On the Hausdorff number of a topological space (Q2862661)

The Hausdorff number of a space \(X\) is \(H(X)\), the minimum cardinal \(\kappa\) such that for every subset \(A\) of \(X\) with size at least \(\kappa\) there are open neighbourhoods \(U_a\) for all the points \(a\in A\) so that \(\bigcap\{U_a:a\in A\}=\emptyset\). Of course, \(X\) is Hausdorff whenever \(H(X)\leq2\).NEWLINENEWLINEIn the paper there are many results studying relationships between \(H(X)\) and other well known cardinal functions on topological spaces. Some open questions are also posed; for instance, the author asks whether is it true that \(| X|\leq2^{2^{\ell(X)t(X)\psi(X)}}\).