New bounds on the cardinality of n-Hausdorff and n-Urysohn spaces

TL;DR

This paper introduces new cardinal functions for n-Hausdorff and n-Urysohn spaces, providing bounds on their cardinalities and exploring properties of n-Urysohn n-H-closed spaces.

Contribution

It defines new cardinal functions extending pseudocharacter concepts and derives bounds on the size of n-Urysohn spaces, advancing the understanding of their structure.

Findings

Bounds on the cardinality of n-Urysohn spaces are established.

Properties of n-Urysohn n-H-closed spaces are proved.

New cardinal functions generalize existing concepts in topology.

Abstract

Two new cardinal functions defined in the class of $n$-Hausdorff and $n$-Urysohn spaces that extend pseudocharacter and closed pseudocharacter respectively are introduced. Through these new functions bounds on the cardinality of $n$-Urysohn spaces that represent variations of known results are given. Also properties of $n$-Urysohn $n$-H-closed spaces are proved.

Equations

$$\psi (X)\le {\psi }_c(X)\le H\psi (X)\le \chi (X).$$

$$\psi (X)\le {\psi }_c(X)\le H\psi (X)\le U\psi (X)\le \chi (X).$$

Full text

Keywords: $n$-Hausdorff spaces, $n$-Urysohn spaces, $n$-H-closed space, pseudocharacter.

AMS Subject Classification: 54A25, 54D10, 54D20, 03E99.

1 Introduction

Throughout the paper $n$ denote an integer greater than or equal to $2$. In [7] Bonanzinga defined the Hausdorff number $H(X)$ of a topological space $X$ as the least cardinal number $\kappa$ such that for every subset $A\subseteq X$ with $|A|\geq\kappa$ there exists an open neighbourhood $U_{a}$ for every $a\in A$ such that $\bigcap_{a\in A}U_{a}=\emptyset$. A space $X$ is said to be $n$-Hausdorff if $H(X)\leq n$. Of course, with $|X|\geq 2$, $X$ is Hausdorff iff $H(X)=2$. The Urysohn number $U(X)$ of $X$ is the least cardinal number $\kappa$ such that for every subset $A\subseteq X$ with $|A|\geq\kappa$ there exists an open neighbourhood $U_{a}$ for every $a\in A$ such that $\bigcap_{a\in A}\overline{U_{a}}=\emptyset$. A space $X$ is said $n$-Urysohn if $U(X)\leq n$. Of course, with $|X|\geq 2$, $X$ is Urysohn iff $U(X)=2$ (see [9, 10]).

For a subset $A$ of a topological space $X$ we will denote by $[A]^{<\lambda}$ $([A]^{\lambda})$ the family of all subsets of $A$ of cardinality $<\lambda(=\lambda)$.

We consider cardinal invariants of topological spaces (see [17, 22]) and except for the Hausdorff number $H(X)$ and the Urysohn number $U(X)$, all the other cardinal functions are multiplied by $\omega$. In particular, given a topological space $X$, we will denote with $d(X)$ its density, $\chi(X)$ its character, $\pi\chi(X)$ its $\pi$-character, $t(X)$ its tightness, $\psi(X)$ its pseudocharacter, $\psi_{c}(X)$ its closed pseudocharacter, $nw(X)$ its net-weight and $l(X)$ its Lindelöf number. Let $(X,\tau)$ be a topological space, a family $\mathcal{B}\subseteq\tau$ is called $\psi$-base (or pseudobase) if $\{x\}=\bigcap\{B\in{\cal B}:x\in B\}$ for every $x\in X$. If $X$ is $T_{1}$ we define the cardinal function “pseudo weight” as follows $\psi w(X)=\min\{|{\cal B}|\,\,:\,\,{\cal B}\hbox{ is a }\psi\hbox{-base for }X\}+\omega$.

Recall that, given a space $X$, the cardinal invariant $d_{\theta}(X)$, closely related to the density $d(X)$, is defined as follows. A subspace $D\subseteq X$ is $\theta$-dense in $X$ if $D\cap\overline{U}\neq\emptyset$ for every non-empty open set $U$ of $X$. The $\theta$-density $d_{\theta}(X)$ is the least cardinality of a $\theta$-dense subspace of $X$. Observe that $d_{\theta}(X)\leq d(X)$ for any space $X$ (see [8] and [13] for more details about $\theta$-density).

In [8, Example 14] it is shown that in the inequality $|X|\leq d(X)^{\chi(X)}$ which is true for all Hausdorff spaces, one can not replace $d$ with $d_{\theta}$ (again, for Hausdorff spaces) and the following is proved.

Theorem 1.1**.**

[8] If $X$ is $n$-Urysohn space, then $|X|\leq d_{\theta}(X)^{\chi(X)}$.

For a space $X$, the $n$-Hausdorff pseudocharacter of $X$ (where $n\geq 2$), denoted by $n$-$H\psi(X)$ is the minimum infinite cardinal $\kappa$ such that for each point $x$ there is a collection $\{V(\alpha,x):\alpha<\kappa\}$ of open neighbourhoods of $x$ such that if $x_{1},...,x_{n}$ are different points in $X$ then there exist $\alpha_{1},...,\alpha_{n}<\kappa$ such that $V(\alpha_{1},x_{1})\cap...\cap V(\alpha_{n},x_{n})=\emptyset$ [7]. This function is defined for $n$-Hausdorff spaces. Of course, with $|X|\geq 2$, $2$-Hausdorff pseudocharacter of $X$ is exactly the Hausdorff pseudocharacter of $X$, denoted by $H\psi(X)$ [21].

It is well known that the following chain holds for every Hausdorff space $X$

[TABLE]

Of course, $H\psi(X)\leq\kappa\Rightarrow n$-$H\psi(X)\leq\kappa$. Also there exist spaces having countable $n$-Hausdorff pseudocharacter which are not first countable [7].

Recall the following proposition

Theorem 1.2**.**

[7, Proposition 30] If $X$ is an $n$-Hausdorff space then $|X|\leq d(X)^{n\hbox{-}H\psi(X)}$.

For a space $X$ the $n$-Urysohn pseudocharacter of $X$, denoted by $n$-$U\psi(X)$ is the minimum infinite cardinal $\kappa$ such that for each point $x$ there is a collection $\{V(\alpha,x):\alpha<\kappa\}$ of open neighbourhoods of $x$ such that if $x_{1},...,x_{n}$ are different points in $X$ then there exist $\alpha_{1},...,\alpha_{n}<\kappa$ such that $\overline{V(\alpha_{1},x_{1})}\cap...\cap\overline{V(\alpha_{n},x_{n})}=\emptyset$ [12]. This function is defined for $n$-Urysohn spaces. Of course, $2$-Urysohn pseudocharacter of $X$ is exactly the Urysohn pseudocharacter of $X$, denoted by $U\psi(X)$, and introduced in [26]. The following holds for every Urysohn space:

[TABLE]

Of course, $U\psi(X)\leq\kappa\Rightarrow n$-$U\psi(X)\leq\kappa$.

