Variations on known and recent cardinality bounds

الأصل · EN

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|≤ 2aLc(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-aL(X), hθ-aL(X), θ-aLc(X) and θ-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-aLc(X)ψ(X), for a Urysohn space X.