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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

##### MSC:
 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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