The space of ultrafilters on N covered by nowhere dense sets. (English) Zbl 0568.54004

This is an interesting paper concerning \(N^*=\beta N-N\) (where N denotes the countable discrete space of natural numbers), and the Baire or Novák number \(n(N^*)\), then smallest cardinal number of a family of nowhere dense subsets of \(N^*\) which cover \(N^*\). The authors introduce the concept of a shattering matrix for \(N^*\), and prove some relations between \(n(N^*)\) and the number \(\kappa (N^*)\) which is the smallest cardinal number of a shattering matrix for \(N^*\). A collection \(\theta =\{{\mathcal G}\}\), where each \({\mathcal G}\) in \(\theta\) is a family of pairwise disjoint open subsets of \(N^*\), is called a matrix for \(N^*\), and is called a shattering matrix provided that for each non- void open set \(U\subset N^*\), there exists \({\mathcal G}\) in \(\theta\) such that U meets at least two members of \({\mathcal G}\). The number \(\kappa (N^*)\) is also the least \(\kappa\) such that the Boolean algebra \({\mathcal P}(N)/fin\) is not (\(\kappa\),\({\mathfrak c})\)-distributive. To see that \(\kappa (N^*)\) is related to \(n(N^*)\) it is helpful to mention a result in a preprint ”Sequential compactness and trees” by the second author, P. Nyikos and the third author which states that \(\kappa (N^*)\) equals the smallest cardinal number of a family of nowhere dense subsets of \(N^*\) whose union is dense in \(N^*\). Thus \(\kappa (N^*)\leq n(N^*)\). In the paper under review it is proved that \(\aleph_ 1\leq \kappa (N^*)\leq cf({\mathfrak c}),\) and that \(\kappa (N^*)\) is regular. Further, if \(\kappa (N^*)<{\mathfrak c},\) then \(\kappa (N^*)\leq n(N^*)\leq k(N^*)^+.\) Let \({\mathfrak b}\) denote the smallest cardinal number of an unbounded family of functions in \(^{\omega}\omega\) (mod finite order), then \(\kappa (N^*)\leq {\mathfrak b}.\) There are also results about shattering matrices, and Gleason spaces. The aleph value of the cardinal numbers \(\kappa (N^*)\) and \(n(N^*)\) are worked out in several known models of set theory.
Reviewer: J.E.Vaughan


54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
06E15 Stone spaces (Boolean spaces) and related structures
54A35 Consistency and independence results in general topology
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