For a Hausdorff space $X$, Hajnal and Juhász showed in 1967, that $|X|\le 2^{c(X)\chi (X)}$ and $|X|\le 2^{2^{s(X)}}$, where $c(X)$ is the cellularity, $\chi (X)$ is the character and $s(X)$ is the spread of $X$; Arhangel’skii, in 1969, proved that $|X|\le 2^{\chi (X)L(X)}$, where $\chi (X)$ is the character and $L(X)$ is the Lindelӧf degree of $X$; and, in 1974, Arhangel’skiĭ and Šapirovskiĭ strengthened Arhangel’skiĭ’s inequality by showing that $|X|\le 2^{t(X)\psi (X)L(X)}$, where $t(X)$ is the tightness and $\psi (X)$ is the pseudocharacter of $X$.

It has been an open question for a long time if Arhangel’skiĭ’s inequality is true for every $T_1$-space $X$.

In this talk we will mention what is known in relation to the above question and how by using other cardinal functions, some of the above inequalities could be extended to be valid for all $T_1$-spaces and, in some cases, even for all topological spaces.