  We present a bound for the weak Lindel\"of number of the
$G_\delta$-modification of a Hausdorff space which implies various known
cardinal inequalities, including the following two fundamental results in the
theory of cardinal invariants in topology: $|X|\le 2^{L(X)\chi(X)}$
(Arhangel'skii) and $|X|\le 2^{c(X)\chi (X)}$ (Hajnal-Juhasz). This solves a
question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's
survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not
need any separation axiom beyond $T_2$.
