On n-Hausdorff homogeneous and n-Urysohn homogeneous spaces

TL;DR

This paper investigates properties of n-Hausdorff and n-Urysohn homogeneous spaces, providing bounds on their size, examples, and constructions of homogeneous extensions, revealing structural limitations and possibilities.

Contribution

It establishes upper bounds for the cardinality of n-Hausdorff and n-Urysohn homogeneous spaces, and constructs homogeneous extensions from n-Hausdorff spaces.

Findings

No n-Hausdorff 2-homogeneous space exists for n>2

Provides upper bounds for space cardinalities

Constructs homogeneous extensions as unions of n-H-closed spaces

Abstract

In this paper we study $n$-Hausdorff homogeneous and $n$-Urysohn homogeneous spaces. We give some upper bounds for the cardinality of these kind of spaces and give examples. Additionally we show that for every $n>2$, there is no $n$-Hausdorff 2-homogeneous space. Finally, for any $n$-Hausdorff space we construct an $n$-Hausdorff homogeneous extension which is the union of countably many $n$-H-closed spaces.

Equations

$${\mathcal{U}}_{\alpha }=\{V\cap X:p_{\alpha i}\in V\in \tau (Y)for\; some\; 1\le i\le k\alpha \},$$

Full text

Keywords: $n$-Hausdorff spaces, $n$-Urysohn spaces, homogeneous extensions, $n$-Katetov extensions.

AMS Subject Classification: 54A25, 54D10, 54D20, 54D35, 54D80.

1 Introduction

Throughout the paper, $n$ will always denote an integer. Given a topological space $X$, the Hausdorff number $H(X)$ (finite or infinite) of $X$ is the least cardinal number $\kappa$ such that for every subset $A\subseteq X$ with $|A|\geq\kappa$ there exist open neighbourhoods $U_{a}$, $a\in A$, such that $\bigcap_{a\in A}U_{a}=\emptyset$. A space $X$ is said n-Hausdorff, $n\geq 2$, if $H(X)\leq n$. Of course, with $|X|\geq 2$, $X$ is Hausdorff iff $n=2$ [3]; the Urysohn number $U(X)$ (finite or infinite) of $X$ is the least cardinal number $\kappa$ such that for every subset $A\subseteq X$ with $|A|\geq\kappa$ there exist open neighbourhoods $U_{a}$, $a\in A$, such that $\bigcap_{a\in A}\overline{U_{a}}=\emptyset$. A space $X$ is said n-Urysohn, $n\geq 2$, if $U(X)\leq n$. Of course, with $|X|\geq 2$, $X$ is Urysohn iff $n=2$ (see [5, 6]).

A space $X$ is homogeneous if for every $x,y\in X$ there exists a homeomorphism $h:X\rightarrow X$ such that $h(x)=y$ (see [1, 9] for surveys on homogeneous spaces).

Definition 1.1**.**

[12] A space $X$ is $2$-homogeneous if for every $x_{1},x_{2},y_{1},y_{2}\in X$ there exists a homeomorphism $h:X\rightarrow X$ such that $h(x_{1})=y_{1}$ and $h(x_{2})=y_{2}$.

In general one can give the definition of $n$-homogeneous space for any $n$. Notice that 1-homogeneity coincides with the definition of homogeneity. Of course, if a space is $(n+1)$-homogeneous, then it is $n$-homogeneous for every $n\geq 1$.

In this paper we prove that $n$-Hausdorff ($n>2$) non Hausdorff spaces are not $m$-homogeneous ($m>1$) and give an example (Example 2.7) of a 3-Urysohn homogeneous non Urysohn space. Also we show that even in the class of homogeneous spaces $(n+1)$-Hausdorff ($(n+1)$-Urysohn) spaces need not be $n$-Hausdorff (resp., $n$-Urysohn), with $n\geq 2$. Also we present some upper bounds on the cardinality of $n$-Hausdorff homogeneous and $n$-Urysohn homogeneous spaces (see also [3, 7] for other bounds on the cardinality of n-Hausdorff and n-Urysohn spaces). In particular, we prove the analogous version of the following result for $n$-Urysohn spaces and a variation of the same result for $n$-Hausdorff spaces.