The almost Lindelöf degree with respect to closed sets, $aL_{c}(X)$ is the supremum of $aL(E,X)$ where $E$ is a closed subset of the space $X$ and $aL(E,X)$ is the minimum cardinal number $\kappa$ such that for every family $\cal U$ of open subsets of $X$ that covers $E$ there exist a subfamily $\cal V$ of $\cal U$ such that $|\cal V|\leq\kappa$ and $E\subseteq\bigcup\{\overline{V}:V\in{\cal V}\}$ [29]. The weak Lindelöf degree of a space $X$, $wL(X)$, is the minimum cardinal number $\kappa$ such that for every open cover $\cal U$ of $X$ there exist a subfamily $\cal V$ of $\cal U$ such that $|\cal V|\leq\kappa$ and $X=\overline{\bigcup{\cal V}}$.

In Section 2 we introduce a new cardinal function, denoted by $n$-$\psi(\cdot)$ and defined for $n$-Hausdorff spaces, which is an extension of pseudocharacter $\psi(\cdot)$, defined for $T_{1}$ spaces. Note that $n$-Hausdorffness and satisfying $T_{1}$ separation axiom are independent properties. We prove that $|X|\leq nw(X)^{n\hbox{-}\psi(X)}$ for each $n$-Hausdorff space $X$ which is true for $T_{1}$ spaces with $\psi(X)$ in the place of $n\hbox{-}\psi(X)$. Also we obtain that for $T_{1}$ $n$-Hausdorff spaces the inequality $|X|\leq\psi w(X)^{l(X)n\hbox{-}\psi(X)}$ holds.

In Section 3 we introduce a new cardinal function, denoted by $n$-$\psi_{c}(\cdot)$ and defined for $n$-Urysohn spaces which is an extension of the cardinal function $\psi_{c}(\cdot)$, defined for Hausdorff spaces. Using this new cardinal function we prove that $|X|\leq d(X)^{n\hbox{-}\psi_{c}(X)wt(X)}$ for every $n$-Urysohn space $X$ where $wt(\cdot)$ is a weaker form of tightness introduced by Carlson in [14], and the following modification for $n$-Urysohn spaces of the Willard-Dissanayake’s inequality: $|X|\leq d(X)^{\pi\chi(X)n\hbox{-}\psi_{c}(X)}$. Also we show that if $X$ is an $n$-Urysohn space, then $|X|\leq\pi\chi(X)^{c(X)\chi(X)}$ and we give a partial answer to the question whether the inequality $|X|\leq\pi\chi(X)^{c(X)n\hbox{-}\psi_{c}(X)}$ is true for every $n$-Urysohn space.

In Section 4 we deal with the class of $n$-H-closed spaces and we give some properties of $n$-Urysohn $n$-H-closed spaces. Recall that an $n$-Hausdorff space is called $n$-$H$-closed if it is closed in every $n$-Hausdorff space in which it is embedded [1]. In particular we notice that if $X$ is an $n$-Urysohn $n$-H-closed space, then $|X|\leq 2^{n\hbox{-}U\psi(X)}$ hence, $X\leq 2^{\chi(X)}$). In addition, since $|X|\leq 2^{\psi_{c}(X)}$ for every H-closed space $X$ [16], we pose the question whether $|X|\leq 2^{n\hbox{-}\psi_{c}(X)}$ is true for $n$-Urysohn $n$-H-closed spaces. Also we prove that $|X|\leq wL(X)^{\chi(X)}$ holds in the class of $n$-Urysohn locally $n$-H-closed spaces.

2 The $n$-pseudocharacter of an $n$-Hausdorff space

In this section we introduce the following cardinal function which is an extension of pseudocharacter in the class of $n$-Hausdorff spaces.

Definition 2.1**.**

For a space $X$, the $n$-pseudocharacter of $X$ (where $n\geq 2$), denoted by $n$-$\psi(X)$, is the minimum infinite cardinal $\kappa$ such that for each point $x$ there is a collection $\{V(\alpha,x)\,:\,\alpha<\kappa\}$ of open neighbourhoods of $x$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points, then $\bigcap_{\alpha<\kappa}V(\alpha,x_{1})\cap....\cap\bigcap_{\alpha<\kappa}V(\alpha,x_{n})=\emptyset$. In other words, $n$-$\psi(X)$ is the minimum infinite cardinal $\kappa$ such that for each point $x\in X$ there exists a $G_{\kappa}$-set $A_{x}$ containing $x$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points, then $A_{x_{1}}\cap...\cap A_{x_{n}}=\emptyset$. Note that the $2$-pseudocharacter is defined for $T_{1}$ spaces while the $n$-pseudocharacter, $n>2$, is defined for $n$-Hausdorff spaces.

The $2$-pseudocharacter is exactly the pseudocharacter as the following proposition shows.

Proposition 2.2**.**

Let $X$ be a $T_{1}$ space, then $2$-$\psi(X)=\psi(X)$.

{proof}

The proof of $2$-$\psi(X)\leq\psi(X)$ is trivial. For every $x\in X$ let $\{V(\alpha,x)\,:\,\alpha<\kappa\}$ be a family of open neighbourhood of $x$ such that if $x,y\in X$ are two different points, then $\bigcap_{\alpha<\kappa}V(\alpha,x)\cap\bigcap_{\alpha<\kappa}V(\alpha,y)=\emptyset$. We want to prove that $\{x\}=\bigcap_{\alpha<\kappa}V(\alpha,x)$. By contradiction let $y\in\bigcap_{\alpha<\kappa}V(\alpha,x)$ be a point different from $x$. Then if we consider $\bigcap_{\alpha<\kappa}V(\alpha,x)\cap\bigcap_{\alpha<\kappa}V(\alpha,y)$ it is certainly nonempty because it contains $y$. So we have a contradiction because $\bigcap_{\alpha<\kappa}V(\alpha,x)\cap\bigcap_{\alpha<\kappa}V(\alpha,y)=\emptyset$.

It is straightforward to prove the following proposition.

Proposition 2.3**.**

If $X$ is an $n$-Hausdorff space, $n\geq 2$, then $n$-$\psi(X)\leq n\hbox{-}H\psi(X)$.

