The Jónsson Class of Boolean Algebras

The general results of the theory of Jonsson classes, developed in §4, are applied here to the Jonsson class of (proper) Boolean algebras. The α-homogeneous-universal Boolean algebra ℭ α of cardinality a and its Stone space S α , which exist if and only if \(\alpha = {\alpha ^{\underset{\raise0.3em\hbox{

Intersection Systems and Families of Large Oscillation

\smash{\scriptscriptstyle\smile}

Families of Almost Disjoint Sets

}}{\alpha } }}\), are studied in considerable detail and characterized by properties given in terms of the partial order of a Boolean algebra; these properties are similar to the η α -property of ordered sets (studied in §5). We study the Jonsson class of Boolean algebras not mainly to illustrate the general theory of Jonsson classes, but rather in order to describe the space S α itself; in later sections some of its properties will be set in analogy or contrast to those of certain spaces of ultrafilters. There are intriguing parallels between some rather refined properties of S α and corresponding properties of the space U(α + ) (to be defined in § 7) of uniform ultrafilters on α+ the particular case α = ω+=2 ω the space S ω + indeed is homeomorphic to the space (of non-principal ultrafilters on ω) s(ω)\ ω (cf. the introduction and results of § 14 and § 15).