Theorem 1.2**.**

[13] Let $X$ be a homogeneous Hausdorff space. Then $|X|\leq 2^{c(X)\pi\chi(X)}$.

In the last section of the paper, for any $n\geq 2$ and for any $n$-Hausdorff space, we construct an $n$-Hausdorff homogenous extension which is the countable union of $n$-H-closed spaces. Using this result we give an example of $n$-Hausdorff homogeneous space which is not $n$-Urysohn, for every $n\geq 2$.

We consider cardinal invariants of topological spaces (see [14, 18, 19]) and all 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, $\pi w(X)$ its $\pi$-weight and $c(X)$ its cellularity. Recall also that, for any space $X$, $d(X)\pi\chi(X)=\pi w(X)$.

Recall that a family $\mathcal{U}$ of open sets of a space $X$ is point-finite if for every $x\in X$, the set $\{U\in{\cal U}:x\in U\}$ is finite [14]. Tkachuck [21] defined $p(X)=sup\{|\mathcal{U}|:\mathcal{U}$ is a point-finite family in $X\}$. In [3], Bonanzinga introduced the following definition:

Definition 1.3**.**

[3] A family $\mathcal{U}$ of open sets of a space $X$ is point-($\leq n$) finite, where $n\in\mathbb{N}$, if for every $x\in X$, the set $\{U\in{\cal U}:x\in U\}$ has cardinality $\leq n$. For each $n\in\mathbb{N}$, put

$p_{n}(X)=sup\{|\mathcal{U}|:\mathcal{U}$ is a point-($\leq n$) finite family in $X\}.$

Proposition 1.4**.**

[3]

Let $X$ be a topological space. Then $p_{n}(X)=c(X)$ for every $n\in\mathbb{N}$.

2 Examples and positive results

In [3], examples of $(n+1)$-Hausdorff spaces which are not $n$-Hausdorff, for every $n\geq 2$, and an example of a space $X$ such that $H(X)=\omega$ and $H(X)\neq n$, for each $n\geq 2$, are given. Also, in [5] examples of Hausdorff $(n+1)$-Urysohn spaces which are not $n$-Urysohn were given for every $n\geq 2$.

Recall that a hyperconnected (or nowhere Hausdorff) space is a space such that the intersection of two nonempty open sets is nonempty; a space is nowhere Urysohn if there is no pair of nonempty open sets with disjoint closures.

Proposition 2.1**.**

A non Hausdorff $2$-homogeneous space is hyperconnected.

{proof}

Let $X$ be a non Hausdorff $2$-homogeneous space. Suppose that there are two nonempty open subset $V_{1}$ and $V_{2}$ of $X$ such that $V_{1}\cap V_{2}=\emptyset$. Fix two points $y_{1}\in V_{1}$ and $y_{2}\in V_{2}$. Since $X$ is not Hausdorff there exist two points $x_{1},x_{2}\in X$ such that for every open neighbourhood $U_{1}$ of $x_{1}$ and $U_{2}$ of $x_{2}$, one has that $U_{1}\cap U_{2}\not=\emptyset$. Define the homeomorphism $h:X\to X$ such that $h(x_{1})=y_{1}$ and $h(x_{2})=y_{2}$. Of course $h^{\leftarrow}(V_{1})\cap h^{\leftarrow}(V_{2})\not=\emptyset$. Pick a point $x\in h^{\leftarrow}(V_{1})\cap h^{\leftarrow}(V_{2})$, then $h(x)\in V_{1}\cap V_{2}$, a contradiction.

Proposition 2.2**.**

A non Urysohn $2$-homogeneous space is nowhere Urysohn.

{proof}

The proof is similar to the one of Proposition 2.1. One just needs to consider that if $h:X\to X$ is a homeomorphism, then $h(\overline{A})=\overline{h(A)}$ for each $A\subseteq X$.

The following proposition follows directly from the definition.

Proposition 2.3**.**

A space $X$ is hyperconnected if and only if for every finite $A\subseteq X$, $|A|=n$, $n\geq 2$, and for every choice of neighbourhoods $U_{a}$, $a\in A$, $\bigcap_{a\in A}U_{a}\neq\emptyset$.