Recall that any Cantor cube $2^{\kappa}$ has a dense subset $X$ with $|X|=\kappa$ and $\psi(X)=\omega$. For sake of completeness, we give the contruction of the space $X$. Let $\{s_{\alpha}:\alpha<\kappa\}$ be an enumeration of finite nonempty partial functions from $\kappa$ to 2. Recursively over $\alpha$, construct a sequence of countable partial functions $\{p_{\alpha}:\alpha\in\kappa\}$ from $\kappa$ to $2$ such that (1) $p_{\alpha}$ extends $s_{\alpha}$ and (2) $p_{\alpha}^{-1}(1)\setminus s_{\alpha}^{-1}(1)$ is an infinite subset of $\kappa\setminus\bigcup_{\beta<\alpha}p_{\beta}^{-1}(1)$. Now let $f_{\alpha}\in 2^{\kappa}$ be an extension of $p_{\alpha}$ such that $f_{\alpha}(\beta)=0$ for all $\beta\not\in dom(p_{\alpha})$. It is clear that $F=\{f_{\alpha}:\alpha<\kappa\}$ is a dense subset of $2^{\kappa}$ of cardinality $\kappa$. Also $F$ has countable pseudocharacter. Indeed, it follows from (2) that $\{f_{\alpha}\}=F\cap\bigcap\{U_{\beta}:p_{\alpha}(\beta)=1\}$, where $U_{\beta}=\{x\in 2^{\kappa}:x(\beta)=1\}$.

Further, recall that in the inequality $|X|\leq 2^{c(X)\chi(X)}$ for any Hausdorff space $X$ [19, 20], character can be replaced with Hausdorff pseudocharacter in the way similar to the proof of the same theorem (see [7, Theorem 51]).

Then, since cellularity is hereditary with respect to dense subspaces, it follows that $c(X)=\omega$. By the inequality $|X|\leq 2^{c(X)H\psi(X)}$ for any Hausdorff space, we have that if $\kappa>2^{\omega}$ any dense subset $X$ of $2^{\kappa}$, with $|X|=\kappa$ and $\psi(X)=\omega$, is an example of a Tychonoff space having countable pseudocharacter and uncountable Hausdorff pseudocharacter. Note that, $\omega=\psi(X)<H\psi(X)\leq\chi(X)=\kappa$; recently, in [6] Bella, Carlson and Spadaro constructed a Hausdorff space $X$ such that $\psi(X)<H\psi(X)<\chi(X)$ and asked if there exists a regular space distinguishing the three cardinal functions. Our example is Tychonoff and at least distinguishes $\psi(X)$ and $H\psi(X)$, giving a partial answer to this question.

Recall the following result.

Theorem 2.4**.**

[7] For a 3-Hausdorff space X, $|X|\leq 2^{2^{c(X)3-H\psi(X)}}$.

Theorem 2.4 can be extended to any $n$-Haudorff space in the following way.

Theorem 2.5**.**

For a n-Hausdorff space X, $|X|\leq 2^{2^{\cdot^{\cdot^{2^{c(X)n-H\psi(X)}}}}}$, where the power is made ($n-1$)-many times.

Then we obtain the following example.

Example 2.6**.**

A Tychonoff (hence $n$-Hausdorff) space $X$ such that $n$-$\psi(X)<n\hbox{-}H\psi(X)$.

Consider $n=3$. Let $\kappa>2^{2^{\omega}}$. Any dense subset $X$ of $2^{\kappa}$ with $|X|=\kappa$ and $\psi(X)=\omega$, is an example of a Tychonoff space such that $n$-$\psi(X)=\omega$ for any $n$ and, since $c(X)=\omega$, by Theorem 2.4 we have that $3\hbox{-}H\psi(X)>\omega$. A similar proof can be done for every $n$ using Theorem 2.5 and taking $\kappa>2^{2^{\cdot^{\cdot^{2^{\omega}}}}}$, where the power is made ($n-1$)-many times. $\triangle$

Recall the following result

Theorem 2.7**.**

[23, 2.3(a)] Let $X$ be a $T_{1}$ space, then $|X|\leq nw(X)^{\psi(X)}$.

Now we prove (see Theorem 2.8 below) that in the previous result ${\psi(X)}$ can be replaces by $n$-${\psi(X)}$ when $X$ is $n$-Hausdorff instead of $T_{1}$. Note that $n$-Hausdorffness and satisfying $T_{1}$ separation axiom are independent properties (see also [7]). Indeed, consider the set $X=\omega\cup\{p\}\cup\{*\}$ topologized as follows: $\omega$ has the discrete topology, an open basic neighbourhood of $p$ is of the form $U_{p}=\{p\}\cup(\omega\setminus F)$ where $F$ is a finite subset of $\omega$ and an open neighbourhood of $*$ is of the form $U_{*}=\{*\}\cup\{p\}\cup(\omega\setminus F)$, where $F$ is a finite subset of $\omega$. $X$ is a $3$-Hausdorff not $T_{1}$ space. Moreover, consider the set $Y=4\cup(4\times\omega)$, where $4=\{0,1,2,3\}$, topologized as follows: $4\times\omega$ has the discrete topology, a basic neighbourhood of $i=0,1,2,3$ is of the form $U_{i}=\{i\}\cup\{j\}\times(\omega\setminus F)$, where $F$ is a finite subset of $\omega$ and $j\not=i$. $Y$ is a $T_{1}$ not $3$-Hausdorff space.

Theorem 2.8**.**

Let $X$ be an $n$-Hausdorff space, then $|X|\leq nw(X)^{n\hbox{-}\psi(X)}$.

{proof}

Let $\kappa=n\hbox{-}\psi(X)$ and let $\mathcal{N}$ be a network such that $|{\cal N}|=nw(X)$.

Since $n\hbox{-}\psi(X)=\kappa$, for each $x\in X$ we can fix a collection $\{V(\alpha,x)\,:\,\alpha<\kappa\}$ of open neighbourhoods of $x$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points, then $\bigcap_{i=1}^{n}\bigcap\{V(\alpha,x_{i})\,:\,\alpha<\kappa\}=\emptyset$. Let $x\in X$. Then for every $\alpha<\kappa$ there exists $K(\alpha,x)\in\mathcal{N}$ such that $x\in K(\alpha,x)\subseteq V(\alpha,x)$. Denote by ${\cal K}(x)=\{K(\alpha,x)\,\,:\,\,\alpha<\kappa\}$. Consider the mapping $\phi:X\to[{\mathcal{N}}]^{\leq\kappa}$ defined by $x\mapsto{\cal K}(x)$. We want to prove that the cardinality of each fiber is strictly less than $n$. This will imply that $|X|\leq nw(X)^{\kappa}$. Let $x_{1},...,x_{n}$ be $n$ different points in $X$. Then $\bigcap_{i=1}^{n}\bigcap\{V(\alpha,x_{i}):\alpha<\kappa\}=\emptyset$, therefore $\bigcap_{i=1}^{n}\bigcap\{K(\alpha,x_{i})\,:\alpha<\kappa\}=\emptyset$. So we have that there exists a $j\in\{1,..,n\}$ such that ${\cal K}(x_{j})\not={\cal K}(x_{i})$ for some $i\not=j$.

