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Urysohn spaces
Topology and its Applications 267 (2019) 106847
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Topology and its Applications www.elsevier.com/locate/topol
More cardinal inequalities in T1 /U rysohn spaces Alejandro Ramírez-Páramo a , Ricardo Cruz-Castillo b,∗ a
Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y Río Verde, Ciudad Universitaria, San Manuel Puebla, Pue., CP 72570, Mexico b Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, Carr. Pachuca Tulancingo Km. 4.5, Mineral de la Reforma, Hidalgo, CP 42184, Mexico
a r t i c l e
i n f o
Article history: Received 22 January 2019 Received in revised form 8 August 2019 Accepted 13 August 2019 Available online 2 September 2019 MSC: 54A25 03C30
a b s t r a c t In this paper we provide a partial positive answer to the question 2 in [8]: Do we have |X| ≤ 2L(X)Fa (X)ψ(X) , for every T2 –space X? Precisely, in Theorem 2.6 we show that |X| ≤ 2L(X)Fa (X)ψθ (X) for Urysohn spaces. Even more, we use cardinal functions recently introduced by Basile, Bonanzinga and Carlson in [4] to generalize some cardinal inequalities; among others, in the Theorem 2.8, we provide a result related with the well known inequality of Hajnal and Juhász: |X| ≤ 2c(X)χ(X) ; and in the Theorem 2.10, we generalize the inequality |X| ≤ 2wLc (X)χ(X) , for any Urysohn space X; due to Ofelia Alas ([1]). Some proofs of the theorems provided in this paper use the elementary submodels technique. © 2019 Elsevier B.V. All rights reserved.
Keywords: Cardinal functions Cardinal inequalities
1. Introduction and notation During the last years, the theory of cardinal invariants has become an interesting topic for some mathematicians, Bonanzinga, Stavrova and Cammaroto, among others. This is due to the ideas introduced by Bonanzinga, Cammaroto and Mateev in [6], in order to improve already known bounds of the cardinality of topological spaces. In 2013, Bonanzinga ([5]) introduced the Hausdorff and the weak Hausdorff number and obtained, among others, an important upturn for one of the most important results in the theory of cardinal invariants, which is due to Arhangel’skii: For every Hausdorff space X, |X| ≤ 2L(X)χ(X) . * Corresponding author. E-mail addresses: [email protected] (A. Ramírez-Páramo), [email protected] (R. Cruz-Castillo). https://doi.org/10.1016/j.topol.2019.106847 0166-8641/© 2019 Elsevier B.V. All rights reserved.
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In 2017, Bonanzinga, Stavrova and Staynova ([7]), introduced novel cardinal invariants to tackle old problems in the cardinal functions theory; one of them, stated by Arhangel’skii, is: Suppose that X is a T1 –space. Is it true that |X| ≤ 2L(X)χ(X) ? Recently, Basile, Bonanzinga and Carlson, in [4], took the ideas developed in [7] and introduced new cardinal functions with which they generalize several upper bounds; for instance, they show that |X| ≤ πχ(X)U c(X)ψθ (X) , for any Urysohn space X. In this paper we have two goals: The first one is to provide, in the Theorem 2.6, a partial positive answer to the next question, posed in [8]: Is it true that |X| ≤ 2L(X)Fa (X)ψ(X) , for every T2 -space X? The second one is to generalize some cardinal inequalities; particularly, in the Theorems 2.8 and 2.10 we show generalizations to the well known inequalities (respectively): (a) (Hajnal and Juhász) |X| ≤ 2c(X)χ(X) , for any Hausdorff space X. (b) (Alas) |X| ≤ 2wLc (X)χ(X) , for each Urysohn space X. The proofs of the above assertions use the elementary submodels technique (see [9]). Let’s begin introducing some definitions and notations which will be used along this work. If A is a set and κ is a cardinal number, we denote by [A]≤κ the set of all subsets of A whose cardinality is not greater than κ. All topological spaces we consider here, X, are assumed to be infinite and τ represents its topology. We use A and clX (A) to denote the closure of a set A contained in a topological space X. Similarly intX (A) denotes the interior of A in X. The next notion was introduced by Veličko in [12]. The θ-closure of a set A in X is clθ (A) = {x ∈ X: for every U , open neighborhood of x, U ∩ A = ∅}. The (θ, κ)–closure of a set A in X is [A]θκ = {clθ (C) : C ∈ [A]≤κ }. If A = [A]θκ , A is called (θ, κ)–closed. Our notation for cardinal functions is standard: L, F , χ, ψ, ψc and πχ denote the Lindelöf degree, number of free sequence, character, pseudocharacter, closed pseudocharacter and π–character respectively. We refer the reader to [10] for notation and basic facts about cardinal functions, which has not been given explicitly here. We now define additional cardinal functions that are not so well known. From now on, unless we say the opposite, if U is a family of subsets of X, we write U, instead of {U : U ∈ U}. The almost Lindelöf degree of X, denoted by aL(X), is the smallest infinite cardinal κ such that for every open cover U of X, there is V ∈ [U]≤κ such that X = V. The almost Lindelöf degree of X with respect to closed sets, denoted by aLc (X), is the smallest infinite cardinal κ such that for every closed subset A of X and every collection U of open sets in X which covers A, there is V ∈ [U]≤κ such that A ⊆ V. The weak Lindelöf degree of X with respect to closed sets, denoted by wLc (X), is the smallest infinite cardinal κ such that for every closed subset A of X and every collection U of open sets in X which covers A, there is V ∈ [U]≤κ such that A ⊆ clX ( V). Now we present another cardinal functions which were recently introduced and are not widely known. ([4]) If X is a Urysohn space, the θ–pseudocharacter of a point x ∈ X, denoted by ψθ (x, X), is the smallest cardinal κ such that {x} is the intersection of the θ–closure of the closure of a family of open neighborhoods of x having cardinality less or equal to κ. The θ-pseudocharacter of X, denoted by ψθ (X), is sup{ψθ (x, X) : x ∈ X}. The following notions were introduced in [7]. If X is a T1 –space then, for each x ∈ X, we write: Hw(x) =
{U | x ∈ U and U is open in X}.