By Proposition 2.3 one can easily show the following.

Proposition 2.4**.**

Let $n\geq 2$. Any $n$-Hausdorff space is not hyperconnected.

Theorem 2.5**.**

There is no $n$-Hausdorff non Hausdorff $m$-homogeneous space for every $n>2$ and every $m>1$.

{proof}

It follows directly from propositions 2.4 and 2.1.

The following example shows that there exist 3-Hausdorff homogeneous spaces.

Example 2.6**.**

A countable $3$-Hausdorff homogeneous space.

Consider the space $X$ of natural numbers with the topology generated the base $\{\{n,n+1\}:n\hbox{ is even}\}$. $X$ is a $3$-Hausdorff homogeneous space. $\triangle$

Note that the space of the previous example is a homogeneous space which is not $2$-homogeneous.

The analougues Proposition 2.4 and Theorem 2.5 for $n$-Urysohn spaces do not hold, as the following example shows.

Example 2.7**.**

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

Consider the well known “irrational slope space”, also called Bing’s Tripod space (see [20, Example 75]). This space is $n$-homogeneous, $n\geq 1$ [2], and $3$-Urysohn. $\triangle$

Recall that for every $n\geq 2$ there exist examples of $(n+1)$-Hausdorff spaces which are not $n$-Hausdorff [3], and examples of $(n+1)$-Urysohn spaces which are not $n$-Urysohn [6]. Then it is natural to pose the following questions.

Question 2.8**.**

Is every $(n+1)$-Hausdorff homogeneous space $n$-Hausdorff, for each $n\geq 2$?

Question 2.9**.**

Is every $(n+1)$-Urysohn homogeneous space $n$-Urysohn, for each $n\geq 2$?

Examples 2.6 and 2.7 answer negatively to questions 2.8 and 2.9 respectively for $n=2$. Note that the space of Example 2.6 is $3$-Urysohn, and the construction can be generalized to obtain $(n+1)$-Urysohn non $n$-Hausdorff spaces for each $n\geq 2$.

In [3], Bonanzinga gives an example of an $\omega$-Hausdorff space which is not $n$-Hausdorff for every $n\geq 2$. Now we give a countable $\omega$-Hausdorff homogeneous space which is not $n$-Hausdorff for every $n\geq 2$.

Example 2.10**.**

There is a countable $T_{1}$ hyperconnected (hence not $n$-Hausdorff for every $n\geq 2$) space, which is $\omega$-Hausdorff and homogeneous.

In [8], the following space is constructed. Let $X=\mathbb{Z}\times\mathbb{Z}$ and $\mathcal{B}$ $=\{U_{j,k},V_{j,k}:j,k\in\mathbb{Z}\}$ is the subbase for the topology, where

• $U_{j,k}=\{(x,y)\in{\mathbb{Z}}^{2}:x>j$ or $y>k\}$
• $V_{j,k}=\{(x,y)\in{\mathbb{Z}}^{2}:x<j$ or $y<k\}.$

This is a $T_{1}$ hyperconnected, hence not $n$-Hausdorff space for every $n\geq 2$ which is $\omega$-Hausdorff, homogeneous, first countable, Lindelof. $\triangle$

In [6], Bonanzinga, Cammaroto and Matveev constructed an Hausdorff space with extent equal to $\kappa$, $\kappa\geq\omega$, which is not $\kappa$-Urysohn (we give this example for sake of completeness, see Example 2.12 below). The construction of such a space may be considered a modification of irrational slope space [20, Example 75]. Since the irrational slope space is homogeneous, it is natural to ask the following.

Question 2.11**.**

Is the space in Example 2.12 homogeneous?

Example 2.12**.**

For every cardinal $\kappa$ there exists a Hausdorff space with extent equal to $\kappa$, $\kappa\geq\omega$, which is not $\kappa$-Urysohn.

