Regular $${G_\delta}$$-diagonals and some upper bounds for cardinality of topological spaces

We prove that, under CH, any space with a regular \({G_{\delta}}\)-diagonal and caliber \({\omega_1}\) is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space X, we establish the inequality \({|X|\leqq wL{(X)}^{s\Delta_2(X)\cdot{\dot(X)}}}\) which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if X is a Hausdorff space, then \({|X| \leqq {(\pi\chi(X)\cdot d(X))}^{{\rm ot}(X)\cdot\psi_c(X)}}\); this result implies Sapirovskiĭ’s inequality \({|X|\leqq \pi\chi{(X)}^{c(X)\cdot\psi(X)}}\) which only holds for regular spaces. It is also proved that \({|X|\leqq \pi\chi{(X)}^{{\rm ot}(X)\cdot\psi_c(X)\cdot aL_c(X)}}\) for any Hausdorff space X; this gives one more generalization of the famous Arhangel’skii’s inequality \({|X|\le 2^{\chi(X)\cdot L(X)}}\).