Variations on known and recent cardinality bounds

Abstract

Sapirovskii [18] proved that $|X|\leqπχ(X)^{c(X)ψ(X)}$, 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 $c(X)\leqκ$ and on pseudocharacter $ψ(X)\leqκ$ with a bound on Urysohn cellularity $Uc(X)\leqκ$ (which is a weaker conditon because $Uc(X)\leq c(X)$) and on $θ$-pseudocharacter $ψ_θ(X)\leqκ$ respectivly (note that in general $ψ(\cdot)\leqψ_θ(\cdot)$ and in the class of regular spaces $ψ(\cdot)=ψ_θ(\cdot)$). Further, in [6] the authors generalized the Dissanayake and Willard's inequality: $|X|\leq 2^{aL_{c}(X)χ(X)}$, for Hausdorff spaces $X$ [25], in the class of $n$-Hausdorff spaces and de Groot's result: $|X|\leq 2^{hL(X)}$, 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 $UW(X)$, $ψw_θ(X)$, $θ\hbox{-}aL(X)$, $hθ\hbox{-}aL(X)$, $θ\hbox{-}aL_c(X)$ and $θ\hbox{-}aL_θ(X)$. 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 $|X|\leq psw(X)^{L(X)ψ(X)}$ [7]. In this paper, we introduce the Urysohn point separating weight of a space $X$, denoted by $Upsw(X)$, and prove that $|X|\leq Upsw(X)^{θ\hbox{-}aL_{c}(X)ψ(X)}$, for a Urysohn space $X$.