Let $˜\tilde{D}=\{d_{\alpha,n}:\alpha<\kappa,n\in\omega\}$ be a discrete space of cardinality $\kappa$, and $D=\tilde{˜D}\cup\{p\}$ the one point compactification of $\tilde{˜D}$. Put $E=D\cup\{d^{*}\}$ where $d^{*}$ is isolated in $E$ and is not in $D$. Consider $\kappa^{+}$ with order topology, $D\times\kappa^{+}$ with the Tychonoff product topology, and denote $W=\{p\}\times\kappa^{+}$; then $W$ is a subspace of $D\times\kappa^{+}$ homeomorphic to $\kappa^{+}$. Also, for $\alpha<\kappa^{+}$ denote $W_{\alpha}=\{p\}\times[\alpha,\kappa^{+})$. For $\alpha<\kappa$, $\beta<\kappa^{+}$, denote $D_{\alpha}=\{d_{\alpha,n}:n\in\omega\}$, and $T_{\alpha,\beta}=D_{\alpha}\times[\beta,\kappa^{+})\subset D\times\kappa^{+}$. Let $\vec{p}$ be the point in $E^{\kappa^{+}}$ with all coordinates equal to $p$. Let $S=\{x\in E^{\kappa^{+}}:|\{\alpha<\kappa^{+}:x(\alpha)\neq p\}|\leq\kappa\}$ be the $\Sigma_{k}$-product in $E^{\kappa^{+}}$ with center at $\vec{p}$. It can be proved that there is a homeomorphic embedding $f:D\times\kappa^{+}\rightarrow E^{\kappa^{+}}$ such that

• (1)

$f(D\times\kappa^{+})\cap S=f(W)$.

• (2)

$f(W)$ is closed in $S$ and homeomorphic to $\kappa^{+}$ with the order topology.

• (3)

for every distinct $\alpha,\gamma<\kappa$, the sets $f(T_{\alpha,0})$ and $f(T_{\gamma,0})$ can be separated by open neighbourhoods in $E^{\kappa^{+}}$.

• (4)

$\overline{f(T_{\alpha,\beta})}\cap S=f(W_{\beta})$.

Finally, let $L=\{l_{\alpha}:\alpha<\kappa\}$ (where all points $l_{\alpha}$ are distinct) be a set disjoint from $E^{\kappa^{+}}$ and topologize $X=S\cup L$ as follows: $S$, with the topology inherited from $E^{\kappa^{+}}$ is open in $X$; a basic neighbourhood of $l_{\alpha}$ takes the form $\{l_{\alpha}\}\cup(U\cap S)$ where $U$ is arbitrary neighbourhood (in $E^{\kappa^{+}}$) of $f(T_{\alpha,\beta})$ for some $\beta<\kappa^{+}$. We recall that $L$ is closed discrete in this space, so $e(X)\geq\kappa$, and for every family $\{U_{l}:l\in L\}$ of neighbourhoods of points $l\in L$ in $X$, $\bigcap\{\overline{U_{l}}:l\in L\}\neq\emptyset$, so it is not $\kappa$-Urysohn. $\triangle$

3 On the cardinality of n-Hausdorff homogeneous and $n$-Urysohn homogeneous spaces.

In [17], Hajnal and Juhaśz proved that, for every Hausdorff space $X$, $|X|\leq 2^{c(X)\chi(X)}$. In [3] Bonanzinga proved that $|X|\leq 2^{2^{c(X)\chi(X)}}$ for every $3$-Hausdorff space $X$ and asked if $|X|\leq 2^{c(X)\chi(X)}$ holds for every $n$-Hausdorff space $X$, with $n\geq 2$. In [16] Gotchev, using the cardinal function called “non Hausdorff number” introduced independently from [3], gave a positive answer to the previous question.

In [13], Carlson and Ridderbos proved the following result.

Theorem 3.1**.**

[13] Let $X$ be a homogeneous Hausdorff space. Then $|X|\leq 2^{c(X)\pi\chi(X)}$.

In fact, in [13] it is proved that the previous theorem holds for power homogeneous Hausdorff spaces. Recall that a topological space $X$ is power homogeneous if $X^{\mu}$ is homogeneous for some cardinal number $\mu$. Clearly, if a space is homogeneous it is power homogeneous.

Then, it is natural to pose the following question.

