An extension of a result of \(l^ 1(r)\)-embeddings (Q1092442)

The following theorem is proved: Let \(\alpha\) be a cardinal and \((X_ i)_{i\in I}^ a \)family of compact Hausdorff topological spaces such that \(\prod _{i\in I}X_ i\) has caliber \(\alpha\). We suppose that \(\alpha\) is strongly-\(\kappa\)-inaccessible, where \(\kappa =\sup \{S(X_ i):\) \(i\in I\}\) and \(cf\alpha >\kappa\). Then every subspace of C(\(\prod _{i\in I}X_ i)\) of dimension \(\alpha\) contains an isomorphic copy of \(\ell ^ 1_{\alpha}\). This result extends a previous one of Argyros- Tsarpalias in the case of a compact Hausdorff topological space X.