Let $\kappa$ be a cardinal (I'm most interested in $\kappa=\aleph_{\omega+1}$ but I suspect a general answer is known). What is the cofinality of $(P(\kappa),\subseteq)$? By this I mean the least cardinal $\lambda$ such that there exists a subcollection $X\subseteq P(\kappa)$ of size $\lambda$ such that for any $A\in P(\kappa)$ there is $B\in X$ with $A\subseteq B$.

This may be trivial but it's been bugging me all day!

Drake