Question 3.2**.**

Is $|X|\leq 2^{c(X)\pi\chi(X)}$ true for every homogeneous space $X$ such that $H(X)$ is finite?

In the following we give partial answers to the previous question.

Given a set $A$ and a cardinal $\kappa$, $[X]^{\kappa}$ denotes the set of all subsets of $A$ whose cardinality is $\kappa$.

Theorem 3.3**.**

[15] Let $\kappa$ be a cardinal number and $f:[(2^{2^{\kappa}})^{+}]^{3}\to\kappa$ a function, then there exists a subset $H\in[(2^{2^{\kappa}})^{+}]^{\kappa^{+}}$ such that $f\restriction[H]^{3}$ is constant.

Theorem 3.4**.**

Let $X$ be a $3$-Hausdorff homogeneous space. Then

$|X|\leq 2^{2^{c(X)\pi\chi(X)}}$

{proof}

Let $c(X)\pi\chi(X)=\kappa$. Then, by Proposition 1.4, we have $p_{2}(X)\leq\kappa$. Suppose that $|X|\geq(2^{2^{\kappa}})^{+}$. For every triple $x_{1},x_{2},x_{3}\in X$ of distinct points select neighbouroods $U_{i}(x_{1},x_{2},x_{3})$ of $x_{i}$ for $i=1,2,3$ such that

$\bigcap_{i=1}^{3}U_{i}(x_{1},x_{2},x_{3})=\emptyset$. Fix a point $p\in X$ and a local $\pi$-base $\mathcal{B}$ for $p$ with $|\mathcal{B}|=\kappa$. Since the space is homogeneous, there exists a family $\{h_{x}\}_{x\in X}$ of homeomorphisms $h_{x}:X\rightarrow X$ such that $h_{x}(p)=x$ for every $x\in X$. Fix distinct points $x_{1},x_{2},x_{3}\in X$ and observe that the set $\bigcap_{i=1}^{3}h^{\leftarrow}_{x_{i}}(U_{i}(x_{1},x_{2},x_{3}))$ is an open neighbourhood of $p$; since $\mathcal{B}$ is a $\pi$-base, there is a non empty $B(x_{1},x_{2},x_{3})\in\mathcal{B}$ such that $B(x_{1},x_{2},x_{3})$ is contained in it. Consider now the function $f:[X]^{3}\rightarrow\mathcal{B}$ defined by $f(\{x_{1},x_{2},x_{3}\})=B(x_{1},x_{2},x_{3})$. Then by Theorem 3.3 there is $Z\in[X]^{\kappa^{+}}$ and $B\in\mathcal{B}$ such that $f\restriction[Z]^{3}=\{B\}$.

Now, the family $\{h_{z}(B):z\in Z\}$ is point-($\leq 2$) finite in $X$. To see this, suppose by way of contradiction that there exists $x_{0}\in X$ such that $|\{h_{z}(B):x_{0}\in h_{z}(B)\}|=3$. So there are $z_{1},z_{2},z_{3}\in X$ such that $x_{0}\in h_{z_{i}}(B),\hskip 2.84526pti=1,2,3$. This implies $x_{0}\in h_{z_{i}}(B)\subseteq h_{z_{i}}(\bigcap\limits_{i=1}^{3}h_{z_{i}}^{\leftarrow}(U_{i}(z_{1},z_{2},z_{3})))\subseteq h_{z_{i}}(h_{z_{i}}^{\leftarrow}(U_{i}(z_{1},z_{2},z_{3})))=U_{i}(z_{1},z_{2},z_{3})$. Then, $x_{0}\in\bigcap\limits_{i=1}^{3}U_{i}(z_{1},z_{2},z_{3})\neq\emptyset$, a contradiction.

Furthermore, $\{h_{z}(B):z\in Z\}$ has cardinality exactly $\kappa^{+}$. Otherwise there exists $z_{0}\in Z$ s.t. $|\{z\in Z:h_{z}(B)=h_{z_{0}}(B)\}|=\kappa^{+}$. As before, from $h_{z_{0}}(B)\subseteq U_{i}(z_{1},z_{2},z_{3})$ for every triple of elements in $\{z\in Z:h_{z}(B)=h_{z_{0}}(B)\}$ we obtain a contradiction.