Proposition 2.9**.**

[23, Proposition 2.3] For a $T_{1}$ space $X$, $nw(X)\leq\psi w(X)^{l(X)}$.

Combining Propositions 2.8 and 2.9 we obtain

Corollary 2.10**.**

For a $T_{1}$ $n$-Hausdorff space $X$, $|X|\leq\psi w(X)^{l(X)n\hbox{-}\psi(X)}$.

The authors were not able to find the source of the result following result and we give the proof for sake of completness.

Proposition 2.11**.**

For a Hausdorff space $X$, $\psi w(X)\leq d(X)^{t(X)}$.

{proof}

Let $D$ be a dense subset of $X$ such that $|D|\leq d(X)$. Let $\kappa=t(X)$ and ${\cal D}=\{\overline{A}:A\in[D]^{\leq\kappa}\}$. We have that $X=\overline{D}=\bigcup{\cal D}$ and $|{\cal D}|\leq d(X)^{\kappa}$. Put ${\cal B}=\{X\setminus\overline{A}:\overline{A}\in{\cal D}\}$. We will prove that it is a $\psi$-base. Let $x,y\in X$, then there exists an open subset $U$ of $X$ such that $x\not\in\overline{U}$ and $y\in U$. Since $D$ is dense and $t(X)\leq\kappa$, $y\in\overline{U\cap D}$ and there exists a subset $A\in[U\cap D]^{\leq\kappa}$ such that $y\in\overline{A}$. Therefore $y\not\in X\setminus\overline{A}$ and $x\in X\setminus\overline{A}$.

Question 2.12**.**

Is it true that $\psi w(X)\leq d(X)^{t(X)}$ (or, at least, $\psi w(X)\leq d(X)^{t(X)l(X)n\hbox{-}\psi(X)}$), for an n-Hausdorff space $X$?

Remark 2.13**.**

Note that, by Propositions 2.8 and 2.9, a positive answer to the previous question will allow us to prove that $|X|\leq d(X)^{t(X)l(X)n\hbox{-}\psi(X)}$ for every $T_{1}$ $n$-Hausdorff space $X$ and then to obtain a partial answer to the following question posed in [7].

Question 2.14**.**

[7] Is it true that if $X$ is a $T_{1}$ $n$-Hausdorff space, then $|X|\leq 2^{l(X)t(X)\psi(X)}$?

3 $n$-closed-pseudocharacter of an $n$-Urysohn space

In this section we introduce the following cardinal function which is a generalization of the closed pseudocharacter in the class of $n$-Urysohn spaces.

Definition 3.1**.**

For a space $X$, the $n$-closed pseudocharacter of $X$ (where $n\geq 2$) denoted by $n$-$\psi_{c}(X)$, is the minimum infinite cardinal $\kappa$ such that for each point $x$ there is a collection $\{V(\alpha,x)\,:\,\alpha<\kappa\}$ of open neighbourhoods of $x$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points, then $\bigcap\{\overline{V(\alpha,x_{1})}\,:\alpha<\kappa\}\cap....\cap\bigcap\{\overline{V(\alpha,x_{n})}\,:\alpha<\kappa\}=\emptyset$. Note that the $2$-closed pseudocharacter is defined for Hausdorff spaces while the $n$-closed pseudocharacter is defined for $n$-Urysohn spaces.

It is straightforward to prove, as in Proposition 2.2, that the $2$-closed pseudocharacter is exactly the closed pseudocharacter.

Proposition 3.2**.**

$2$-$\psi_{c}(X)=\psi_{c}(X)$, for every Hausdorff space $X$.

Also, it follows directly from the definitions that the following inequality holds.

Proposition 3.3**.**

$n\hbox{-}\psi_{c}(X)\leq n\hbox{-}U\psi(X)$, for every $n$-Uryshon space $X$.

Example 3.4**.**

A Tychonoff (hence $n$-Urysohn) space such that $n\hbox{-}\psi_{c}(X)<n\hbox{-}U\psi(X)$.

Consider the space $X$ of Example 2.6. Recall that pseudocharacter and closed pseudocharacter coincide for regular spaces and that $n$-$H\psi(Y)\leq n$-$U\psi(Y)$, for every space $Y.\hfill\triangle$

Recall the following result.

Theorem 3.5**.**

[3] If $X$ is Hausdorff, then $|X|\leq d(X)^{\psi_{c}(X)t(X)}$.

In [14] Carlson generalized the previous result introducing the following weaker form of tightness.

Definition 3.6**.**

[14] Let $X$ be a space. The weak tightness $wt(X)$ of $X$ is defined as the least infinite cardinal $\kappa$ for which there is a cover ${\cal C}$ of $X$ such that $|{\cal C}|\leq 2^{\kappa}$ and for all $C\in\mathcal{C}$, $t(C)\leq\kappa$ and $X=\bigcup_{B\in[C]^{\leq 2^{\kappa}}}\overline{B}$.

It is clear that $wt(X)\leq t(X)$.

Theorem 3.7**.**

[14] If $X$ is Hausdorff, then $|X|\leq d(X)^{\psi_{c}(X)wt(X)}$.

We prove a variation (Theorem 3.11 below) of the previous theorem in the class of $n$-Urysohn spaces. First we prove the following general result.

Theorem 3.8**.**

Let $\kappa$ and $\lambda$ be two infinite cardinals. Let $X$ be an n-Urysohn space such that

for every $p\in X$ there exists a family ${\mathcal{V}}_{p}$ of open neighbourhoods of $p$ such that $|{\mathcal{V}}_{p}|\leq\kappa$ and for every $x_{1},...,x_{n}\in X$, $(\bigcap_{V\in{\mathcal{V}}_{x_{1}}}\overline{V})\cap....\cap(\bigcap_{V\in{\mathcal{V}}_{x_{n}}}\overline{V})=\emptyset$.

• 2.

there exists a subset $S$ of $X$ such that $|S|\leq\lambda$ and $X=\bigcup_{A\in[S]^{\leq\kappa}}\overline{A}$.

Then $|X|\leq\lambda^{\kappa}$.

