Summary: In this note we first give a conjecture as to a slight generalization of a theorem by Iseki and Kasahara. We also prove the existence of a Hausdorff space \(X\) in which 1) point-\(\omega_\alpha\), open covers have a subcover with cardinality less than \(\omega_\alpha\), 2) \(X\) has a closed-discrete subset with cardinality \(\omega_\alpha\) and finally 3) \(X\) has a point-\(\omega_\alpha\) open cover with cardinality \(\geq\omega_\alpha\). We finally prove that every Hausdorff space \(X\) is a closed nowhere dense subspace of a Hausdorff space \(Z_X\) in which point-finite (resp. point-countable) open covers have finite (resp. countable) subcovers. We also pose a related conjecture.