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We prove that, under CH, any space with a regular G δ -diagonal and caliber ω1 is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space X, we establish the inequality |X| ≤ wL(X) s∆ 2 (X)·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| ≤ (πχ(X) · d(X)) ot(X)·ψc(X) ; this result im-pliesŠapirovskiȋ's inequality |X| ≤ πχ(X) c(X)·ψ(X) which only holds for regular spaces. It is also proved that |X| ≤ πχ(X) ot(X)·ψc(X)·aLc(X) for any Hausdorff space X; this gives one more generalization of the famous Arhangel ′ skii's inequality |X| ≤ 2 χ(X)·L(X) .

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