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The Hausdorff width is HW (X) = sup{|Hw(x)| | x ∈ X}. Moreover, for every x ∈ X, ψw(x) = min{|Ux | | {U | U ∈ Ux } = Hw(x), and Ux is a family of open neighborhoods of x}, and ψw(X) = sup{ψw(x) | x ∈ X}. It is easy to prove that for any T1 –space, X, ψw(X) ≤ χ(X) holds. We will make use of the technique of elementary submodels to prove our main results. For details of this technique see [9]. Let κ be an infinite cardinal. From now on, unless otherwise specified, M is an elementary submodel which reflects the required formulas to satisfy the following conditions: (1) (2) (3) (4) (5) (6)
If C ∈ [M]≤κ , then C ∈ M. For every x ∈ X ∩ M, if Bx is a local base for x, then Bx ∈ M. If B ⊆ X and B ∈ M, then B ∈ M and clθ (B) ∈ M. If A ∈ M, then A ∈ M. If B ⊆ X is such that X ∩ M ⊆ B and B ∈ M, then X = B. For every C ⊆ X such that C ∈ M and |C| ≤ |M|, we have C ⊆ M.
2. Main results In the paper [8], Cammaroto, Catalioto and Porter introduced the notion of a-free sequence: A set {xα : α ∈ κ} ⊆ X is an a–free sequence of length κ if for every β ∈ κ, clθ ({xα : α ≤ β}) ∩ clX ({xα : α > β}) = ∅ and the a–free sequence number of X, denoted by Fa (X), is the supremum of the infinite cardinals κ such that there exists an a–free sequence in X of length κ. In the same article, the authors posed the next question: Question 2.1. Does |X| ≤ 2L(X)Fa (X)ψ(X) or |X| ≤ 2aLc (X)Fa (X)ψ(X) hold for every Hausdorff space? Our first goal is to provide a partial positive answer to the above question, particularly, to the inequality: |X| ≤ 2L(X)Fa (X)ψ(X) . Before that, we present some auxiliary results. Proposition 2.2. If L(X)Fa (X) ≤ κ, where κ is an infinite cardinal, and A ⊆ X is (κ, θ)-closed, then for every open collection in X which covers A, U, there is V ∈ [U]≤κ such that A ⊆ V. Proof. Let A be as in the hypothesis and U an open collection in X such that A ⊆ U. Let {Mα : α ∈ κ+ } be an increasing chain of elementary submodels with |Mα | ≤ κ, for every α < κ+ and such that {X, A, U} ⊆ M0 and Mα ∈ Mα+1 . Suppose that α < κ+ and, even more, that {xβ : β < α} ⊆ Mα ∩ A and {Vβ : β < α} ⊆ U ∩ Mα has been constructed. As |{xβ : β < α}| ≤ κ, then clθ ({xβ : β < α}) ⊆ [A]θκ = A and thus clθ ({xβ : β < α}) ⊆ U; hence, there exists Vα ∈ [U]≤κ such that clθ ({xβ : β < α}) ⊆ Vα and Vβ ⊆ Vα , for every β < α < κ+ . So, A \ Vα = ∅ and let xα ∈ A \ Vα . Now note that Vα and xα can be chosen in Mα+1 . Finally, the set {xα : α ∈ κ+ } is an a-free sequence, which is a contradiction. 2 Theorem 2.3. Let X be a T1 –space, and let κ be an infinite cardinal such that: 1. L(X)Fa (X) ≤ κ; 2. ψw(X) ≤ 2κ , and 3. for each A ∈ [X]≤κ , we have |clθ (A)| ≤ 2κ . Then |X| ≤ HW (X)2κ .
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Proof. For every x ∈ X, let Ux be a family of open neighborhoods of x in X such that |Ux | ≤ 2κ and {U | U ∈ Ux } = Hw(x). Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M and |M| = 2κ . Claim 2.4. X ∩ M is a (κ, θ)–closed set in X. To see this, it is enough to show that [X ∩ M]θκ ⊆ X ∩ M. If x ∈ [X ∩ M]θκ , then there is A ∈ [X ∩ M]≤κ such that x ∈ clθ (A). Now, given that A ∈ [X ∩ M]≤κ , we have A ∈ M and so clθ (A) ∈ M. Then, by the third condition, we have |clθ (A)| ≤ 2κ ; so, clθ (A) ⊆ M. Therefore, x ∈ X ∩ M and so X ∩ M is a (κ, θ)–closed set. Now, it is clear that |(X ∩ M)∗ | ≤ HW (X)2κ , where (X ∩ M)∗ = {Hw(x) : x ∈ X ∩ M}. So, the proof will be complete if we show that X ⊆ (X ∩ M)∗ . Suppose not and fix p ∈ X \ (X ∩ M)∗ . Then, for every y ∈ X ∩ M there is Uy ∈ Uy such that p ∈ / U y. Clearly, the collection {Ux : x ∈ X ∩ M} covers X ∩ M. Hence, by Proposition 2.2, there exists E ∈ [X ∩ M]≤κ such that X ∩ M ⊆ {Ux : x ∈ E} = B. By the properties of M, we have that B ∈ M, and by (5), we conclude that X = B, a contradiction with p∈ / B. Therefore X ⊆ (X ∩ M)∗ and thus |X| ≤ HW (X)2κ . 2 Lemma 2.5. If X is an Urysohn space with ψθ (X) ≤ κ and D ∈ [X]≤κ , then |clθ (D)| ≤ 2κ . Proof. Let M be an elementary submodel closed under κ sequences such that |M| ≤ 2κ , X ∈ M and D ⊆ M. For every x ∈ X, let Bx be a collection of open neighborhoods of x in X with |Bx | ≤ κ, such that {clθ (B) : B ∈ Bx } = {x}. Notice that for every x ∈ X, Bx ⊆ M. Now, if a ∈ clθ (D), then {a} = {clθ (B∩D) : B ∈ Ba } ∈ M. Hence, clθ (D) ⊆ M and so |clθ (A)| ≤ 2κ . 2 From the previous lemma, we have that if A ⊆ X, then |[A]θκ | ≤ 2κ |A|κ . Note that for any Hausdorff space X, ψw(X) ≤ ψc (X) holds and for each Urysohn space X, we have ψc (X) ≤ ψθ (X). Now we can establish and prove the result promised at the beginning of this section. Theorem 2.6. If X is an Urysohn space, then |X| ≤ 2L(X)Fa (X)ψθ (X) . Proof. Let κ = L(X)Fa (X)ψθ (X). By the previous lemma, we have that for each A ∈ [X]≤κ , |clθ (A)| ≤ 2κ . Hence, by the Theorem 2.3, |X| ≤ HW (X)2κ = 2κ (because by the definition of HW (X), we have HW (X) = 1). 2 Now we use some cardinal functions defined recently by Basile, Bonanzinga and Carlson (see [4]). For comfort of the reader, we reproduce them here: If X is a T1 –space then, for each x ∈ X, we write: U w(x) =
{clθ (U ) | x ∈ U and U is open in X}.
