# On the distributive number $\mathfrak h$

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?

• There's a trick. Admittedly, I cannot recall it, but there is trick to move from a "non-dense" intersection to an empty one. – 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 $$B\to\omega$$.)