Variations on known and recent cardinality bounds

Sapirovskii proved that |X|leqpichi(X)^{c(X)psi(X)}, for a regular space X. We introduce the θ-pseudocharacter of a Urysohn space X, denoted by psi_{ heta}(X), and prove that the previous inequality holds for Urysohn spaces replacing the bounds on cellularity c(X)leq k and on pseudocharacter psi(X)leq k with a bound on Urysohn cellularity Uc(X)leq k (which is a weaker condition because Uc(X)leq c(X) ) and on θ-pseudocharacter respectively psi_{ heta}(X)leq k (note that in general psi(.)leq psi_{ heta}(.) and in the class of regular spaces psi(.)= psi_{ heta}(.) ). Further, in [6] the authors generalized the Dissanayake and Willard's inequality: |X|leq 2^{aL_c(X)ci(X)} , for Hausdorff spaces X[, in the class of n-Hausdorff spaces and de Groot's result: |X|leq 2^{hL(X)} , for Hausdorff spaces, 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). psiw_{theta}(X), ehta-aL(X). h heta-aL(X), heta-aL_c(X) and heta-aL_{ heta}(X) . In [5] the authors introduced the Hausdorff point separating weight of a space X denoted by and proved a Hausdorff version of Charlesworth's inequality |X|leq Upsw(X). In this paper, we introduce the Urysohn point separating weight of a space X, denoted by Upsw(X) , and prove that |X|leq Upsw(X)^{ heta-aLc(X)psi(X)}, for a Urysohn space X.