The Urysohn width is U W (X) = sup{|U w(x)| | x ∈ X}. Moreover, for every x ∈ X, ψwθ (x) = min{|Ux | | {clθ (U ) | U ∈ Ux } = U w(x), and Ux is a family of open neighborhoods of x}, and ψwθ (X) = sup{ψwθ (x) | x ∈ X}. It is easy to prove that if X is a T1 –space, then ψwθ (X) ≤ χ(X). Another cardinal invariant we use, called Urysohn-cellularity of a space X, and denoted by U c(X), is defined as follows:
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U c(X) = sup{|U| : U is an Urysohn-cellular family} + ω where a family of non empty open sets, U, is called Urysohn-cellular, if for every U, V ∈ U, with U = V , V ∩ U = ∅. This cardinal invariant was defined by Schröder in [11]. Proposition 2.7. Let X be a topological space and κ an infinite cardinal. Then U c(X) ≤ κ if and only if for every collection of non empty open sets, U, there is V ∈ [U]≤κ such that U ⊆ clθ ( V). Proof. (⇒) Suppose not and let U be a collection of non empty open sets in X, such that for all V ∈ [U]≤κ , ( U)\clθ ( V) = ∅. Now we will construct a collection {Uα : α < κ+ } ⊆ U and a family {Vα : α < κ+ } of non empty open sets in X such that, for any α < κ+ : 1. Vα ⊆ Uα , and 2. for every β < α, V β ∩ ( {V ρ : ρ < β}) = ∅. Take α < κ+ and suppose that for every β < α we have constructed Vβ and Uβ such that 1 and 2 hold. Let us build Uα and Vα . Given that |{Uβ : β < α}| ≤ κ, we have ( U)\clθ ( {V β : β < α}) = ∅; so, there is Uα ∈ U such that Uα \clθ ( {V β : β < α}) = ∅. Then, there exists Vα , an open set in X, which satisfy Vα ⊆ Uα and V α ∩ ( {V β : β < α}) = ∅. Hence, the collection {Vα : α < κ+ } is an Urysohn cellular family in X which is a contradiction. (⇐) It follows immediately. 2 In [2] the authors settled down (see Theorem 2) that if X is an Urysohn space, then |X| ≤ 2U c(X)k(X) ; where k(X) is the smallest cardinal number κ which satisfies that for each point x ∈ X, there is a collection Bx of closed neighborhoods of x such that |Bx | ≤ κ and if U is a closed neighborhood of x, then U contains a member of Bx . A natural question is if this inequality holds in the class of T2 –spaces and the next theorem provides a partial positive answer to it. Theorem 2.8. If X is a T1 –space, then |X| ≤ U W (X)2U c(X)k(X) . Proof. Let κ = U c(X)k(X) and for every x ∈ X, let Bx be a witness of k(X) ≤ κ, and so, a family of closed neighborhoods of x in X. Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M and |M| = 2κ . It is easily seen that |(X ∩ M)∗ | ≤ U W (X)2κ ; where (X ∩ M)∗ = {U w(x) : x ∈ X ∩ M}. So, the proof will be complete if we show that X ⊆ (X ∩ M)∗ . Suppose not and fix a x0 ∈ X \ (X ∩ M)∗ . As we have |Bx0 | ≤ κ, we can write Bx0 = {Bγ : 0 ≤ γ < κ}. For every γ < κ let Wγ = {B : B ∈ Bx , x ∈ X ∩ M and B ∩ Bγ = ∅} and let W∗γ = {intX (B) : B ∈ Wγ }. Note that for every x ∈ X ∩ M, there is γ < κ such that x ∈ W∗γ . Indeed, let x ∈ X ∩ M. Since x0 ∈ / U w(x), there is U , an open neighborhood of x in X such that x0 ∈ / clθ (U ); then, there is an open neighborhood of x0 , let us say V , such that V ∩ U = ∅. Furthermore, there are γ < κ and B ∈ Bx such that Bγ ⊆ V and B ⊆ U ; which implies Bγ ∩ B = ∅. So, we conclude that x ∈ intX (B) ⊆ W∗γ . By Proposition 2.7, for any γ < κ there is Fγ ∈ [Wγ ]≤κ such that W∗γ ⊆ clθ ( {intX (B) : B ∈ Fγ }. Finally X ∩ M ⊆ { W∗γ : γ < κ} ⊆ clθ ( {intX (B) : B ∈ Fγ } ⊆ {clθ ( Fγ ) : γ < κ} = B. Furthermore x0 ∈ / B and, by the properties of M, we have that B ∈ M, and by (5), we conclude that X = B, a contradiction with x0 ∈ / B. Therefore X ⊆ (X ∩ M)∗ and thus |X| ≤ U W (X)2κ . 2 Corollary 2.9. Let X be a topological space.
