\(n\)-Luzin gaps
type: Item
instance of: scholarly article
title: \(n\)-Luzin gaps
zbMATH Open document ID: 1283.03078
author: Michael Hrušák
author: Osvaldo Guzmán
published in: Topology and its Applications
publication date: 2014-01-08T00:00:00Z
reviewed by: Judith Roitman
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 6243378
zbMATH Keywords: AD family
zbMATH Keywords: Luzin gap
zbMATH Keywords: Mrówka-Isbell space
zbMATH Keywords: normal
MaRDI profile type: Publication
full work available at URL: j.topol.2013.05.015
OpenAlex ID: W2094623745
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DOI: 10.1016/J.TOPOL.2013.05.015
review text: This paper introduces the notion of \(n\)-Luzin gaps in subfamilies of \(\mathcal P(\omega)\), and explores its relation to (potential) separation properties of almost disjoint families and (potential) normality (of the Mrówka-Isbell space), where ``potential'' means: there is a ccc partial order which forces the family to have that property.NEWLINENEWLINEDefinition. For each \(i < \omega\) let \(\mathcal B_i = \{B^i_{\alpha}: \alpha < \omega_1\}\), where \(\bigcup_{i < \omega}B_i\) is almost disjoint (i.e., no two elements have infinite intersection). Then \(\{\mathcal B_i: i < n\}\) is an \(n\)-Luzin gap iff there is \(m \in \omega\) with {\parindent=6mm \begin{itemize}\item[1.] \(B^i_{\alpha} \cap B^j_{\alpha} \subseteq m\) for all \(i \neq j\), \(\alpha < \omega_1\) and \item[2.] \(\bigcup_{i \neq j}(B^i_{\alpha} \cap B^j_{\beta}) \not\subseteq m\) for \(\alpha \neq \beta< \omega_1\). NEWLINENEWLINE\end{itemize}} Given an almost disjoint family \(\mathcal A \subset \mathcal P(\omega)\), the Mrówka-Isbell space \(\Psi(\mathcal A)\) is defined as follows: {\parindent=6mm \begin{itemize}\item[--] as a set, \(\Psi(\mathcal A) = \omega \cup \mathcal A\), where \item[--] \(\omega\) is a discrete subset and \item[--] a basic open neighborhood of \(A \in \mathcal A\) has the form \(\{A\} \cup A \setminus F\), where \(F\) is a finite subset of \(\omega\). NEWLINENEWLINE\end{itemize}} Normality of \(\Psi(\mathcal A)\) essentially means that, given any \(\mathcal B \subset \mathcal A\) there is \(B \subseteq \omega\) so that \(\forall A \in \mathcal A \;A \subseteq^* B\) and \(\forall A \in \mathcal A \setminus \mathcal B \;A \cap B = \emptyset\). I.e., \(B\) separates \(\mathcal B \subseteq \mathcal A\) from \(\mathcal A \setminus \mathcal B\).NEWLINENEWLINEA strong negation of this property is:NEWLINENEWLINEDefinition. An almost disjoint family \(\mathcal A\) is inseparable if, for every uncountable disjoint \(\mathcal B, \mathcal C \subset \mathcal A\), there is no \(B\…
description: scientific article; zbMATH DE number 6243378