{proof}

For every $p\in X$ fix $A_{p}\in[S]^{\leq\kappa}$ such that $p\in\overline{A}_{p}$. For every $V\in{\mathcal{V}}_{p}$, we have that $p\in\overline{A_{p}\cap V}$ and $A_{p}\cap V\in[S]^{\leq\kappa}$. Define the function $\phi:X\to[[S]^{\leq\kappa}]^{\leq\kappa}$ such that $\phi(p)=\{A_{p}\cap V:V\in{\mathcal{V}}_{p}\}$ for every $p\in X$. We will show that $\phi$ is a $n-1$ to $1$ map; this implies that $|X|\leq|[[S]^{\leq\kappa}]^{\leq\kappa}|\leq(|S|^{\kappa})^{\kappa}\leq\lambda^{\kappa}$. Assume, by contradiction, there exist $x_{1},...,x_{n}\in X$ such that $\phi(x_{1})=...=\phi(x_{n})$. Then $\{A_{x_{1}}\cap V:V\in{\mathcal{V}}_{x_{1}}\}=...=\{A_{x_{n}}\cap V:V\in{\mathcal{V}}_{x_{n}}\}$. Therefore $(\bigcap_{V\in{\mathcal{V}}_{x_{1}}}\overline{V})\cap....\cap(\bigcap_{V\in{\mathcal{V}}_{x_{n}}}\overline{V})\not=\emptyset$; a contradiction.

In [24], Juhász and van Mill introduced the notion of a $\mathcal{C}$-saturated subset of a space $X$.

Definition 3.9**.**

Given a cover $\mathcal{C}$ of $X$, a subset $A\subseteq X$ is $\mathcal{C}$-saturated if $A\cap C$ is dense in $A$ for every $C\in\mathcal{C}$.

It is clear that the union of $\mathcal{C}$-saturated subsets is $\mathcal{C}$-saturated. The following is given in [24] in the case $\kappa=\omega$, and extended to the general case in [14].

Lemma 3.10**.**

[24, 14] Let $X$ be a space, $wt(X)=\kappa$, and let $\mathcal{C}$ be a cover witnessing that $wt(X)=\kappa$. Then for all $x\in X$ there exists $S_{x}\in[X]^{\leq 2^{\kappa}}$ such that $x\in S_{x}$ and $S_{x}$ is $\mathcal{C}$-saturated.

Theorem 3.11**.**

If $X$ is an $n$-Urysohn space, then $|X|\leq d(X)^{n\hbox{-}\psi_{c}(X)wt(X)}$.

{proof}

Put $\kappa=n$-$\psi_{c}(X)wt(X)$. Let $\mathcal{C}$ be a cover witnessing that $wt(X)\leq\kappa$ and $D$ be a dense subset of $X$ such that $|D|=d(X)$. By Lemma 3.10, for all $x\in X$ there exists a $\mathcal{C}$-saturated set $S_{x}$ such that $x\in S_{x}$ and $|S_{x}|\leq 2^{\kappa}$. Let $S=\bigcup_{d\in D}S_{d}$. Then $S$ is $\mathcal{C}$-saturated, as $S$ is the union of $\mathcal{C}$-saturated sets, and $|S|\leq|D|\cdot 2^{\kappa}\leq|D|^{\kappa}$. Observe that as $D\subseteq S$, we have that $S$ is dense in $X$. Since $n$-$\psi_{c}(X)\leq\kappa$, for every $p\in X$ fix a collection $\{V(\alpha,p)\,:\,\alpha\leq\kappa\}$ of open neighbourhoods of $p$ such that if $x_{1},...,x_{n}\in X$ are $n$ different points, then $\bigcap\{\overline{V(\alpha,x_{1})}\,:\alpha<\kappa\}\cap....\cap\bigcap\{\overline{V(\alpha,x_{n})}\,:\alpha<\kappa\}=\emptyset$. Fix $x\in X$ and let $C\in\mathcal{C}$ such that $x\in C$. We will show that for each $\alpha<\kappa$ we have $x\in\overline{V(\alpha,x)\cap S\cap C}^{C}$. As $S$ is $\mathcal{C}$-saturated (hence $S\cap C$ is dense in $S$) and $S$ is dense in $X$, we have $S\cap C$ is dense in $X$ and then $x\in V(\alpha,x)\subseteq\overline{V(\alpha,x)}\subseteq\overline{V(\alpha,x)\cap C\cap S}.$ Therefore $x\in C\cap\overline{V(\alpha,x)\cap S\cap C}=\overline{V(\alpha,x)\cap S\cap C}^{C}$. As $t(C)\leq\kappa$, there exists $A_{\alpha}\subseteq V(\alpha,x)\cap S\cap C$ such that $x\in\overline{A_{\alpha}}^{C}\subseteq\overline{A_{\alpha}}$ and $|A_{\alpha}|\leq\kappa$. Now let $A_{x}=\bigcup_{\alpha<\kappa}A_{\alpha}$ and note $x\in\overline{A_{x}}$, $|A_{x}|\leq\kappa\cdot\kappa=\kappa$, and $A_{x}\subseteq S$.

Unfixing $x$, we see that $X=\bigcup_{x\in X}\overline{A_{x}}=\bigcup_{A\in[S]^{\leq\kappa}}\overline{A}$. Letting $\lambda=d(X)^{\kappa}$, we see that the conditions of Theorem 3.8 hold and thus $|X|\leq\lambda^{\kappa}=(d(X)^{\kappa})^{\kappa}=d(X)^{\kappa}$.

Corollary 3.12**.**

If $X$ is an $n$-Urysohn space, then $|X|\leq d(X)^{n\hbox{-}\psi_{c}(X)t(X)}$.

Since $n\hbox{-}\psi_{c}(X)t(X)\leq\chi(X)$, we can obtain the following corollary.

Corollary 3.13**.**

If $X$ is an $n$-Urysohn space, then $|X|\leq d(X)^{\chi(X)}$.

Considering Theorem 1.1, it is natural to pose the following question. Recall that the $\theta$-closure of a subset $A$ of a space $X$, denoted by $cl_{\theta}(A)$, is the subset $\{x\in X:\overline{U}\cap A\not=\emptyset\hbox{ for every open neighbourhood }U\hbox{ of }x\}$ and the $t_{\theta}(X)$ is the minimum cardinal number $\kappa$ such that for every $x\in X$ and every subset $A$ of $X$ such that $x\in cl_{\theta}(A)$ there exists a subset $B$ such that $x\in cl_{\theta}(B)$ and $|B|\leq\kappa$.

Question 3.14**.**

Is it true that $|X|\leq d_{\theta}(X)^{n\hbox{-}\psi_{c}(X)t_{\theta}(X)}$ for every $n$-Urysohn space $X$?

Now we give an $n$-Urysohn generalization of the following Willard-Dissanayake result.

Theorem 3.15**.**

[29] If $X$ is a Hausdorff space then $|X|\leq d(X)^{\pi\chi(X)\psi_{c}(X)}$.

