A generalization of some cardinal function inequalities (Q1731348)

If $A$ is a set and $\kappa$ is a cardinal, we denote by $[A]^{\leqslant \kappa}$ the family of all subsets of $A$ whose cardinality is not greater than $\kappa$. \par In what follows, we assume the reader is familiar with the standard terminology and notation regarding cardinal functions (including the definitions of character, pseudocharacter, Lindelöf degree and free sequence number, denoted, respectively, by $\chi$, $\psi$, $L$ and $F$). The following cardinal functions are not so well known (and are defined for $T_1$ spaces): the \textit{Hausdorff width} of a topological space $X$ is $HW(X) = \sup\{|Hw(x)|: x \in X\}$, where, for any $x \in X$, $Hw(x) = \bigcap \{\overline{U}: x \in U\,\,\text{and}\,\,U\,\,\text{is open in}\,\,X\}$. For any $x \in X$, $\psi w(x)$ is the smallest cardinal $\kappa$ such that there is a family $\mathcal{U}_x$ of open neighbourhoods of $x$ with $|\mathcal{U}_x| = \kappa$ and $\bigcap \{\overline{U}: U \in \mathcal{U}_x \} = Hw(x)$; and then $\psi w(X) = \sup\{\psi w(x): x \in X\}$. For any $T_1$ space one has $\psi w(X) \leqslant \chi(X)$. \par The best known results about cardinal functions are those that give upper bounds for the cardinality of a topological space in terms of the values of some of its cardinal functions. For instance, we may refer to Arhangel'skii's widely famous inequality: \[ \text{For every Hausdorff space, }|X| \leqslant 2^{L(X)\chi(X)}.\] It is very natural to ask: if $X$ is a $T_1$ space, is it true that $|X| \leqslant 2^{L(X)\chi(X)}$? In [Houston J. Math. 39, No. 3, 1013--1030 (2013; Zbl 1290.54003)], \textit{M. Bonanzinga} has introduced a pair of new cardinal functions: the Hausdorff number and the weak Hausdorff number of a topological space $X$, denoted by $H(X)$ and $H^*(X)$ respectively, and has extended Arhangel'skii's inequality to $T_1$ spaces with $H^*(X) \leqslant \aleph_0$. Accordingly to the author, ``Bonanzinga's idea have given new impetus to the theory of topological cardinal invariants and today many authors have returned to the problems in this field''. \par In the paper under review, the author generalizes a number of recently obtained cardinal inequalities. A typical result of the paper is the following: \par Theorem. Let $X$ be a $T_1$ space, and let $\kappa$ be an infinite cardinal such that \par $1.$ $L(X)F(X) \leqslant \kappa$; \par $2.$ $\psi w(X) \leqslant 2^\kappa$; and \par $3.$ for each $A \in [X]^{\leqslant \kappa}$, one has $|\overline{A}| \leqslant 2^\kappa$. \par Under these assumptions, $|X| \leqslant HW(X)\,2^\kappa$. \par The preceding theorem generalizes the following inequality, which is itself a generalization of Arhangel'skii's inequality, and was proved, independently, by \textit{S. Spadaro} [Topology Appl. 158, No. 16, 2091--2093 (2011; Zbl 1229.54004)]: \[\text{For every Hausdorff space, }|X| \leqslant 2^{L(X)F(X)\psi(X)}.\] All proofs of the paper use the elementary submodels technique.