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1. If X is a T1 –space, then |X| ≤ U W (X)2c(X)χ(X) . 2. ([2]) If X is an Urysohn space, then |X| ≤ 2U c(X)k(X) . In [1], Ofelia Alas proved that |X| ≤ 2wLc (X)χ(X) , for any Urysohn space. Of course, a natural and immediate question is if this inequality holds in the class of Hausdorff spaces. Until this moment, the authors do not know an answer to this question; however, it follows from the next theorem that |X| ≤ U W (X)2wLc (X)χ(X) , for any T1 –space X with finite Urysohn number. Where the Urysohn number of a topological space X ([6]), denoted by U (X), is the smallest cardinal κ (finite or infinite) such that for every subset A ⊆ X, with |A| ≥ κ one can pick open neighborhoods Ua of a in X, for each a ∈ A, such that {U a : a ∈ A} = ∅. Theorem 2.10. Let X be a T1 –space and let κ be an infinite cardinal such that 1. wLc (X)χ(X) ≤ κ; 2. if A ∈ [X]≤κ , then |clθ (A)| ≤ 2κ . Then |X| ≤ U W (X)2κ . Proof. For every x ∈ X, let Bx be a local base of x in X such that it is closed under finite intersections and |Bx | ≤ κ. Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M, with |M| = 2κ . It is not difficult to show, using the hypothesis 2, that X ∩ M = clθ (X ∩ M). Given that |(X ∩ M)∗ | ≤ U W (X)2κ , where (X ∩ M)∗ = {U w(x) : x ∈ X ∩ M}; it is enough to show that X ⊆ (X ∩ M)∗ . Suppose not and fix p ∈ X \ (X ∩ M)∗ . Since p ∈ / X ∩ M, there is Bp ∈ Bp such that B p ∩ (X ∩ M) = ∅. Now, for each x ∈ X ∩ M, p ∈ / U w(x); given that Bx is closed under finite intersections, there is Bx ∈ Bx such that Bx ∩ Bp = ∅ and p ∈ / clθ (B x ). Clearly, {Bx : x ∈ X ∩ M} is an open cover of X ∩ M. Hence, there exists E ∈ [X ∩ M]≤κ such that X ∩ M ⊆ {Bx : x ∈ E} = B. By the properties of M, we have B ∈ M, and by (5), we conclude X = B, a contradiction, because p ∈ / B. Therefore |X| ≤ U W (X)2κ . 2 Corollary 2.11. If X is a T1 –space and U (X) < ω, then |X| ≤ U W (X)2wLc (X)χ(X) . Finally, in the next definition we are going to introduce a notion inspired in the following facts. Alas and Kočinac defined in [2] the θ–quasi–Menger number, denoted by qMθ , as follows: For every topological space X, qMθ (X) is the smallest cardinal number κ such that for any non empty closed subset A of X, and every collection {Uα : α ∈ κ} of open subsets of X with A ⊆ { Uα : α ∈ κ}, there is Vα ∈ [Uα ]≤ω such that A ⊆ {clθ ( Vα ) : α ∈ κ}. Due to for any open set U , of a space X, clθ (U ) = clX (U ), the definition of this cardinal invariant is more restrictive than the next notion, introduced by Arhangel’skiˇı in [3]: A topological space X is called strictly–quasi-κ-Lindelöf if for every non empty closed subset P of X and any collection {Uα | α ∈ κ} of families of open subsets in X such that P ⊆ { Uα | α ∈ κ}, there exists, for each α ∈ κ, Vα ∈ [Uα ]≤κ such that P ⊆ {clX ( Vα ) | α ∈ κ}. Hence, if X is a space which satisfies qMθ (X) = κ, we have that X is strictly–quasi-κ-Lindelöf. Even more, it is not difficult to see that every Lindelöf space and any space with numerable cellularity is strictly–quasi-ω-Lindelöf. Definition 2.12. Let X be a topological space and let κ be an infinite cardinal, we say that X is strictly quasi-(κ, θ)-Lindelöf if for every non empty subset P of X and any collection {Uα | α ∈ κ} of families of open subsets in X such that clθ (P ) ⊆ { Uα | α ∈ κ}, there exists, for each α ∈ κ, Vα ∈ [Uα ]≤κ such that clθ (P ) ⊆ {clθ Vα | α ∈ κ}.
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Clearly, if X is a topological space with Uc (X) ≤ κ, then X is strictly quasi-(κ, θ)-Lindelöf. Theorem 2.13. Let X be a T1 –space and κ an infinite cardinal such that: (i) χ(X) ≤ κ, (ii) X is strictly quasi-(κ, θ)-Lindelöf, and (iii) |clθ (A)| ≤ 2κ . Then |X| ≤ U W (X)2κ . Proof. For every x ∈ X, let Bx be a local base of x in X with |Bx | ≤ κ. Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M, with |M| = 2κ . Using the hypothesis 2, it is not difficult to show that X ∩ M = clθ (X ∩ M). Since |(X ∩ M)∗ | ≤ U W (X)2κ , it remains to show that X ⊆ (X ∩ M)∗ , where (X ∩ M)∗ = {U w(x) : x ∈ X ∩ M}. Suppose not and choose p ∈ X \ (X ∩ M)∗ . With out loss of generality, suppose that Bp = {Bα : α ∈ κ} and let B = {Bx : x ∈ X ∩ M} and Wα = {B ∈ B : B ∩ B α = ∅}, for any α ∈ κ. Note that X ∩ M ⊆ { Wα : α ∈ κ}. Now, as X is strictly quasi-(κ, θ)-Lindelöf, for each α < κ, there exists Vα ∈ [Wα ]≤κ such that X ∩ M ⊆ {clθ Vα | α ∈ κ} = S. By the properties of M, we have S ∈ M and, by (5), X = S, a contradiction with p ∈ / S. 2 An immediate consequence of the last theorem is the next corollary. Corollary 2.14. Let X be an Urysohn space with χ(X) ≤ κ. If X is strictly quasi-(κ, θ)-Lindelöf, then |X| ≤ 2κ . To conclude this paper we will pose some interesting question. Question 2.15. Is it true that |X| ≤ 2aLc (X)Fa (X)ψθ (X) , for any Urysohn space X? Question 2.16. Does |X| ≤ HW (X)2U c(X)χ(X) hold, for every Urysohn space X? Question 2.17. The authors do not know if |X| ≤ HW (X)2wLc (X)χ(X) , for any T1 –space X with H(X) < ω. References [1] O.T. Alas, More topological cardinal inequalities, Colloq. Math. 85 (2) (1993) 165–168. [2] O.T. Alas, L.D. Kočinac, More cardinal inequalities on Urysohn spaces, Math. Balk. 14 (2) (2000) 247–251. [3] A.V. Arhangel’skii, A generic theorem in the theory of cardinal invariants of topological spaces, Comment. Math. Univ. Carol. 36 (2) (1995) 305–327. [4] F.A. Basile, M. Bonanzinga, N. Carlson, Variations on known and recent cardinality bounds, Topol. Appl. 240 (2018) 228–237. [5] M. Bonanzinga, On the Hausdorff number of a topological space, Houst. J. Math. 39 (3) (2013) 1013–1030. [6] M. Bonanzinga, F. Cammaroto, M. Matveev, On the Urysohn number of a topological space, Quaest. Math. 34 (2011) 441–446. [7] M. Bonanzinga, D. Stavrova, P. Staynova, Separation and cardinality – some new results and old questions, Topol. Appl. 221 (2017) 556–569. [8] F. Cammaroto, A. Catalioto, J. Porter, On the cardinality of Hausdorff spaces, Topol. Appl. 160 (2013) 137–142. [9] A. Fedeli, S. Watson, Elementary submodels and cardinal functions, Topol. Proc. 20 (1995) 91–110. [10] R.E. Hodel, Cardinal functions I, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 1–61. [11] J. Schröder, Urysohn cellularity and Urysohn spread, Math. Jpn. 38 (6) (1993) 1129–1133. [12] N.V. Velicko, H-closed topological spaces, Mat. Sb. (N.S.) 70 (112) (1) (1966) 98–112 (in Russian); English transl.: Am. Math. Soc. 78 (2) (1968) 103–118.
