Uniform Ultrafilters Research Articles

Spaces homeomorphic to (2^{𝑎})ₐ. II

Topological characterizations and properties of the spaces ( 2 α ) α {({2^\alpha })_\alpha } , where α \alpha is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Čech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces ( 2 α ) α {({2^\alpha })_\alpha } in terms of a simple set of conditions, and (b) the α \alpha -Baire category property of ( 2 α ) α {({2^\alpha })_\alpha } and the stability of the class of spaces homeomorphic to ( 2 α ) α {({2^\alpha })_\alpha } (or to ( α α ) α {({\alpha ^\alpha })_\alpha } ) when taking intersections of at most α \alpha open and dense subsets of ( 2 α ) α {({2^\alpha })_\alpha } . Among the applications of these results are the following. Assuming α + = 2 α {\alpha ^ + } = {2^\alpha } , the class of spaces homeomorphic to ( 2 ( α + ) ) α + {({2^{({\alpha ^ + })}})_{{\alpha ^ + }}} includes the space of uniform ultrafilters on α \alpha with the P α + {P_{{\alpha ^ + }}} -topology ( U ( α ) ) α + {(U(\alpha ))_{{\alpha ^ + }}} , its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming ω + = 2 ω {\omega ^ + } = {2^\omega } (or in some cases only Martin’s axiom), the class of spaces homeomorphic to ( 2 ( ω + ) ) ω + {({2^{({\omega ^ + })}})_{{\omega ^ + }}} includes the following: The space ( β X ∖ X ) ω + {(\beta X\backslash X)_{{\omega ^ + }}} where X is a noncompact locally compact realcompact space such that | C ( X ) | ≤ 2 ω |C(X)| \leq {2^\omega } and its subspaces of P ω + {P_{{\omega ^ + }}} -points of β X ∖ X \beta X\backslash X and/or (if X is in addition a metric space without isolated elements) the remote points. In particular the existence of good and/or Rudin-Keisler minimal ultrafilters and the existence of P-points and/or remote points follows always from a Baire category type of argument.

The growth of subuniform ultrafilters

Some of the results on the topology of spaces of uniform ultrafilters are applied to the space $\Omega ({\alpha ^ + })$ of subuniform ultrafilters (i.e., the set of ultrafilters which are $\alpha$-uniform but not ${\alpha ^ + }$-uniform) on ${\alpha ^ + }$ when $\alpha$ is a regular cardinal. The main object is to find for infinite cardinals $\alpha$, such that $\alpha = {\alpha ^{\underline {a}}}$, a topological property that separates the space $\beta (\Omega ({\alpha ^ + }))\backslash \Omega ({\alpha ^ + })$ (

On nonregular ultrafilters

In this paper we shall construct nonregular ultrafilters showing many of the model-theoretic properties of their regular counterparts. The crucial idea in these constructions is to replace the use of regularity by independent functions. We shall use the notation and terminology of [1], our fundamental concepts being defined as follows:Definition 1.1. (1) A uniform ultrafilter D over a cardinal κ is regular if there is a family {Xα ∣ α < κ} so that every infinite intersection of these Xα's is empty.(2) A filter F over a cardinal κ is ω1-saturated if there is no family of sets {aα ∣ α < ω1} so that for every α < β < ω1For more on regular ultrafilters, see [2]. The germinal theorem on the subject of nonregular ultrafilters is the following well-known result of Jack Silver:Theorem 1.2 [1, Theorem 1.39]. If F is an ω1-saturated κ-complete filter over κ, then any ultrafilter extending F is nonregular.This result will be the cornerstone of our constructions. Of course, the existence of a filter of the above type depends on large cardinal axioms. For more on un ω1-saturated κ-complete filters, see [1].

The cardinality of ultrapowers—an example

Assume the axiom of measurable cardinals. If D D is an ω \omega -incomplete uniform ultrafilter on I I , and A A is infinite, it is still not necessarily the case that A I / D {A^I}/D has the same cardinality as A I {A^I} .