Two cardinal inequalities about bidiscrete systems (Q321399)

A sequence \(\langle(x^0_\alpha,x^1_\alpha)\rangle_{\alpha<\kappa}\) of pairs of points of a topological space \(X\) is a bidiscrete system if there is a sequence \(\langle f_\alpha\rangle_{\alpha<\kappa}\) of continuous real-valued functions on \(X\) such that for each \(\alpha,\beta<\kappa\), \(f_\alpha(x_\alpha^\ell)=\ell\) for \(\ell=0,1\) and \(f_\alpha(x^0_\beta)=f_\alpha(x^1_\beta)\) if \(\alpha\not=\beta\). Then \(bd(X)=\sup\{|S|\;\mid S \text{ is a bidiscrete system in } X\}\). It is shown that when \(X\) is compact and Hausdorff then \(w(X)\leq bd(X)hL(X)^+\) and \(\pi(X)\leq bd(X)\); in particular if \(w(X)\) is a strong limit cardinal then \(w(X)=bd(X)\). Applications to semibiorthogonal systems on the Banach space \(C(X)\) are also given.