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Topology and its Applications www.elsevier.com/locate/topol
More cardinal inequalities in T1 /U rysohn spaces Alejandro Ramírez-Páramo a , Ricardo Cruz-Castillo b,∗ a
Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y Río Verde, Ciudad Universitaria, San Manuel Puebla, Pue., CP 72570, Mexico b Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, Carr. Pachuca Tulancingo Km. 4.5, Mineral de la Reforma, Hidalgo, CP 42184, Mexico
a r t i c l e
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Article history: Received 22 January 2019 Received in revised form 8 August 2019 Accepted 13 August 2019 Available online 2 September 2019 MSC: 54A25 03C30
a b s t r a c t In this paper we provide a partial positive answer to the question 2 in [8]: Do we have |X| ≤ 2L(X)Fa (X)ψ(X) , for every T2 –space X? Precisely, in Theorem 2.6 we show that |X| ≤ 2L(X)Fa (X)ψθ (X) for Urysohn spaces. Even more, we use cardinal functions recently introduced by Basile, Bonanzinga and Carlson in [4] to generalize some cardinal inequalities; among others, in the Theorem 2.8, we provide a result related with the well known inequality of Hajnal and Juhász: |X| ≤ 2c(X)χ(X) ; and in the Theorem 2.10, we generalize the inequality |X| ≤ 2wLc (X)χ(X) , for any Urysohn space X; due to Ofelia Alas ([1]). Some proofs of the theorems provided in this paper use the elementary submodels technique. © 2019 Elsevier B.V. All rights reserved.
Keywords: Cardinal functions Cardinal inequalities
1. Introduction and notation During the last years, the theory of cardinal invariants has become an interesting topic for some mathematicians, Bonanzinga, Stavrova and Cammaroto, among others. This is due to the ideas introduced by Bonanzinga, Cammaroto and Mateev in [6], in order to improve already known bounds of the cardinality of topological spaces. In 2013, Bonanzinga ([5]) introduced the Hausdorff and the weak Hausdorff number and obtained, among others, an important upturn for one of the most important results in the theory of cardinal invariants, which is due to Arhangel’skii: For every Hausdorff space X, |X| ≤ 2L(X)χ(X) . * Corresponding author. E-mail addresses: [email protected] (A. Ramírez-Páramo), [email protected] (R. Cruz-Castillo). https://doi.org/10.1016/j.topol.2019.106847 0166-8641/© 2019 Elsevier B.V. All rights reserved.
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In 2017, Bonanzinga, Stavrova and Staynova ([7]), introduced novel cardinal invariants to tackle old problems in the cardinal functions theory; one of them, stated by Arhangel’skii, is: Suppose that X is a T1 –space. Is it true that |X| ≤ 2L(X)χ(X) ? Recently, Basile, Bonanzinga and Carlson, in [4], took the ideas developed in [7] and introduced new cardinal functions with which they generalize several upper bounds; for instance, they show that |X| ≤ πχ(X)U c(X)ψθ (X) , for any Urysohn space X. In this paper we have two goals: The first one is to provide, in the Theorem 2.6, a partial positive answer to the next question, posed in [8]: Is it true that |X| ≤ 2L(X)Fa (X)ψ(X) , for every T2 -space X? The second one is to generalize some cardinal inequalities; particularly, in the Theorems 2.8 and 2.10 we show generalizations to the well known inequalities (respectively): (a) (Hajnal and Juhász) |X| ≤ 2c(X)χ(X) , for any Hausdorff space X. (b) (Alas) |X| ≤ 2wLc (X)χ(X) , for each Urysohn space X. The proofs of the above assertions use the elementary submodels technique (see [9]). Let’s begin introducing some definitions and notations which will be used along this work. If A is a set and κ is a cardinal number, we denote by [A]≤κ the set of all subsets of A whose cardinality is not greater than κ. All topological spaces we consider here, X, are assumed to be infinite and τ represents its topology. We use A and clX (A) to denote the closure of a set A contained in a topological space X. Similarly intX (A) denotes the interior of A in X. The next notion was introduced by Veličko in [12]. The θ-closure of a set A in X is clθ (A) = {x ∈ X: for every U , open neighborhood of x, U ∩ A = ∅}. The (θ, κ)–closure of a set A in X is [A]θκ = {clθ (C) : C ∈ [A]≤κ }. If A = [A]θκ , A is called (θ, κ)–closed. Our notation for cardinal functions is standard: L, F , χ, ψ, ψc and πχ denote the Lindelöf degree, number of free sequence, character, pseudocharacter, closed pseudocharacter and π–character respectively. We refer the reader to [10] for notation and basic facts about cardinal functions, which has not been given explicitly here. We now define additional cardinal functions that are not so well known. From now on, unless we say the opposite, if U is a family of subsets of X, we write U, instead of {U : U ∈ U}. The almost Lindelöf degree of X, denoted by aL(X), is the smallest infinite cardinal κ such that for every open cover U of X, there is V ∈ [U]≤κ such that X = V. The almost Lindelöf degree of X with respect to closed sets, denoted by aLc (X), is the smallest infinite cardinal κ such that for every closed subset A of X and every collection U of open sets in X which covers A, there is V ∈ [U]≤κ such that A ⊆ V. The weak Lindelöf degree of X with respect to closed sets, denoted by wLc (X), is the smallest infinite cardinal κ such that for every closed subset A of X and every collection U of open sets in X which covers A, there is V ∈ [U]≤κ such that A ⊆ clX ( V). Now we present another cardinal functions which were recently introduced and are not widely known. ([4]) If X is a Urysohn space, the θ–pseudocharacter of a point x ∈ X, denoted by ψθ (x, X), is the smallest cardinal κ such that {x} is the intersection of the θ–closure of the closure of a family of open neighborhoods of x having cardinality less or equal to κ. The θ-pseudocharacter of X, denoted by ψθ (X), is sup{ψθ (x, X) : x ∈ X}. The following notions were introduced in [7]. If X is a T1 –space then, for each x ∈ X, we write: Hw(x) =
{U | x ∈ U and U is open in X}.
