Oscillations of sets of integers
type: Item
instance of: scholarly article
title: Oscillations of sets of integers
zbMATH Open document ID: 0929.03049
published in: Advances in Applied Mathematics
publication date: 2000-01-11T00:00:00Z
review text: This very rich paper extends the combinatorial notion of oscillation from the author's previous introduction of it on \({\mathbb N}^{\mathbb N}\) to \({\mathcal P}({\mathbb N})\). Applications are then given to filters (as spaces) and to the Cantor-Bendixson derivative. Consider this review a necessarily inadequate summary; there is much more in this paper than a review can encapsulate. First, some definitions: Let \(a, b \subset {\mathbb N}, n < m \in {\mathbb N}\). Define \(n \sim_{a,b} m\) iff either (1) \(n, m \in a \setminus b\) and the closed interval \([n,m] \cap b \setminus a = \emptyset\), or (2) \(n, m \in b \setminus a\) and the closed interval \([n,m] \cap a \setminus b = \emptyset\). Oscillation is defined by: osc\((a,b) = | a \Delta b/\sim_{a,b}| \). Which \(X \subset \mathcal P(\omega)\) satisfy: \(\forall i \in {\mathbb N}\) \(\exists a,b \in X \) osc\((a,b) = i\)? Section 2 takes up this and related questions. Section 3 turns its attention to \(X \subset [\omega]^{<\omega}\) and answers bounded versions of the question in section 2. E.g., for certain \(X\) you can guarantee that \(\forall i \in [2, 2k] \;\exists a,b \in X\) osc\((a,b) = i\). Section 4 exploits the results of section 2, and the easy fact that osc\((a,b) = 1\) iff either \(a \subset b\) or \(b \subset a\), to explore and exploit the topologies of filters as subspaces of \([\omega]^{\omega}\) under the topology exp\(({\mathbb N})\) (also called the Ellentuck topology. For example, certain filters \(F\) are generated by \(L\)-spaces \(A \subset \exp({\mathbb N})\). For another example, a selective ultrafilter contains a Luzin subset (in exp\(({\mathbb N}))\) iff it is generated by a set of size \(\omega_1\). Finally, section 5 examines spaces \(X\) for which it is possible to predict the Cantor-Bendixson sequence of derivatives of a set \(A \subset X\). In particular, for regular or \(\sigma\)-discrete metric spaces \(X\), when is there \(p\) and a continuous function \(f: [X]^p \righta…
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 1221621
zbMATH Keywords: oscillation
zbMATH Keywords: \(\exp(\mathbb N)\)
zbMATH Keywords: filters
zbMATH Keywords: Cantor-Bendixson derivative
zbMATH Keywords: Ellentuck topology
zbMATH Keywords: selective ultrafilter
zbMATH Keywords: metric spaces
author: Stevo Todorčević
reviewed by: Judith Roitman
MaRDI profile type: Publication
full work available at URL: aama.1997.0572
OpenAlex ID: W2046660375
cites work: Partition relations for countable topological spaces
cites work: Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions
cites work: Q3791663
cites work: Q5186348
cites work: L-spaces and S-spaces in \(P(\omega)\)
cites work: Borel sets and Ramsey's theorem
cites work: On the existence of large p-ideals
cites work: Q4171268
cites work: Q5541832
cites work: Q4069502
cites work: Happy families
cites work: Compacts de fonctions mesurables et filtres non mesurables
cites work: Real functions on the family of all well-ordered subsets of a partially ordered set
cites work: Q3778759
cites work: Partition Problems in Topology
cites work: Some partitions of three-dimensional combinatorial cubes
cites work: Combinatorial set theory
DOI: 10.1006/AAMA.1997.0572
description: scientific article; zbMATH DE number 1221621