Sapirovskii [18] proved that |X| ≤ πχ(X) c(X)ψ(X) , for a regular space X. We introduce the θ-pseudocharacter of a Urysohn space X, denoted by ψ θ (X), and prove that the previous inequality holds for Urysohn spaces replacing the bounds on celluarity c(X) ≤ κ and on pseudocharacter ψ(X) ≤ κ with a bound on Urysohn cellularity U c(X) ≤ κ (which is a weaker conditon because U c(X) ≤ c(X)) and on θ-pseudocharacter ψ θ (X) ≤ κ respectivly (Note that in general ψ(·) ≤ ψ θ (·) and in the class of regular spaces ψ(·) = ψ θ (·)). Further, in [6] the authors generalized the Dissanayake and Willard's inequality: |X| ≤ 2 aLc(X)χ(X) , for Hausdorff spaces X [25], in the class of n-Hausdorff spaces and de Groot's result: |X| ≤ 2 hL(X) , for Hausdorff spaces [11], in the class of T 1 spaces (see Theorems 2.22 and 2.23 in [6]). In this paper we restate Theorem 2.22 in [6] in the class of n-Urysohn spaces and give a variation of Theorem 2.23 in [6] using new cardinal functions, denoted by U W (X), ψw θ (X), θ-aL(X), hθ-aL(X), θ-aL c (X) and θ-aL θ (X). In [5] the authors introduced the Hausdorff point separating weight of a space X denoted by Hpsw(X) and proved a Hausdorff version of Charlesworth's inequality |X| ≤ psw(X) L(X)ψ(X) [7]. In this paper, we introduce the Urysohn point separating weight of a space X, denoted by U psw(X), and prove that |X| ≤ U psw(X) θ-aLc(X)ψ(X) , for a Urysohn space X.

We introduce the cardinal invariant θ-aL ′ (X), related to θ-aL(X), and show that if X is Urysohn, then |X| 2 θ-aL ′ (X)χ(X). As θ-aL ′ (X) aL(X), this represents an improvement of the Bella-Cammaroto inequality. We also introduce the classes of firmly Urysohn spaces, related to Urysohn spaces, strongly semiregular spaces, related to semiregular spaces, and weakly H-closed spaces, related to H-closed spaces.

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