Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 2 / a / ii / Solution

For a regular cardinal $\kappa$,Again $q\le p$ means $q\supseteq p$, and the maximal element is the empty function. The regularity of $\kappa$ ensures that the union of a descending sequence of fewer than $\kappa$ conditions still has domain of cardinality below $\kappa$ whenever the conditions form a compatible increasing chain of partial functions.