Maximal independent families and a topological consequence

For κ⩾ω and X a set, a family A⊆P(X) is said to be κ-independent on X if |⋂A∈F Af(A)|⩾κ for each F∈[A]<ω and f∈{−1,+1}F; here A+1=A and A−1=X⧹A. m 3.6 ω, some A⊆P(κ) with |A|=2κ is simultaneously maximal κ-independent and maximal ω-independent on κ. The family A may be chosen so that every two elements of κ are separated by 2κ-many elements of A. ary 5.4 ω there is a dense subset D of {0,1}2κ such that each nonempty open U⊆D satisfies |U|=d(U)=κ and no subset of D is resolvable. The set D may be chosen so that every two of its elements differ in 2κ-many coordinates. s eorem 3.6 answers affirmatively a question of Eckertson [Topology Appl. 79 (1997) 1–11]. Two proofs are given here. (b) Parts of Corollary 5.4 have been obtained by other methods by Feng [Topology Appl. 105 (2000) 31–36] and (for κ=ω) by Alas et al. [Topology Appl. 107 (2000) 259–273].