ON $\omega$-INDEPENDENCE AND THE KUNEN–SHELAH PROPERTY

Abstract

We prove that spaces with an uncountable ω-independent family fail the Kunen-Shelah property. Actually, if {x i } i∈I is an uncountable ω-independent family, there exists an uncountable subset J ⊂ I such that x j / ∈ conv({x i } i∈J\{j} ) for every j ∈ J. This improves a previous result due to Sersouri, namely that every uncountable ω-independent family contains a convex right-separated subfamily.

On the Kunen–Shelah properties in Banach spaces

“…The implication KS 4 ⇒ KS 3 was proved in [3]. It also follows from Proposition 3.2 and from Proposition 7.3 and a result of Sersouri [12].…”