Generalized Kurepa and Mad Families and Topology

Closing a Kurepa family under finite intersection yields a Kurepa family of the same cardinality, so we may assume N = {Nα : α ∈ μ} is closed under finite intersection. For each N ∈ N let m(N) = {α : Nα ⊂ N}. Since N is a Kurepa family, m(N) is a countable subset of μ. Also, m(N1 ∩N2) = m(N1) ∩m(N2) and so K = {m(N) : N ∈ N and m(N) is infinite} is a Kurepa family of cardinality no greater than μ. To show that K is cofinal in [μ] and hence of cardinality ≥ μ, let M be a countable subset of μ and let N(M) be a member of N containing ⋃ {Nα : α ∈M}. Then clearly M ⊂ m(N(M)).