Ultrafilters and almost disjoint sets

Let μκ denote the space of uniform ultrafilters on κ, and let λ be a cardinal. Uε{lunate}μκ is said to be a λ-point of μκ if U is a boundary point of λ pairwise disjoint open subsets of μκ. We prove that if κ is a successor cardinal, 2^κ= κ⁺, and Kurepa's hypothesis for κ holds, then each U ε{lunate} μκ is a 2^κ-point of μκ.