The author gives a topological proof of the following combinatorial theorem which easily implies a theorem of \textit{M. Talagrand} [Isr. J. Math. 40, 324-330 (1981; Zbl 0488.46016)] concerning the existence of copies of \(\ell^ 1_{\alpha}\) in closed subspaces of C(K), separating the points of K, for a compact space K which can be mapped continuously onto \([0,1]^{\alpha}\). If \(\alpha\) is a cardinal with uncountable cofinality, \(\{(A_{\xi},B_{\xi}):\xi <\alpha\}\) an independent family of sets, n a natural number and \(\{(A^ i_{\xi},B^ i_{\xi}):1\leq i\leq n\}\) sets with \(A^ i_{\xi}\subset A_{\xi},\quad B^ i_{\xi}\subset B_{\xi}\) and \(A_{\xi}\times B_{\xi}=\cup^{n}_{1}A^ i_{\xi}\times B^ i_{\xi},\) then there exist \(i_ 0\) and \(I\subset\alpha \) with \(| I| =\alpha\) such that \(\{(A_{\xi}^{i_ 0},B_{\xi}^{i_ 0}):\xi\in I\}\) is independent. Using Stone's duality theorem, this theorem is translated into a topological claim which is proved by induction on n.