In 1969, Arhangel’skiĭ proved that if $X$ is a Hausdorff space, then $|X|\le 2^{\chi (X)L(X)}$, where $\chi (X)$ is the character and $L(X)$ is the Lindelöf degree of $X$. Since then it has been an open question if his inequality is true for every $T_1$-space $X$. In 2013, we proved that if $X$ is a $T_1$-space, then $|X|\le nh(X{)}^{\chi (X)L(X)}$, where $nh(X)$ is the non-Hausdorff number of $X$. In that way we were able to positively answer this question for every $T_1$-space for which $nh(X)\le 2^{\chi (X)L(X)}$, and, in particular, when $nh(X)$ is not grater than the cardinality of the continuum. A simple example shows that our inequality is not always true for $T_0$-spaces.

Arhangel’skiĭ and Šapirovskiĭ strengthened Arhangel’skiĭ’s inequality in 1974 by showing that if $X$ is a Hausdorff space, then $|X|\le 2^{t(X)\psi (X)L(X)}$, where $t(X)$ is the tightness and $\psi (X)$ is the pseudocharacter of $X$.

In this talk we will show how Arhangel’skiĭ–Šapirovskiĭ’s inequality, and therefore, Arhangel’skiĭ’s inequality, could be extended to be valid for all topological spaces.