The distributive number $\mathfrak h$ is defined as the least cardinal $\kappa$ such that there exists a family of $\kappa$ open dense subsets in the preordered set $([\omega]^\omega,\subset^*)$ whose intersection is empty.

In the chapter Combinatorial Cardinal Characteristics of the Continuum, in Kanamori's Handbook of Set Theory (p. 426) Blass states that $\mathfrak h$ is also the least cardinal $\kappa$ such that $([\omega]^\omega,\subset^*)$ is not $\kappa$-distributive. However, all characterizations of $\kappa$-distributivity I have found, for instance, Jech (third edition) Lemma 7.16, state that a complete Boolean algebra is $\kappa$-distributive if and only if the intersection of $\kappa$ open dense subsets is dense.

I cannot fill the gap between "dense" and "non-empty". Is it true that $\mathfrak h$ is also the least cardinal $\kappa$ such that there exist $\kappa$ open dense subsets in $([\omega]^\omega,\subset^*)$ whose intersection is not dense?

  • $\begingroup$ There's a trick. Admittedly, I cannot recall it, but there is trick to move from a "non-dense" intersection to an empty one. $\endgroup$ – Asaf Karagila Jul 15 at 12:22

If you had $\kappa$ open dense sets $\mathcal D_i$ whose intersection is not dense, fix some $B\in[\omega]^\omega$ such that the intersection has no elements $\subseteq^*B$. Then the sets $\mathcal D_i\cap\mathcal P(B)$ (where $\mathcal P$ means power set) are dense open subsets of $([B]^\omega,\subseteq^*)$, and their intersection is empty. (If you insist on $[\omega]^\omega$ instead of $[B]^\omega$, just apply a bijection $B\to\omega$.)

  • $\begingroup$ Ok. Thank you! It was easier than I thought. $\endgroup$ – Carlos Jul 15 at 21:36

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