On the cardinality of Hausdorff spaces (Q1928424)

It is shown that \(|X|\leq2^\kappa\) for Hausdorff \(X\) and an infinite cardinal \(\kappa\) with \(\psi_c(X)\leq\kappa\) and \(aL_\kappa(X)\leq\kappa\). Here \(\psi_c(X)\) is the closed pseudo-character, the least infinite cardinal \(\lambda\) such that for each \(x\in X\) there is a collection of size \(\lambda\) of open neighbourhoods of \(x\) the intersection of whose closures is \(\{x\}\), while the \(\kappa\)-almost Lindelöf degree of \(X\), \(aL_\kappa(X)\), is \(\sup\{aL(\text{cl}_\kappa A,X)| A\subset X\}\), where \(aL(Y,X)\) is the almost Lindelöf degree of \(Y\) relative to \(X\) and \(\text{cl}_\kappa A=\cup_{B\in[A]^{\leq\kappa}}\overline{B}\).