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The Hausdorff width is HW (X) = sup{|Hw(x)| | x ∈ X}. Moreover, for every x ∈ X, ψw(x) = min{|Ux | | {U | U ∈ Ux } = Hw(x), and Ux is a family of open neighborhoods of x}, and ψw(X) = sup{ψw(x) | x ∈ X}. It is easy to prove that for any T1 –space, X, ψw(X) ≤ χ(X) holds. We will make use of the technique of elementary submodels to prove our main results. For details of this technique see [9]. Let κ be an infinite cardinal. From now on, unless otherwise specified, M is an elementary submodel which reflects the required formulas to satisfy the following conditions: (1) (2) (3) (4) (5) (6)
If C ∈ [M]≤κ , then C ∈ M. For every x ∈ X ∩ M, if Bx is a local base for x, then Bx ∈ M. If B ⊆ X and B ∈ M, then B ∈ M and clθ (B) ∈ M. If A ∈ M, then A ∈ M. If B ⊆ X is such that X ∩ M ⊆ B and B ∈ M, then X = B. For every C ⊆ X such that C ∈ M and |C| ≤ |M|, we have C ⊆ M.
2. Main results In the paper [8], Cammaroto, Catalioto and Porter introduced the notion of a-free sequence: A set {xα : α ∈ κ} ⊆ X is an a–free sequence of length κ if for every β ∈ κ, clθ ({xα : α ≤ β}) ∩ clX ({xα : α > β}) = ∅ and the a–free sequence number of X, denoted by Fa (X), is the supremum of the infinite cardinals κ such that there exists an a–free sequence in X of length κ. In the same article, the authors posed the next question: Question 2.1. Does |X| ≤ 2L(X)Fa (X)ψ(X) or |X| ≤ 2aLc (X)Fa (X)ψ(X) hold for every Hausdorff space? Our first goal is to provide a partial positive answer to the above question, particularly, to the inequality: |X| ≤ 2L(X)Fa (X)ψ(X) . Before that, we present some auxiliary results. Proposition 2.2. If L(X)Fa (X) ≤ κ, where κ is an infinite cardinal, and A ⊆ X is (κ, θ)-closed, then for every open collection in X which covers A, U, there is V ∈ [U]≤κ such that A ⊆ V. Proof. Let A be as in the hypothesis and U an open collection in X such that A ⊆ U. Let {Mα : α ∈ κ+ } be an increasing chain of elementary submodels with |Mα | ≤ κ, for every α < κ+ and such that {X, A, U} ⊆ M0 and Mα ∈ Mα+1 . Suppose that α < κ+ and, even more, that {xβ : β < α} ⊆ Mα ∩ A and {Vβ : β < α} ⊆ U ∩ Mα has been constructed. As |{xβ : β < α}| ≤ κ, then clθ ({xβ : β < α}) ⊆ [A]θκ = A and thus clθ ({xβ : β < α}) ⊆ U; hence, there exists Vα ∈ [U]≤κ such that clθ ({xβ : β < α}) ⊆ Vα and Vβ ⊆ Vα , for every β < α < κ+ . So, A \ Vα = ∅ and let xα ∈ A \ Vα . Now note that Vα and xα can be chosen in Mα+1 . Finally, the set {xα : α ∈ κ+ } is an a-free sequence, which is a contradiction. 2 Theorem 2.3. Let X be a T1 –space, and let κ be an infinite cardinal such that: 1. L(X)Fa (X) ≤ κ; 2. ψw(X) ≤ 2κ , and 3. for each A ∈ [X]≤κ , we have |clθ (A)| ≤ 2κ . Then |X| ≤ HW (X)2κ .
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Proof. For every x ∈ X, let Ux be a family of open neighborhoods of x in X such that |Ux | ≤ 2κ and {U | U ∈ Ux } = Hw(x). Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M and |M| = 2κ . Claim 2.4. X ∩ M is a (κ, θ)–closed set in X. To see this, it is enough to show that [X ∩ M]θκ ⊆ X ∩ M. If x ∈ [X ∩ M]θκ , then there is A ∈ [X ∩ M]≤κ such that x ∈ clθ (A). Now, given that A ∈ [X ∩ M]≤κ , we have A ∈ M and so clθ (A) ∈ M. Then, by the third condition, we have |clθ (A)| ≤ 2κ ; so, clθ (A) ⊆ M. Therefore, x ∈ X ∩ M and so X ∩ M is a (κ, θ)–closed set. Now, it is clear that |(X ∩ M)∗ | ≤ HW (X)2κ , where (X ∩ M)∗ = {Hw(x) : x ∈ X ∩ M}. So, the proof will be complete if we show that X ⊆ (X ∩ M)∗ . Suppose not and fix p ∈ X \ (X ∩ M)∗ . Then, for every y ∈ X ∩ M there is Uy ∈ Uy such that p ∈ / U y. Clearly, the collection {Ux : x ∈ X ∩ M} covers X ∩ M. Hence, by Proposition 2.2, there exists E ∈ [X ∩ M]≤κ such that X ∩ M ⊆ {Ux : x ∈ E} = B. By the properties of M, we have that B ∈ M, and by (5), we conclude that X = B, a contradiction with p∈ / B. Therefore X ⊆ (X ∩ M)∗ and thus |X| ≤ HW (X)2κ . 2 Lemma 2.5. If X is an Urysohn space with ψθ (X) ≤ κ and D ∈ [X]≤κ , then |clθ (D)| ≤ 2κ . Proof. Let M be an elementary submodel closed under κ sequences such that |M| ≤ 2κ , X ∈ M and D ⊆ M. For every x ∈ X, let Bx be a collection of open neighborhoods of x in X with |Bx | ≤ κ, such that {clθ (B) : B ∈ Bx } = {x}. Notice that for every x ∈ X, Bx ⊆ M. Now, if a ∈ clθ (D), then {a} = {clθ (B∩D) : B ∈ Ba } ∈ M. Hence, clθ (D) ⊆ M and so |clθ (A)| ≤ 2κ . 2 From the previous lemma, we have that if A ⊆ X, then |[A]θκ | ≤ 2κ |A|κ . Note that for any Hausdorff space X, ψw(X) ≤ ψc (X) holds and for each Urysohn space X, we have ψc (X) ≤ ψθ (X). Now we can establish and prove the result promised at the beginning of this section. Theorem 2.6. If X is an Urysohn space, then |X| ≤ 2L(X)Fa (X)ψθ (X) . Proof. Let κ = L(X)Fa (X)ψθ (X). By the previous lemma, we have that for each A ∈ [X]≤κ , |clθ (A)| ≤ 2κ . Hence, by the Theorem 2.3, |X| ≤ HW (X)2κ = 2κ (because by the definition of HW (X), we have HW (X) = 1). 2 Now we use some cardinal functions defined recently by Basile, Bonanzinga and Carlson (see [4]). For comfort of the reader, we reproduce them here: If X is a T1 –space then, for each x ∈ X, we write: U w(x) =
{clθ (U ) | x ∈ U and U is open in X}.
