On the Urysohn number of a topological space

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 ψ θ (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 U c(X) ≤ κ (which is a weaker conditon because U c(X) ≤ c(X)) and on θ-pseudocharacter ψ θ (X) ≤ κ respectivly (Note that in general ψ(·) ≤ ψ θ (·) and in the class of regular spaces ψ(·) = ψ θ (·)). 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| ≤ 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 U W (X), ψw θ (X), θ-aL(X), hθ-aL(X), θ-aL c (X) and θ-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| ≤ psw(X) L(X)ψ(X) [7]. In this paper, we introduce the Urysohn point separating weight of a space X, denoted by U psw(X), and prove that |X| ≤ U psw(X) θ-aLc(X)ψ(X) , for a Urysohn space X.

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“…Considering the fact that if X is a n-Urysohn space we have that for every A ⊆ X, |[A] θ | ≤ |A| χ(X) [4] we…”

Section: A Generalization Of Sapirovskii's Inequalitymentioning

confidence: 99%

Variations on known and recent cardinality bounds

Basile

Bonanzinga

Carlson

2018

Topology and its Applications

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show abstract

“…Of course, a space is 2-Hausdorff iff it is Hausdorff. For every finite , -Hausdorff implies ( + 1)-Hausdorff, but there are ( + 1)-Hausdorff spaces which are not -Hausdorff [4]. The notion of Hausdorff number was also used in [10].…”

Section: Introductionmentioning

confidence: 99%

“…In [6] the Urysohn number (finite or infinite) U(X ) was introduced as the smallest cardinal τ such that for every subset A ⊂ X such that |A| ≥ τ one can pick neighborhoods U for all ∈ A so that ∈A U = ∅. A space X is -Urysohn (where ≥ 2 is finite) if U(X ) = ; of course, a space is 2-Urysohn iff it is Urysohn.…”

Section: Introductionmentioning

confidence: 99%

“…In [4] the Hausdorff number (finite or infinite) H(X ) of a topological space X is defined as the smallest cardinal τ such that for every subset A ⊂ X , |A| ≥ τ, there exist neighborhoods U , ∈ A, such that ∈A U = ∅. A space X is said to be -Hausdorff if H(X ) = (where ≥ 2 is finite).…”

Section: Introductionmentioning

confidence: 99%

“…A space X is -Urysohn (where ≥ 2 is finite) if U(X ) = ; of course, a space is 2-Urysohn iff it is Urysohn. The notion of Urysohn number was also used in [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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