Theorem 3.16**.**

If $X$ is an $n$-Urysohn space then $|X|\leq d(X)^{\pi\chi(X)n\hbox{-}\psi_{c}(X)}$.

{proof}

Let $\kappa=\pi\chi(X)n\hbox{-}\psi_{c}(X)$ and let $D$ be a dense set such that $|D|=d(X)$. For all $x\in X$, let $\{V(\alpha,x):\alpha<\kappa\}$ be a closed $n$-pseudobase at $x$ and let ${\cal U}_{x}$ be a local $\pi$-base at $x$ such that $|{\cal U}_{x}|\leq\kappa$. Fix $x\in X$. Let ${\cal U}(\alpha,x)=\{U\in{\cal U}_{x}:U\subseteq V(\alpha,x)\}$. For all $U\in{\cal U}(\alpha,x)$, let $d(U,\alpha,x)\in U\cap D$. Let $D(\alpha,x)=\{d(U,\alpha,x):U\in{\cal U}_{x}\}$. Note $|D(\alpha,x)|\leq|{\cal U}_{x}|\leq\kappa$. We show that $x\in\overline{D(\alpha,x)}$ for every $\alpha<\kappa$. Let $x\in W$, where $W$ is open. Then $x\in W\cap V(\alpha,x)$ for every $\alpha<\kappa$. As ${\cal U}_{x}$ is a local $\pi$-base at $x$, there exists $U_{\alpha}\in{\cal U}_{x}$ such that $U_{\alpha}\subseteq W\cap V(\alpha,x)$ and thus $U_{\alpha}\in{\cal U}(\alpha,x)$ for every $\alpha<\kappa$. It follows that $d(U_{\alpha},\alpha,x)\in U_{\alpha}\subseteq W\cap V(\alpha,x)$ and thus $W\cap D(\alpha,x)\not=\emptyset$. This shows that $x\in\overline{D(\alpha,x)}\subseteq\overline{V(\alpha,x)}$ for every $\alpha<\kappa$. Consider the function $\phi:X\to[[D]^{\leq\kappa}]^{\leq\kappa}$ such that for each $x\in X$, $\phi(x)=\{D(\alpha,x):\alpha<\kappa\}$. This is a $(n-1)\hbox{-}1$ map, in fact, suppose, by contradiction, that there exist $x_{1},...,x_{n}$ different points in $X$ such that $\{D(\alpha,x_{1})\}_{\alpha<\kappa}=...=\{D(\alpha,x_{n})\}_{\alpha<\kappa}$. Since $\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{i})}$ is not empty for each $i=1,...,n$, $\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{1})}=...=\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{n})}=E$ and $\bigcap_{\alpha<\kappa}\overline{D(\alpha,x_{i})}\subseteq\bigcap_{\alpha<\kappa}\overline{V(\alpha,x_{i})}$ for every $i=1,...,n$, we have that $\emptyset\not=E\subseteq\bigcap_{i=1}^{n}\bigcap_{\alpha<\kappa}\overline{V(\alpha,x_{i})}$. That is a contradiction.

Recall the following result.

Theorem 3.17**.**

[27] If X is a Hausdorff space, then $|X|\leq\pi\chi(X)^{c(X)\psi_{c}(X)}$.

Carlson proved the following.

Theorem 3.18**.**

[13] For any space $X$, $d_{\theta}(X)\leq\pi\chi(X)^{c(X)}$.

Then by Theorems 1.1 and 3.18, we have the following result.

Theorem 3.19**.**

If $X$ is $n$-Urysohn, then $|X|\leq\pi\chi(X)^{c(X)\chi(X)}$.

Note that recently in [11] it is proved that $|X|\leq 2^{c(X)\pi\chi(X)}$, for every $n$-Urysohn homogeneous space $X$. It is natural to pose the following question.

Question 3.20**.**

Is $|X|\leq\pi\chi(X)^{c(X)n\hbox{-}\psi_{c}(X)}$ true for every $n$-Urysohn space?

Theorem 3.23 below gives a partial answer to the previous question in the class of $n$-Urysohn quasiregular spaces. Recall that a space is said to be quasiregular if for each open set $U$ there exists a open subset $V$ of $U$ contained with its closure in $U$. The following examples show that quasiregularity and $n$-Urysohness are independent properties.

Example 3.21**.**

A quasiregular space which is not $3$-Urysohn.

Consider a space similar to the one of Example 2.4. $X=3\cup(\omega\times 3)$, where $3=\{0,1,2\}$. The points from $\omega\times 3$ has the discrete topology, instead, a basic open neighbourhood of $i=0,1,2$ is $U(i,F)=\{i\}\cup\{(\omega\times j)\setminus(F\times j)\,\,:\,\,j\in\{0,1,2\}\,\,,\,\,j\not=i\}$, where $F$ is a finite subset of $\omega$. $X$ is not $3$-Urysohn since $0,1$ and $2$ belong to the closure of each open neighbourhood of them. $\triangle$

Carlson and Ridderbos in [15], under $(\mathfrak{c}^{+}=2^{\mathfrak{c}})$, constructed an example of a Urysohn ccc space which has the $\pi$-character equal to the continuum and the density equal to the successor of the continuum. Then such space cannot be quasiregular since the inequality $d(X)\leq\pi\chi(X)^{c(X)}$, proved by Sapirovski for regular spaces [26], holds also for quasiregular spaces, that is

Theorem 3.22**.**

[13] For a quasiregular space $X$, $d(X)\leq\pi\chi(X)^{c(X)}$.

The previous theorem follows from Theorem 3.18 and the fact that $d$ and $d_{\theta}$ coincide in the class of quasiregular spaces.

Then, by Theorems 3.11 and 3.22, we obtain the following partial answer to Question 3.20.

Theorem 3.23**.**

If $X$ is an $n$-Urysohn quasiregular space, then $|X|\leq\pi\chi(X)^{c(X)wt(X)n\hbox{-}\psi_{c}(X)}$.

Recall that every Hausdorff space having a compact $\pi$-base is quasiregular. In [28], Tkachenko introduced the o-tightness of a space, and in [5] Bella, Carlson and Gotchev used it to give an improvement of Theorem 3.17 in the class of Hausdorff spaces having a compact $\pi$-base. We recall that the o-tightness of a space $X$ does not exceed $\kappa$, or $ot(X)\leq\kappa$, if for every family $\mathcal{U}$ of open subsets of $X$ and for every point $x\in X$ with $x\in\overline{\bigcup{\mathcal{U}}}$ there exists a subfamily $\mathcal{V}\subseteq\mathcal{U}$ such that $|{\mathcal{V}}|\leq\kappa$ and $x\in\overline{\bigcup{\mathcal{V}}}$. We have that $ot(X)\leq t(X)$ and $ot(X)\leq c(X)$. Also $wL(X)ot(X)\leq c(X)$. Moreover $ot(X)\leq wt(X)$, for any space $X$ [5].

