Separation and cardinality - some new results and old questions (Q2400894)

In [Sov. Math., Dokl. 10, 951--955 (1969; Zbl 0191.20903); translation from Dokl. Akad. Nauk SSSR 187, 967--970 (1969)], \textit{A. V. Arkhangel'skij} published the solution of the classical question by Alexandrov asking for the cardinality of a Hausdorff compact first countable space. He also posed questions about the cardinality of a \(T_1\) first countable Lindelöf space and about the cardinality of a Hausdorff, Lindelöf space with countable pseudocharacter. The literature about both questions is extensive. In the paper under review the authors study variants of these problems using modified definitions; for instance that of \(n\)-\textit{Hausdorffness}. They rise many interesting questions that will lead to a better understanding of these problems and hopefully to their soon solution.