Stevo Todorcevic

Trees and Linearly Ordered Sets

Publisher Summary This chapter presents an introduction to several problems concerning trees and linearly ordered sets and the dualities between them. Several classes of trees and linearly ordered sets are presented and considered as set-theoretical, topological, and algebraical structures in the chapter. The chapter also discusses many fundamental problems concerning trees and linearly ordered sets, which are set-theoretical in nature. Most of the problems are undecidable on the basis of the usual axioms of set theory. Thus, a great number of results presented in the chapter are consistency results.

Forcing positive partition relations

We show how to force two strong positive partition relations on u, and use them in considering several well-known open problems. In [32] Sierpiñski proved that the well-known Ramsey Theorem [27] does not generalize to the first uncountable cardinal by constructing a partition [ío,]2 = KQ U Kx with no uncountable homogeneous sets. Sierpinskis partition has been analyzed in several directions. One direction was to improve this relation so as to get much stronger negative partition relations on ux. The direction taken in this paper is to prove stronger and stronger positive relations on <*>, which do not appear to be refutable by Sierpinskis partition. The first result of this kind is due to Dushnik and Miller [9] who proved

Compact subsets of the first Baire class

Perhaps the earliest results about pointwise compact sets of Baire class-1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire class-1 functions known today as Helly space, the space of all nondecreasing functions from the unit interval I = [0,1] into itself. More recently, the notion of Baire class-1 function turned out to also be important in some areas of functional analysis (see [R3]). For example, Odell and Rosenthal [OR] showed that the double dual of a separable Banach space E with the weak* topology consists only of Baire class-1 functions defined on the unit ball of E* if and only if the space E contains no subspace isomorphic to ?1. This resulted in a renewed interest in this class of spaces. For example, building on the work of Rosenthal [R2], Bourgain, FYemlin and Talagrand [BFT] proved analogues of the two theorems of Helly for the whole first Baire class. Using their results Godefroy [Go] showed that this class of spaces enjoys some interesting permanence properties. For example, if a compact space K is representable as a compact set of Baire class-1 functions, then so is P(K), the space of all Radon probability measures on K with the weak* topology. Some further permanence properties of this class of spaces were obtained by Marciszewski ([Ml], [M2]) and an excellent survey of the early results is given by R. Pol [Po2]. Our paper is an attempt towards a fine structure theory of compact subsets of first Baire class. The first result that we give is a positive answer to a natural question one usually asks in such a context.

Katetov’s problem

In 1948 Miroslav Katetov showed that if the cube X 3 of a compact space X satisfies the separation axiom T 5 then X must be metrizable. He asked whether X 3 can be replaced by X 2 in this metrization result. In this note we prove the consistency of this implication.

Chain-condition methods in topology

Abstract The special role of countability in topology has been recognized and commented upon very early in the development of the subject. For example, especially striking and insightful comments in this regard can be found already in some works of Weil and Tukey from the 1930s (see, e.g., Weil (1938) and Tukey (1940, p. 83)). In this paper we try to expose the chain condition method as a powerful tool in studying this role of countability in topology. We survey basic countability requirements starting from the weakest one which originated with the famous problem of Souslin (1920) and going towards the strongest ones, the separability and metrizability conditions. We have tried to expose the rather wide range of places where the method is relevant as well as some unifying features of the method.

A new Löwenheim-Skolem theorem

This paper establishes a refinement of the classical Lowenheim-Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this.

A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This extends the Pudlak-Rodl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U1: Every ultrafilter which is Tukey reducible to U1 is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show that there is exactly one Tukey type of nonprincipal ultrafilters strictly below that of U1, namely the Tukey type of a Ramsey ultrafilter.

Combinatorial dichotomies in set theory

We give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further research.

Cellularity of covariant functors

Abstract We compute the cellularity of F ( X ) in terms of the cellularity of X for a class of covariant functors F including exp, λ and P . We also give a number of examples to test the sharpness of our results.

Cofinal types of ultrafilters

Abstract We study Tukey types of ultrafilters on ω , focusing on the question of when Tukey reducibility is equivalent to Rudin–Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ ω 1 ] ω . We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.