Variations on known and recent cardinality bounds

TL;DR

This paper extends known cardinality bounds in topology by introducing new functions and inequalities for Urysohn spaces, broadening the scope of classical results and providing new tools for analyzing space sizes.

Contribution

It introduces the $ heta$-pseudocharacter and Urysohn point separating weight, and generalizes existing cardinality bounds to $n$-Urysohn and $n$-Hausdorff spaces using these new functions.

Findings

Established a generalized inequality for Urysohn spaces involving $ heta$-pseudocharacter.

Introduced the Urysohn point separating weight and proved a new cardinality bound using it.

Extended classical bounds to broader classes of spaces with new cardinal functions.

Abstract

Sapirovskii [18] proved that $|X|\leq\pi\chi(X)^{c(X)\psi(X)}$, for a regular space $X$. We introduce the $\theta$-pseudocharacter of a Urysohn space $X$, denoted by $\psi_\theta (X)$, and prove that the previous inequality holds for Urysohn spaces replacing the bounds on celluarity $c(X)\leq\kappa$ and on pseudocharacter $\psi(X)\leq\kappa$ with a bound on Urysohn cellularity $Uc(X)\leq\kappa$ (which is a weaker conditon because $Uc(X)\leq c(X)$) and on $\theta$-pseudocharacter $\psi_\theta (X)\leq\kappa$ respectivly (note that in general $\psi(\cdot)\leq\psi_\theta (\cdot)$ and in the class of regular spaces $\psi(\cdot)=\psi_\theta(\cdot)$). Further, in [6] the authors generalized the Dissanayake and Willard's inequality: $|X|\leq 2^{aL_{c}(X)\chi(X)}$, 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 $UW(X)$, $\psi w_\theta(X)$, $\theta\hbox{-}aL(X)$, $h\theta\hbox{-}aL(X)$, $\theta\hbox{-}aL_c(X)$ and $\theta\hbox{-}aL_{\theta}(X)$. 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 $|X|\leq psw(X)^{L(X)\psi(X)}$ [7]. In this paper, we introduce the Urysohn point separating weight of a space $X$, denoted by $Upsw(X)$, and prove that $|X|\leq Upsw(X)^{\theta\hbox{-}aL_{c}(X)\psi(X)}$, for a Urysohn space $X$.