Two Cardinal inequalities about bidiscrete systems

We consider the cardinal invariant bd defined by M. Džamonja and I. Juhász concerning bidiscrete systems. Using the relation between bidiscrete systems and irredundance for a compact Hausdorff space K  , we prove that w(K)≤bd(K)⋅hL(K)+w(K)≤bd(K)⋅hL(K)+, generalizing a result of S. Todorcevic concerning the irredundance in Boolean algebras and we prove that for every maximal irredundant family F⊂C(K)F⊂C(K), there is a π  -base BB for K   with |F|=|B||F|=|B|, a result analogous to the McKenzie Theorem for Boolean algebras in the context of compact spaces. In particular, it is a consequence of the latter result that π(K)≤bd(K)π(K)≤bd(K) for every compact Hausdorff space K  . From the relation between bidiscrete systems and biorthogonal systems, we obtain some results about biorthogonal systems in Banach spaces of the form C(K)C(K).