Oscillations of sets of integers
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 \(f: [X]^p \rightarrow {\mathbb N}\) so that \(\forall A \subset X\) the length of the Cantor-Bendixson sequence on \(A\) can be read off from \(f[A^p]\)? This section relies on the results of section 3.
scientific article; zbMATH DE number 1221621
scientific article