Theorem 3.24**.**

[5] If X is a Hausdorff space with a compact $\pi$-base, then $|X|\leq\pi\chi(X)^{wL(X)ot(X)\psi_{c}(X)}$.

It is natural to pose the following question.

Question 3.25**.**

Is $|X|\leq\pi\chi(X)^{wL(X)ot(X)n\hbox{-}\psi_{c}(X)}$ true for every $n$-Urysohn (quasiregular) space?

4 On $n$-Urysohn $n$-H-closed spaces

In this section we present some properties of $n$-Urysohn $n$-H-closed spaces motivated by similar properties that hold in the class of H-closed spaces. Recall that a space is said to be H-closed if it is Hausdorff and it is closed in every Hausdorff space in which it is embedded. An $n$-Hausdorff space is called $n$-$H$-closed if it is closed in every $n$-Hausdorff space in which it is embedded [1].

Theorem 4.1**.**

[1] For an $n$-Hausdorff space $X$ the following are equivalent:

$X$ is $n$-H-closed;

• 2.

for each open ultrafilter $\mathcal{U}$ on $X$, $|a({\cal U})|=n-1$, where $a(\cal U)$ denotes the adherence of $\cal U$;

• 3.

for every open filter ${\cal F}$ on $X$, $|a({\cal F})|\geq n-1$;

• 4.

for every $A\in[X]^{<n-1}$ and for each family of open subsets ${\cal U}$ of $X$ such that $X\setminus A\subseteq\bigcup{\cal U}$, there exists a finite subfamily ${\cal V}$ of ${\cal U}$ such that $X=\bigcup_{V\in{\cal V}}\overline{V}$.

For a topological space $(X,\tau)$ and $U\in\tau$, let $r(U)$ denote $int_{X}(\overline{U})$ and $X(s)$ denote the space whose set is $X$ and whose topology is generated by the base $\{r(U):U\in\tau\}$. A function $f:X\to Y$ is called $\theta$-continuous at $x_{0}$ if for every open neighbourhood $V$ of $f(x_{0})$ in $Y$, there exists an open neighbourhood $U$ of $x_{0}$ in $X$ such that $f(\overline{U}^{X})\subseteq\overline{V}^{Y}$.

Recall the following result.

Proposition 4.2**.**

[25] Let $f:X\to Y$ be a $\theta$-continuous surjection. If $X$ is a H-closed space, then $Y$ is H-closed.

Now we prove the following.

Proposition 4.3**.**

Let $f:X\to Y$ be a $\theta$-continuous bijection. If $X$ is an $n$-H-closed space then $Y$ is an $n$-H-closed space.

{proof}

Fix $A\in[Y]^{<n-1}$ and a family $\mathcal{U}$ of open subsets of $Y$ with $Y\setminus A\subseteq\bigcup\mathcal{U}$. For every $x\in X\setminus f^{-1}(A)$ there exists an open subset $V(x)\in\mathcal{U}$ such that $f(x)\in V(x)$. Since $f$ is $\theta$-continuous, there is an open neighbourhood $U(x)$ of $x$ in $X$ such that $f(\overline{U(x)}^{X})\subseteq\overline{V(x)}^{Y}$. Considering that the space $X$ is $n$-H-closed, there exists a finite subset $F$ of $X$ such that $X=\bigcup_{x\in F}\overline{U(x)}^{X}$. Since $Y=f(X)=f(\bigcup_{x\in F}\overline{U(x)}^{X})=\bigcup_{x\in F}f(\overline{U(x)}^{X})\subseteq\bigcup_{x\in F}\overline{V(x)}^{Y}$ holds, $Y$ is $n$-H-closed.

In [25] it is proved that the identity $id:X(s)\to X$ is a $\theta$-homeomorphism. This fact allows to prove the following proposition.

Proposition 4.4**.**

[25] A space $X$ is H-closed iff $X(s)$ is H-closed.

So, using Proposition 4.3 and the same fact, we can prove the following.

Proposition 4.5**.**

A space $X$ is $n$-H-closed iff $X(s)$ is $n$-H-closed.

Proposition 4.6**.**

[25] Let $X$ be a H-closed space and $U$ be an open subset of $X$. Then $\overline{U}$ is H-closed.

We can prove that the previous proposition can be extended to $n$-H-closed spaces.

Proposition 4.7**.**

Let $X$ be an $n$-H-closed space and $U$ be an open subset of $X$. Then $\overline{U}$ is $n$-H-closed.

{proof}

Note that the $n$-Hausdorff property is hereditary. Let $A=\overline{U}^{X}$ and $\mathcal{F}$ an open filter on $A$. Then $\{F\cap U:F\in\mathcal{F}\}$ is an open filter base on $X$. Let ${\cal G}=\{W\subseteq X:W\hbox{ is open in }X\hbox{ and }F\cap U\subseteq W\hbox{ for some }F\in\mathcal{F}\}$. Then $\mathcal{G}$ is an open filter on $X$, therefore $|a_{X}({\cal G})|\geq n-1$. Since $a_{X}({\cal G})\subseteq\bigcap_{F\in{\cal F}}\overline{(F\cap U)}^{X}=\bigcap_{F\in{\cal F}}\overline{(F\cap U)}^{A}\subseteq\bigcap_{F\in{\cal F}}\overline{F}^{A}=a_{A}({\cal F})$, we have that $|a_{A}({\cal F})|\geq|a_{X}({\cal G})|\geq n-1$.

We prove the following surprising result.

Proposition 4.8**.**

Every $n$-Hausdorff $T_{1}$ space with at least one isolated point is not $n$-H-closed.

{proof}

Suppose by contradiction that for each $A\in[X]^{n-1}$ and every family of open subsets $\cal U$ of $X$ with $X\setminus A\subseteq\bigcup{\cal U}$ there exists a finite subfamily $\cal V$ of $\cal U$ such that $X=\bigcup\{\overline{V}:V\in{\cal V}\}$. Suppose there exists one isolated point $x_{1}$. Let $A=\{x_{1}\}$. Since $X$ is $T_{1}$ there exists a family $\cal U$ of open subsets such that $X\setminus A=\bigcup{\cal U}$. Since $x_{1}$ is isolated then it does not belong to $\overline{U}$ for each $U\in{\cal U}$, therefore one cannot select a subfamily $\cal V$ of $\cal U$ such that $X=\bigcup\{\overline{V}:V\in{\cal V}\}$.

