Given a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact (...) case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide. (shrink)

We study classes of ultrafilters on ω defined by a natural property of the Loeb measure in the Nonstandard Universe corresponding to the ultrafilter. This class, the Property M ultrafilters, is shown to contain all ultrafilters built up by taking iterated products over collections of pairwise nonisomorphic selective ultrafilters. Results on Property M ultrafilters are applied to the construction of extensions of probability measures, and to the study of measurable reductions between ultrafilters.

We study relations between measure-theoretic classes of ultrafilters, such as the Property M ultrafilters of [4], with other well-known ultrafilter classes. We define several classes of measure theoretic ultrafilters, of which the Property M ultrafilters are the strongest. We show which containments are provable in ZFC between these measure-theoretic ultrafilters and boolean combinations of well-known ultrafilters such as the selective, semi-selective, and P-point ultrafilters. We also list some of the containment results between measure-theoretic ultrafilters and several other ultrafilter classes, such (...) as the Arrow and Property C ultrafilters. (shrink)

We prove various results on the notion of ordinal ultrafilters introduced by J. Baumgartner. In particular, we show that this notion of ultrafilter complexity is independent of the more familiar Rudin-Keisler ordering.

A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base (...) of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\). (shrink)

On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.

We study the I-ultrafilters on ω, where I is a collection of subsets of a set X, usually R or ω 1 . The I-ultrafilters usually contain the P-points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist.

We study ultrafilters produced by forcing, obtaining different combinatorics and related Rudin-Keisler ordering; in particular we answer a question of Baumgartner and Taylor regarding tensor products of ultrafilters. Adapting a method of Blass and Mathias, we show that in most cases the combinatorics satisfied by the ultrafilters recapture the forcing notion in the Lévy model.

In this article we, given a free ultrafilter p on ω, consider the following classes of ultrafilters:(1) T(p) - the set of ultrafilters Rudin-Keisler equivalent to p,(2) S(p)={q ∈ ω*:∃ f ∈ ω ω , strictly increasing, such that q=f β (p)},(3) I(p) - the set of strong Rudin-Blass predecessors of p,(4) R(p) - the set of ultrafilters equivalent to p in the strong Rudin-Blass order,(5) P RB (p) - the set of Rudin-Blass predecessors of p, and(6) P RK (p) (...) - the set of Rudin-Keisler predecessors of p,and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p ∈ ω* for which P RB (p)=P RK (p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its α-th left power α p is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order ⊴ in T(p) by r⊴q if and only if r ∈ S(q). We prove that (S(p),⊴) is always downwards directed, (R(p),⊴) is always downwards and upwards directed, and (T(p),⊴) is linear if and only if p is selective.We also characterize rapid ultrafilters as those ultrafilters p ∈ ω* for which R(p)∖S(p) is a dense subset of ω*.A space X is M-pseudocompact (for ) if for every sequence (U n ) n < ω of disjoint open subsets of X, there are q ∈ M and x ∈ X such that x=q-lim (U n ); that is, for every neighborhood V of x. The P RK (p)-pseudocompact spaces were studied in [ST].In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P RB (p) and P RK (p). We prove that every Frolik space is S(p)-pseudocompact for every p ∈ ω*, and determine when a subspace with is M-pseudocompact. (shrink)

In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.

The class of 2-square ultrafilters on ω equals the union, for n ≥ 1, of the classes of strictly n Ramsey ultrafilters on ω.

We discuss saturating ultrafilters on N, relating them to other types of nonprincipal ultrafilter. (a) There is an (ω,c)-saturating ultrafilter on $\mathbf{N} \operatorname{iff} 2^\lambda \leq \mathfrak{c}$ for every $\lambda and there is no cover of R by fewer than c nowhere dense sets. (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafilter on N is (ω,κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω,κ)-saturating, and (ii) the set of (ω,κ)-saturating ultrafilters is invariant under (...) homeomorphisms of $\beta\mathbf{N\backslash N}$ . (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω,c)-saturating. (shrink)

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω , using ultrafilters on a countable index set, which we also assume to be ω . Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have no maximal element, (...) and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors. (shrink)