We prove that spaces with an uncountable $\omega$-independent family fail the Kunen–Shelah property. Actually, if $\{x_i\}_{i\in I}$ is an uncountable $\omega$-independent family, there exists an uncountable subset $J\subset I$ such that $x_j\notin\overline{\co}(\{x_i\}_{i\in J\setminus\{j\}})$ for every $j\in J$. This improves a previous result due to Sersouri, namely that every uncountable $\omega$-independent family contains a convex right-separated subfamily.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B26