Recall that $\psi_{c}(X)=\psi_{c}(X(s))$, for any Hausdorff space $X$. Now we prove the following result.

Proposition 4.9**.**

If $X$ is an $n$-Urysohn space then $n\hbox{-}\psi_{c}(X)=n\hbox{-}\psi_{c}(X(s))$.

{proof}

Since $\overline{U}=\overline{r(U)}=\overline{r(U)}^{X(s)}$ (see [25]), if $X$ is $n$-Urysohn, then $X(s)$ is $n$-Urysohn and $n\hbox{-}\psi_{c}(X)=n\hbox{-}\psi_{c}(X(s))$.

The following result is assumed without proof in [16]; we give the proof for sake of completeness.

Lemma 4.10**.**

Let $X$ be a H-closed space. Then $\chi(X(s))\leq\psi_{c}(X)$.

{proof}

Let $\kappa=\psi_{c}(X)$ and $x\in X$. There is a family $\mathcal{U}$ of open neighbourhood of $x$ of such that $x\in\bigcap_{U\in\mathcal{U}}U\subseteq\bigcap_{U\in\mathcal{U}}\overline{U}=\{x\}$ and $\kappa=|\mathcal{U}|$. Without loss of generality we can assume that $\mathcal{U}$ is closed under finite intersections. We want to show that $\{int(\overline{U}):U\in\mathcal{U}\}$ is a neighborhood base of $x$ in $X(s)$. Let $T$ be an open neighborhood of $x$ in $X(s)$. As $X(s)$ is semiregular, we can assume that $T=int(\overline{U})$ is regular open. So, $\{x\}=\bigcap_{U\in\mathcal{U}}\overline{U}\subseteq T$ and then $X\backslash T\subseteq X\backslash\bigcap_{U\in\mathcal{U}}\overline{U}=\bigcup_{U\in\mathcal{U}}X\backslash\overline{U}$. Thus, $\{X\backslash\overline{U}:U\in\mathcal{U}\}$ is a family of regular open sets of $X$ that cover $X\backslash T$. Since $X\backslash T$ is a H-set (i.e. a regular closed subset in a H-closed space), there is a finite subset $\mathcal{G}\subseteq\mathcal{U}$ such that $X\backslash T\subseteq\bigcup_{U\in\mathcal{G}}\overline{X\backslash\overline{U}}=\bigcup_{U\in\mathcal{G}}X\setminus int(\overline{U})=X\setminus\bigcap_{U\in\mathcal{G}}int(\overline{U})$. Then, $\bigcap_{U\in\mathcal{G}}int(\overline{U})\subseteq T$ implying $x\in int(\overline{\bigcap_{U\in\mathcal{G}}U})\subseteq int(\bigcap_{U\in\mathcal{G}}\overline{U})=\bigcap_{U\in\mathcal{G}}int(\overline{U})\subseteq T$. By the arbitrarity of $x$, we conclude that $\chi(X(s))\leq\kappa$.

Since the inequalities $\psi_{c}(X)\leq\psi_{c}(X(s))\leq\chi(X(s))$ are true for every space $X$, by the previous lemma, we obtain the following proposition.

Proposition 4.11**.**

[16] Let $X$ be a H-closed space. Then $\chi(X(s))=\psi_{c}(X)$.

Then it is natural to pose the following question.

Question 4.12**.**

Let $X$ be an $n$-Urysohn $n$-H-closed space. Is $\chi(X(s))\leq n$-$\psi_{c}(X)$ true?

Recall the following amazing Dow and Porter’s result that strongly improved the Gryzlov’s theorem: if $X$ is a H-closed space with $\psi_{c}(X)\leq\omega$, then $|X|\leq 2^{\omega}$[18].

Theorem 4.13**.**

[16] Let $X$ be a H-closed space. Then $|X|\leq 2^{\psi_{c}(X)}$.

The previous theorem was proved using Propositions 4.4, 4.11 and the next result:

Theorem 4.14**.**

[16] Let $X$ be a H-closed space. Then $|X|\leq 2^{\chi(X)}$.

Theorem 4.14 can be extended to the class of $n$-H-closed spaces.

Theorem 4.15**.**

[2] Let $X$ be an $n$-H-closed space. Then $|X|\leq 2^{\chi(X)}$.

Then it is natural to pose also the following question.

Question 4.16**.**

Is it true that $|X|\leq 2^{n\hbox{-}\psi_{c}(X)}$ (or at least $|X|\leq 2^{n\hbox{-}\psi_{c}(X)t(X)}$) for every $n$-Urysohn $n$-H-closed space?

Using the following result, we give a partial answer (Theorem 4.18 below) to Question 4.16 in the class of $n$-Urysohn $n$-H-closed spaces.

Theorem 4.17**.**

[12] If $X$ is an $n$-Urysohn space. Then $|X|\leq 2^{aL(X)n\hbox{-}U\psi(X)}$.

Theorem 4.18**.**

Let $X$ be an $n$-Urysohn $n$-H-closed space. Then $|X|\leq 2^{n\hbox{-}U\psi(X)}$ (hence $|X|\leq 2^{\chi(X)}$).

Remark 4.19**.**

Note that, by Theorems 4.5 and 4.15, a positive answer to Question 4.12 allows us to give a positive answer to Question 4.16 too.

Finally, we give a bound for the cardinality of $n$-Urysohn, locally $n$-H-closed spaces. This result is an analogue of Theorem 4.2 in [4] which involves locally H-closed spaces. Recall that a space is called locally $n$-H-closed if for every point $x$ there exists an open neighbourhood which closure is $n$-H-closed.

Theorem 4.20**.**

Let $X$ be an $n$-Urysohn locally $n$-H-closed space. Then $|X|\leq wL(X)^{\chi(X)}$.

{proof}

For every $x\in X$ there exists an open neighbourhood $U_{x}$ of $x$ such that $\overline{U_{x}}$ is $n$-H-closed. By Theorem 4.15, $|\overline{U_{x}}|\leq 2^{\chi(\overline{U_{x}})}\leq 2^{\chi(X)}$. Clearly, ${\cal U}=\{U_{x}:x\in X\}$ is an open cover of $X$, then there exists ${\cal V}\in[{\cal U}]^{\leq wL(X)}$ such that $X=\overline{\bigcup{\cal V}}$. The set $\bigcup{\cal V}$ is dense in $X$ and $|\bigcup{\cal V}|\leq wL(X)2^{\chi(X)}\leq wL(X)^{\chi(X)}$. By Corollary 3.13 we can conclude the proof.

Acknowledgement: The authors express gratitude to I. Juhasz and L. Zdomskyy for useful discussions. The research was supported by “National Group for Algebric and Geometric Structures, and their Applications” (GNSAGA-INdAM).