The Urysohn width is U W (X) = sup{|U w(x)| | x ∈ X}. Moreover, for every x ∈ X, ψwθ (x) = min{|Ux | | {clθ (U ) | U ∈ Ux } = U w(x), and Ux is a family of open neighborhoods of x}, and ψwθ (X) = sup{ψwθ (x) | x ∈ X}. It is easy to prove that if X is a T1 –space, then ψwθ (X) ≤ χ(X). Another cardinal invariant we use, called Urysohn-cellularity of a space X, and denoted by U c(X), is defined as follows:
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U c(X) = sup{|U| : U is an Urysohn-cellular family} + ω where a family of non empty open sets, U, is called Urysohn-cellular, if for every U, V ∈ U, with U = V , V ∩ U = ∅. This cardinal invariant was defined by Schröder in [11]. Proposition 2.7. Let X be a topological space and κ an infinite cardinal. Then U c(X) ≤ κ if and only if for every collection of non empty open sets, U, there is V ∈ [U]≤κ such that U ⊆ clθ ( V). Proof. (⇒) Suppose not and let U be a collection of non empty open sets in X, such that for all V ∈ [U]≤κ , ( U)\clθ ( V) = ∅. Now we will construct a collection {Uα : α < κ+ } ⊆ U and a family {Vα : α < κ+ } of non empty open sets in X such that, for any α < κ+ : 1. Vα ⊆ Uα , and 2. for every β < α, V β ∩ ( {V ρ : ρ < β}) = ∅. Take α < κ+ and suppose that for every β < α we have constructed Vβ and Uβ such that 1 and 2 hold. Let us build Uα and Vα . Given that |{Uβ : β < α}| ≤ κ, we have ( U)\clθ ( {V β : β < α}) = ∅; so, there is Uα ∈ U such that Uα \clθ ( {V β : β < α}) = ∅. Then, there exists Vα , an open set in X, which satisfy Vα ⊆ Uα and V α ∩ ( {V β : β < α}) = ∅. Hence, the collection {Vα : α < κ+ } is an Urysohn cellular family in X which is a contradiction. (⇐) It follows immediately. 2 In [2] the authors settled down (see Theorem 2) that if X is an Urysohn space, then |X| ≤ 2U c(X)k(X) ; where k(X) is the smallest cardinal number κ which satisfies that for each point x ∈ X, there is a collection Bx of closed neighborhoods of x such that |Bx | ≤ κ and if U is a closed neighborhood of x, then U contains a member of Bx . A natural question is if this inequality holds in the class of T2 –spaces and the next theorem provides a partial positive answer to it. Theorem 2.8. If X is a T1 –space, then |X| ≤ U W (X)2U c(X)k(X) . Proof. Let κ = U c(X)k(X) and for every x ∈ X, let Bx be a witness of k(X) ≤ κ, and so, a family of closed neighborhoods of x in X. Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M and |M| = 2κ . It is easily seen that |(X ∩ M)∗ | ≤ U W (X)2κ ; where (X ∩ M)∗ = {U w(x) : x ∈ X ∩ M}. So, the proof will be complete if we show that X ⊆ (X ∩ M)∗ . Suppose not and fix a x0 ∈ X \ (X ∩ M)∗ . As we have |Bx0 | ≤ κ, we can write Bx0 = {Bγ : 0 ≤ γ < κ}. For every γ < κ let Wγ = {B : B ∈ Bx , x ∈ X ∩ M and B ∩ Bγ = ∅} and let W∗γ = {intX (B) : B ∈ Wγ }. Note that for every x ∈ X ∩ M, there is γ < κ such that x ∈ W∗γ . Indeed, let x ∈ X ∩ M. Since x0 ∈ / U w(x), there is U , an open neighborhood of x in X such that x0 ∈ / clθ (U ); then, there is an open neighborhood of x0 , let us say V , such that V ∩ U = ∅. Furthermore, there are γ < κ and B ∈ Bx such that Bγ ⊆ V and B ⊆ U ; which implies Bγ ∩ B = ∅. So, we conclude that x ∈ intX (B) ⊆ W∗γ . By Proposition 2.7, for any γ < κ there is Fγ ∈ [Wγ ]≤κ such that W∗γ ⊆ clθ ( {intX (B) : B ∈ Fγ }. Finally X ∩ M ⊆ { W∗γ : γ < κ} ⊆ clθ ( {intX (B) : B ∈ Fγ } ⊆ {clθ ( Fγ ) : γ < κ} = B. Furthermore x0 ∈ / B and, by the properties of M, we have that B ∈ M, and by (5), we conclude that X = B, a contradiction with x0 ∈ / B. Therefore X ⊆ (X ∩ M)∗ and thus |X| ≤ U W (X)2κ . 2 Corollary 2.9. Let X be a topological space.
