Generalizations of two cardinal inequalities of Hajnal and Juh\'asz

Abstract

A non-empty subset $A$ of a topological space $X$ is called \emph{finitely non-Hausdorff} if for every non-empty finite subset $F$ of $A$ and every family $\{U_x:x\in F\}$ of open neighborhoods $U_x$ of $x\in F$, $\cap\{U_x:x\in F\}\ne\emptyset$ and \emph{the non-Hausdorff number $nh(X)$ of $X$} is defined as follows: $nh(X):=1+\sup\{|A|:A\subset X$ is finitely non-Hausdorff$\}$. Clearly, if $X$ is a Hausdorff space then $nh(X)=2$. We define the \emph{non-Urysohn number of $X$ with respect to the singletons}, $nu_s(X)$, as follows: $nu_s(X):=1+\sup\{\mathrm{cl}_\theta(\{x\}):x\in X\}$. In 1967 Hajnal and Juh\'asz proved that if $X$ is a Hausdorff space then: (1) $|X|\le 2^{c(X)\chi(X)}$; and (2) $|X|\le 2^{2^{s(X)}}$; where $c(X)$ is the cellularity, $\chi(X)$ is the character and $s(X)$ is the spread of $X$. In this paper we generalize (1) by showing that if $X$ is a topological space then $|X|\le nh(X)^{c(X)\chi(X)}$. Immediate corollary of this result is that (1) holds true for every space $X$ for which $nh(X)\le 2^\omega$ (and even for spaces with $nh(X)\le 2^{c(X)\chi(X)}$). This gives an affirmative answer to a question posed by M. Bonanzinga in 2013. A simple example of a $T_1$, first countable, ccc-space $X$ is given such that $|X|>2^\omega$ and $|X|=nh(X)^\omega=nh(X)$. This example shows that the upper bound in our inequality is exact and that $nh(X)$ cannot be omitted (in particular, $nh(X)$ cannot always be replaced by $2$ even for $T_1$-spaces). In this paper we also generalize (2) by showing that if $X$ is a $T_1$-space then $|X|\le 2^{nu_s(X)\cdot 2^{s(X)}}$. It follows from our result that (2) is true for every $T_1$-space for which $nu_s(X)\le 2^{s(X)}$. A simple example shows that the presence of the cardinal function $nu_s(X)$ in our inequality is essential.