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On densities of box products. (English) Zbl 0926.03060

Let \(\lambda\) be a strong limit, cf(\(\lambda) = \omega, 2^{\lambda} > \lambda^+\). This paper explores the density of the space \(d_{<\mu}(\lambda)\) whose underlying set is \(^{\lambda}2\) with a base of all \([g] = \{f: f\supset g\}\) where \(g \in ^A2\) for some \(A \in [\lambda]^{< u}\).
Using large cardinals (e.g., hyperextendibility), the authors construct models of set theory where, say \(d_{\omega_1}(\lambda) = \lambda^+\), or \(d_{\omega_1}(\lambda) = \lambda^{++} < 2^{\lambda}\), for some \(\lambda\) a strong limit (it starts out larger). Using a huge cardinal they also construct, in an intermediate model, a normal filter on a cardinal \(\kappa\) which is generated by \(\kappa^+\) sets, \(\kappa < 2^{\kappa}\). A more complicated construction gives two models in which GCH holds below \(\aleph_{\omega}, 2^{\aleph_{\omega}} = \aleph_{\omega+2}\), and in one model \(d_{<\omega_1}\aleph_{\omega} = \aleph_{\omega+1}\); in the other \(d_{<\omega_1}\aleph_{\omega} = \aleph_{\omega+2}\).

MSC:

03E35 Consistency and independence results
03E55 Large cardinals
03E45 Inner models, including constructibility, ordinal definability, and core models
54B10 Product spaces in general topology
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