# $\kappa$-complete

A structured set $S$ (typically a filter or a Boolean algebra) is $\kappa$-complete if, given any $K\subseteq S$ with $|K|<\kappa$, $\bigcap K\in S$. It is complete if it is $\kappa$-complete for all $\kappa$.

Similarly, a partial order is $\kappa$-complete if any sequence of fewer than $\kappa$ elements has an upper bound within the partial order.

A $\aleph_{1}$-complete is called countably complete.

Title $\kappa$-complete kappacomplete 2013-03-22 12:53:07 2013-03-22 12:53:07 Henry (455) Henry (455) 11 Henry (455) Definition msc 03E10 kappa-complete kappa complete Filter BooleanAlgebra countably complete