2017/09/29 by Basile, Fortunata Aurora, Bonanzinga, Maddalena, Carlson, Nathan
#54A25 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.1709.10497
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 ψθ(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 Uc(X)≤κ (which is a weaker conditon because Uc(X)≤ c(X)) and on θ-pseudocharacter ψθ(X)≤κ respectivly (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|≤ 2^aLc(X)χ(X), for Hausdorff spaces X [25], in the class of n-Hausdorff spaces and de Groot's result: |X|≤ 2hL(X), for Hausdorff spaces [11], 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\hbox-aL(X), hθ\hbox-aL(X), θ\hbox-aLc(X) and θ\hbox-aLθ(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 Upsw(X), and prove that |X|≤ Upsw(X)^θ\hbox-aLc(X)ψ(X), for a Urysohn space X.