Two Cardinal Inequalities about Bidiscrete Systems

TL;DR

This paper explores inequalities involving bidiscrete systems in compact spaces, linking them to irredundance and biorthogonal systems, and generalizes known results in topology and Banach space theory.

Contribution

It establishes new inequalities relating weight, bidiscrete systems, and hereditary Lindelöf degree, and extends McKenzie's theorem to compact spaces.

Findings

Proves that w(K) ≤ bd(K) · hL(K)^+ for compact Hausdorff spaces.

Shows that for maximal irredundant families in C(K), there exists a π-base with the same cardinality.

Concludes that π(K) ≤ bd(K), connecting bidiscrete systems to π-characteristics.

Abstract

We consider the cardinal invariant $bd$ defined by M. D\v{z}amonja and I. Juh\'asz concerning bidiscrete systems. Using the relation between bidiscrete systems and irredundance for a compact Hausdorff space $K$, we prove that ${w(K)\leq bd(K)\cdot hL(K)^+}$, 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 $\pi(K)\leq bd(K)$ 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 $C(K)$.