A topological partition theorem and open covers

AbstractA set A in a topological space X is called κ-closed if B⊂A whenever B⊂A and |B|<κ. A κ-hole in X is a maximal centered family of κ-closed sets which is both κ-complete and free. A family S of finite subsets of X is called κ-closed if u∪{x:u∪{x}∈S} is κ-closed in X for every u∈[X]<ω.Theorem 1. If the T1 space X has a κ-hole and S⊂[X]<ω is κ-closed for an uncountable regular cardinal κ, then either there is a set Y∈[X]κ with [Y]<ω∩S=∅ or there are n∈ω and Z∈[X]κ with [Z]n⊂S.Theorem 2. If κ=cf(κ)>ω, X is an initially <κ-compact T1 space and U is an open cover of X such that for every A∈[X]κ there is a set B∈[A]<ω with o(B,U)<κ (i.e., |{U∈U:B⊂U}|