Two Cardinal Inequalities about Bidiscrete Systems

Abstract

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 $w(K)\le bd(K)\cdot hL(K{)}^{+}$, generalizing a result of S. Todorcevic concerning the irredundance in Boolean algebras and we prove that for every maximal irredundant family $\mathcal{F}\subset C(K)$, there is a $\pi$-base $\mathcal{B}$ for $K$ with $|\mathcal{F}|=|\mathcal{B}|$, 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 $\pi (K)\le bd(K)$ 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 $C(K)$.