\(n\)-Luzin gaps
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 \(B\) separating \(\mathcal B\) from \(\mathcal C\).NEWLINENEWLINETwo major theorems of this paper are:NEWLINENEWLINETheorem. Under MA, if \(\mathcal A\) is an almost disjoint subfamily of \(\mathcal P(\omega)\), then \(\Psi(\mathcal A)\) is normal iff \(|\mathcal A| < \mathfrak c\) and \(\mathcal A\) does not contain \(n\)-Luzin gaps for any \(n \in \omega\).NEWLINENEWLINETheorem. Under the \(P\)-ideal dichotomy, if \(\mathcal A\) is an almost disjoint family size \(\omega_1\) then either \(\mathcal A\) is partially separated (not defined here) or it contains a Luzin gap.NEWLINENEWLINEA number of results relate to what ccc forcing can do to an almost disjoint family.NEWLINENEWLINETheorem. The following are equivalent: {\parindent=6mm \begin{itemize}\item[1.] \(\mathcal A\) does not contain \(n\)-Luzin gaps for any \(n < \omega\). \item[2.] \(\Psi(\mathcal A)\) is potentially normal. \item[3.] \(\Psi(\mathcal A)\) is potentially embeddable in \(\mathbb R\). \item[4.] \(\mathcal A\) is potentially partially separated. NEWLINENEWLINE\end{itemize}} Defining an almost disjoint family \(\mathcal A\) to be schizophrenic iff it is inseparable and contains no \(n\)-Luzin gaps for any \(n\):NEWLINENEWLINETheorem. It is consistent with MA(\(\sigma\)-centered) that there is a schizophrenic family of size \(\omega_1\).NEWLINENEWLINETheorem. \(\diamondsuit\) implies the existence of a schizophrenic family.
scientific article; zbMATH DE number 6243378
scientific article