Two Cardinal inequalities about bidiscrete systems

摘要

We consider the cardinal invariant bd defined by M. Dzamonja and I. Juhasz 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)(+), generalizing a result of S. Todorcevic concerning the irredundance in Boolean algebras and we prove that for every maximal irredundant family F subset of C(K), there is a pi-base B for K with vertical bar F vertical bar = vertical bar B vertical bar, 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 pi(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) 2016 Elsevier B.V. All rights reserved.