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1. If X is a T1 –space, then |X| ≤ U W (X)2c(X)χ(X) . 2. ([2]) If X is an Urysohn space, then |X| ≤ 2U c(X)k(X) . In [1], Ofelia Alas proved that |X| ≤ 2wLc (X)χ(X) , for any Urysohn space. Of course, a natural and immediate question is if this inequality holds in the class of Hausdorff spaces. Until this moment, the authors do not know an answer to this question; however, it follows from the next theorem that |X| ≤ U W (X)2wLc (X)χ(X) , for any T1 –space X with finite Urysohn number. Where the Urysohn number of a topological space X ([6]), denoted by U (X), is the smallest cardinal κ (finite or infinite) such that for every subset A ⊆ X, with |A| ≥ κ one can pick open neighborhoods Ua of a in X, for each a ∈ A, such that {U a : a ∈ A} = ∅. Theorem 2.10. Let X be a T1 –space and let κ be an infinite cardinal such that 1. wLc (X)χ(X) ≤ κ; 2. if A ∈ [X]≤κ , then |clθ (A)| ≤ 2κ . Then |X| ≤ U W (X)2κ . Proof. For every x ∈ X, let Bx be a local base of x in X such that it is closed under finite intersections and |Bx | ≤ κ. Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M, with |M| = 2κ . It is not difficult to show, using the hypothesis 2, that X ∩ M = clθ (X ∩ M). Given that |(X ∩ M)∗ | ≤ U W (X)2κ , where (X ∩ M)∗ = {U w(x) : x ∈ X ∩ M}; it is enough to show that X ⊆ (X ∩ M)∗ . Suppose not and fix p ∈ X \ (X ∩ M)∗ . Since p ∈ / X ∩ M, there is Bp ∈ Bp such that B p ∩ (X ∩ M) = ∅. Now, for each x ∈ X ∩ M, p ∈ / U w(x); given that Bx is closed under finite intersections, there is Bx ∈ Bx such that Bx ∩ Bp = ∅ and p ∈ / clθ (B x ). Clearly, {Bx : x ∈ X ∩ M} is an open cover of X ∩ M. Hence, there exists E ∈ [X ∩ M]≤κ such that X ∩ M ⊆ {Bx : x ∈ E} = B. By the properties of M, we have B ∈ M, and by (5), we conclude X = B, a contradiction, because p ∈ / B. Therefore |X| ≤ U W (X)2κ . 2 Corollary 2.11. If X is a T1 –space and U (X) < ω, then |X| ≤ U W (X)2wLc (X)χ(X) . Finally, in the next definition we are going to introduce a notion inspired in the following facts. Alas and Kočinac defined in [2] the θ–quasi–Menger number, denoted by qMθ , as follows: For every topological space X, qMθ (X) is the smallest cardinal number κ such that for any non empty closed subset A of X, and every collection {Uα : α ∈ κ} of open subsets of X with A ⊆ { Uα : α ∈ κ}, there is Vα ∈ [Uα ]≤ω such that A ⊆ {clθ ( Vα ) : α ∈ κ}. Due to for any open set U , of a space X, clθ (U ) = clX (U ), the definition of this cardinal invariant is more restrictive than the next notion, introduced by Arhangel’skiˇı in [3]: A topological space X is called strictly–quasi-κ-Lindelöf if for every non empty closed subset P of X and any collection {Uα | α ∈ κ} of families of open subsets in X such that P ⊆ { Uα | α ∈ κ}, there exists, for each α ∈ κ, Vα ∈ [Uα ]≤κ such that P ⊆ {clX ( Vα ) | α ∈ κ}. Hence, if X is a space which satisfies qMθ (X) = κ, we have that X is strictly–quasi-κ-Lindelöf. Even more, it is not difficult to see that every Lindelöf space and any space with numerable cellularity is strictly–quasi-ω-Lindelöf. Definition 2.12. Let X be a topological space and let κ be an infinite cardinal, we say that X is strictly quasi-(κ, θ)-Lindelöf if for every non empty subset P of X and any collection {Uα | α ∈ κ} of families of open subsets in X such that clθ (P ) ⊆ { Uα | α ∈ κ}, there exists, for each α ∈ κ, Vα ∈ [Uα ]≤κ such that clθ (P ) ⊆ {clθ Vα | α ∈ κ}.
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Clearly, if X is a topological space with Uc (X) ≤ κ, then X is strictly quasi-(κ, θ)-Lindelöf. Theorem 2.13. Let X be a T1 –space and κ an infinite cardinal such that: (i) χ(X) ≤ κ, (ii) X is strictly quasi-(κ, θ)-Lindelöf, and (iii) |clθ (A)| ≤ 2κ . Then |X| ≤ U W (X)2κ . Proof. For every x ∈ X, let Bx be a local base of x in X with |Bx | ≤ κ. Let A = 2κ ∪ {2κ , τ, X}, and let M be an elementary submodel such that A ⊆ M, with |M| = 2κ . Using the hypothesis 2, it is not difficult to show that X ∩ M = clθ (X ∩ M). Since |(X ∩ M)∗ | ≤ U W (X)2κ , it remains to show that X ⊆ (X ∩ M)∗ , where (X ∩ M)∗ = {U w(x) : x ∈ X ∩ M}. Suppose not and choose p ∈ X \ (X ∩ M)∗ . With out loss of generality, suppose that Bp = {Bα : α ∈ κ} and let B = {Bx : x ∈ X ∩ M} and Wα = {B ∈ B : B ∩ B α = ∅}, for any α ∈ κ. Note that X ∩ M ⊆ { Wα : α ∈ κ}. Now, as X is strictly quasi-(κ, θ)-Lindelöf, for each α < κ, there exists Vα ∈ [Wα ]≤κ such that X ∩ M ⊆ {clθ Vα | α ∈ κ} = S. By the properties of M, we have S ∈ M and, by (5), X = S, a contradiction with p ∈ / S. 2 An immediate consequence of the last theorem is the next corollary. Corollary 2.14. Let X be an Urysohn space with χ(X) ≤ κ. If X is strictly quasi-(κ, θ)-Lindelöf, then |X| ≤ 2κ . To conclude this paper we will pose some interesting question. Question 2.15. Is it true that |X| ≤ 2aLc (X)Fa (X)ψθ (X) , for any Urysohn space X? Question 2.16. Does |X| ≤ HW (X)2U c(X)χ(X) hold, for every Urysohn space X? Question 2.17. The authors do not know if |X| ≤ HW (X)2wLc (X)χ(X) , for any T1 –space X with H(X) < ω. References [1] O.T. Alas, More topological cardinal inequalities, Colloq. Math. 85 (2) (1993) 165–168. [2] O.T. Alas, L.D. Kočinac, More cardinal inequalities on Urysohn spaces, Math. Balk. 14 (2) (2000) 247–251. [3] A.V. Arhangel’skii, A generic theorem in the theory of cardinal invariants of topological spaces, Comment. Math. Univ. Carol. 36 (2) (1995) 305–327. [4] F.A. Basile, M. Bonanzinga, N. Carlson, Variations on known and recent cardinality bounds, Topol. Appl. 240 (2018) 228–237. [5] M. Bonanzinga, On the Hausdorff number of a topological space, Houst. J. Math. 39 (3) (2013) 1013–1030. [6] M. Bonanzinga, F. Cammaroto, M. Matveev, On the Urysohn number of a topological space, Quaest. Math. 34 (2011) 441–446. [7] M. Bonanzinga, D. Stavrova, P. Staynova, Separation and cardinality – some new results and old questions, Topol. Appl. 221 (2017) 556–569. [8] F. Cammaroto, A. Catalioto, J. Porter, On the cardinality of Hausdorff spaces, Topol. Appl. 160 (2013) 137–142. [9] A. Fedeli, S. Watson, Elementary submodels and cardinal functions, Topol. Proc. 20 (1995) 91–110. [10] R.E. Hodel, Cardinal functions I, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 1–61. [11] J. Schröder, Urysohn cellularity and Urysohn spread, Math. Jpn. 38 (6) (1993) 1129–1133. [12] N.V. Velicko, H-closed topological spaces, Mat. Sb. (N.S.) 70 (112) (1) (1966) 98–112 (in Russian); English transl.: Am. Math. Soc. 78 (2) (1968) 103–118.