MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}_\kappa^\downarrow(P)$ be \mathcal{P}_\kappa(P)$denote the poset of$\kappa$-small downward-closed subposets of$P$. P$ and let $\mathcal{P}_\kappa^\downarrow(P)\subseteq\mathcal{P}_\kappa(P)$ be the subposet consisting of those subposets that are downward-closed. Then according to a reliable source, when $\kappa$ is regular, we can show that $\mathcal{P}^\downarrow_\kappa(P)$ is $\kappa$-filtered by showing that because given some $\kappa$-small family of $\kappa$-small subposets, $$A_i:I\to \mathcal{P}^\downarrow_\kappa(P)\quad mathcal{P}_\kappa(P)\quad |I|<\kappa$$

the downward closure of the union over this family, $\operatorname{Cl}^\downarrow(\bigcup_{i\in I}A_i)$, is $\kappa$-small (which gives a majorant for the family $A_i$).

However, since I have no experience at all working with regular cardinals, I'm not really sure how to make heads or tails of this. Why does the regularity of $\kappa$ imply that the downward-closure of that union is $\kappa$-small?

1

# The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal?

Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}_\kappa^\downarrow(P)$ be the poset of $\kappa$-small downward-closed subposets of $P$. Then according to a reliable source, when $\kappa$ is regular, we can show that $\mathcal{P}^\downarrow_\kappa(P)$ is $\kappa$-filtered by showing that given some $\kappa$-small family $$A_i:I\to \mathcal{P}^\downarrow_\kappa(P)\quad |I|<\kappa$$

the downward closure of the union over this family, $\operatorname{Cl}^\downarrow(\bigcup_{i\in I}A_i)$, is $\kappa$-small (which gives a majorant for the family $A_i$).

However, since I have no experience at all working with regular cardinals, I'm not really sure how to make heads or tails of this. Why does the regularity of $\kappa$ imply that the downward-closure of that union is $\kappa$-small?