We consider the cardinal invariant $bd$ defined by M. D\v{z}amonja and I.Juh\'asz concerning bidiscrete systems. Using the relation between bidiscretesystems and irredundance for a compact Hausdorff space $K$, we prove that${w(K)\leq bd(K)\cdot hL(K)^+}$, generalizing a result of S. Todorcevicconcerning the irredundance in Boolean algebras and we prove that for everymaximal irredundant family $\mathcal{F}\subset C(K)$, there is a $\pi$-base$\mathcal{B}$ for $K$ with $|\mathcal{F}|=|\mathcal{B}|$, a result analogous tothe McKenzie Theorem for Boolean algebras in the context of compact spaces. Inparticular, it is a consequence of the latter result that $\pi(K)\leq bd(K)$for every compact Hausdorff space $K$. From the relation between bidiscretesystems and biorthogonal systems, we obtain some results about biorthogonalsystems in Banach spaces of the form $C(K)$.
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机译：我们考虑由M. D \ v {z} amonja和I.Juh \'asz定义的关于双凸混凝土系统的基本不变式$ bd $。利用紧凑的Hausdorff空间$ K $中bidiscrete系统与不冗余之间的关系，我们证明了$ {w（K）\ leq bd（K）\ cdot hL（K）^ +} $，从而推广了S. Todorcevic关于布尔代数中的非冗余度，我们证明对于每个最大的非冗余族$ \ mathcal {F} \ subset C（K）$，对于$ K $和$ | \，都有一个$ \ pi $ -base $ \ mathcal {B} $ mathcal {F} | = | \ mathcal {B} | $，类似于紧凑空间中布尔代数的McKenzie定理。特别地，这是后者结果的结果，即每个紧凑Hausdorff空间$ K $的$ \ pi（K）\ leq bd（K）$。从双体系统与双正交系统之间的关系中，我们获得了有关形式为$ C（K）$的Banach空间中双正交系统的一些结果。
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