Suppose $(P,\leq)$ is a poset without maximal elements. For $X\subseteq P$ we set $X^u = \{p\in P: p \geq x \text{ for all } x\in X\}$ and call this the set of upper bounds of $X$. We say that $B\subseteq P$ is unbounded if $B^u = \emptyset$. Moreover we say $D\subseteq P$ is dominating if for all $p\in P$ there is $d\in D$ such that $p\leq d$. We set

  • ${\frak b}(P) = \min\{|B|: B\subseteq P\text{ is unbounded}\}$, and
  • ${\frak d}(P) = \min\{|D|: B\subseteq P\text{ is dominating}\}$.

It is easy to see that for all posets $P$ without maximal elements we have ${\frak b}(P) \leq {\frak d}(P)$. What is an example of a poset $P$ in which we can prove in $\textsf{ZFC}$ that ${\frak b}(P) < {\frak d}(P)$?


$\omega\times \omega_1$ with the product (pointwise) order.

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  • $\begingroup$ O frabjous day! Callooh! Callay! (The 6th upvote for this answer lifted me above the 10k mark. Not my deepest answer, though.) $\endgroup$ – Goldstern Oct 17 '17 at 18:05
  • $\begingroup$ (But a very concise one - and congrats!) $\endgroup$ – Dominic van der Zypen Nov 5 '17 at 17:42

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