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On the ideal \((v^{0})\). (English) Zbl 1151.03027

A segment is a set of the form \(\langle A,B\rangle= \{X\in[\omega]^\omega:A\subseteq X\subseteq B\}\) where \(A\subseteq B\subseteq\omega\) and \(B\setminus A\) is infinite. A set \(S\subseteq[\omega]^\omega\) is called a \(v\)-set if for each segment \(\langle A,B\rangle\) there exists a segment \(\langle C,D\rangle\subseteq\langle A,B\rangle\) such that \(\langle C,D\rangle\subseteq S\) or \(\langle C,D\rangle\cap S=\emptyset\). A \(v\)-set \(S\) is a \(v^0\)-set if there is no segment \(\langle A,B\rangle\subseteq S\). Similarly, uncountable sets of the form \(\langle A,B\rangle^*= \{X\in[\omega]^\omega:A\subseteq^*X\subseteq^*B\}\) are called \(*\)-segments. The authors show that for \(*\)-segments ordered by inclusion a variant of the Base Matrix Lemma holds and the least height of a \(v\)-base matrix is equal to \(\text{add}(v^0)\). If \(\mathfrak t=\min\{\text{cf}(\mathfrak c),\mathfrak r\}\), then \(\text{cov}(v^0)=\text{add}(v^0)\).
The authors introduce segment topologies and topologies generated by base \(v\)-matrices and prove that nowhere dense sets in every such topology are exactly \(v^0\)-sets. Some more cardinal inequalities and combinatorial properties for the ideal \((s^0)\) are proved.

MSC:

03E35 Consistency and independence results
03E17 Cardinal characteristics of the continuum
03E50 Continuum hypothesis and Martin’s axiom
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
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