Variations on known and recent cardinality bounds

Sapirovskii [16] 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 ψθ(X), and prove that the previous inequality holds for Urysohn spaces replacing the bounds on cellularity c(X)≤κ and on pseudocharacter ψ(X)≤κ with a bound on Urysohn cellularity Uc(X)≤κ (which is a weaker condition because Uc(X)≤c(X)) and on θ-pseudocharacter ψθ(X)≤κ respectively (note that in general ψ(⋅)≤ψθ(⋅) and in the class of regular spaces ψ(⋅)=ψθ(⋅)). Further, in [6] the authors generalized the Dissanayake and Willard's inequality: |X|≤2aLc(X)χ(X), for Hausdorff spaces X[21], in the class of n-Hausdorff spaces and de Groot's result: |X|≤2hL(X), for Hausdorff spaces [10], in the class of T1 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 UW(X), ψwθ(X), θ-aL(X), hθ-aL(X), θ-aLc(X) and θ-aLθ(X). In [5] the authors introduced the Hausdorff point separating weight of a spaceX 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 spaceX, denoted by Upsw(X), and prove that |X|≤Upsw(X)θ-aLc(X)ψ(X), for a Urysohn space X.