Interpolation of -compactness and Pcf

We call a topological space κ-compact if every subset of size κ has a complete accumulation point in it. Let Φ(μ, κ, λ) denote the following statement: μ < κ < λ = cf(λ) and there is {Sξ : ξ < λ} ⊂ [κ] μ such that |{ξ : |Sξ ∩ A| = μ}| < λ whenever A ∈ [κ]. We show that if Φ(μ, κ, λ) holds and the space X is both μ-compact and λ-compact then X is κ-compact as well. Moreover, from PCF theory we deduce Φ(cf(κ), κ, κ) for every singular cardinal κ. As a corollary we get that a linearly Lindelöf and אω-compact space is uncountably compact, that is κ-compact for all uncountable cardinals κ.