Thus $p_{2}(X)=\kappa^{+}$, a contradiction with $p_{2}(X)\leq\kappa$. This concludes the proof.

Recall the following result.

Theorem 3.5**.**

Let $\kappa$ be a cardinal number, $n\geq 3$ and $f:[(2^{2^{.^{.^{.^{2^{\kappa}}}}}})^{+}]^{n}\to\kappa$ a function (where the power is made $(n-1)$-many times), then there exists a subset $H\in[(2^{2^{.^{.^{.^{2^{\kappa}}}}}})^{+}]^{\kappa^{+}}$ such that $f\restriction[H]^{n}$ is constant.

Theorem 3.6**.**

Let $X$ be an $n$-Hausdorff homogeneous space, with $n\geq 2$. Then

$|X|\leq 2^{2^{.^{.^{.^{2^{c(X)\pi\chi(X)}}}}}}$

where the power is made $(n-1)$-many times.

{proof}

Similar to the proof of the previous theorem using Theorem 3.5 instead of Theorem 3.3.

Next Theorem 3.12 shows that Question 3.2 has a positive answer if $H(X)$ is replaced by $U(X)$.

In [11], Carlson, Porter and Ridderbos the following result.

Theorem 3.7**.**

[11] If $X$ is an $n$-Hausdorff homogeneous space, with $n\geq 2$, then $|X|\leq d(X)^{\pi\chi(X)}$.

Also recall that a space is quasiregular if every nonempty open set contains a nonempty regular closed set.

Theorem 3.8**.**

If $X$ is an $n$-Hausdorff quasiregular homogeneous space with $n\geq 2$, then $|X|\leq 2^{c(X)\pi\chi(X)}$.

Definition 3.9**.**

[23] Let $X$ be a space. For $A\subseteq X$ , the $\theta$-closure of $A$ is defined by

$cl_{\theta}(A)=\{x\in X:\overline{V}\cap A\neq\emptyset\hbox{ for every open set }V\hbox{containing }x\}$

A set $A\subseteq X$ is $\theta$-dense if $cl_{\theta}(A)=X$ . The $\theta$-density $d_{\theta}(X)$ of $X$ is defined as the least cardinality of a $\theta$-dense subset of $X$.

Theorem 3.10**.**

[11]

Let $X$ be an $n$-Urysohn homogeneous space, where $n\geq 2$. Then $|X|\leq d_{\theta}(X)^{\pi\chi(X)}$.

Theorem 3.11**.**

[10] Let $X$ be a space. Then $d_{\theta}(X)\leq\pi\chi(X)^{c(X)}$.

By Theorems 3.10 and 3.11, we obtain the following result.

Theorem 3.12**.**

Let $X$ be an $n$-Urysohn homogeneous space, where $n\geq 2$. Then $|X|\leq 2^{c(X)\pi\chi(X)}$.

{proof}

As $X$ is $n$-Hausdorff and homogeneous, we have $|X|\leq d(X)^{\pi\chi(X)}$ by Theorem 3.7. As $X$ is quasiregular, it follows that $d_{\theta}(X)=d(X)$. By Theorem 3.11, we have $d(X)\leq\pi\chi(X)^{c(X)}$. Thus, $|X|\leq d(X)^{\pi\chi(X)}\leq\left(\pi\chi(X)^{c(X)}\right)^{\pi\chi{X}}=2^{c(X)\pi\chi(X)}$.

4 A homogenous extension of an $n$-Hausdorff space

In [12] Carlson, Porter and Ridderbos proved the following result.

Theorem 4.1**.**

[12] Let $X$ be a Hausdorff space. Then $X$ can be embedded in a homogeneous space that is the countable union of $H$-closed spaces.

In the following we construct (Theorem 4.12 below) a homogeneous extension of an $n$-Hausdorff space, $n\geq 2$, which is a countable union of $n$-H-closed spaces.

Definition 4.2**.**

[4] Let $n\geq 2$. An $n$-Hausdorff space $X$ is called n-H-closed if $X$ is closed in every $n$-Hausdorff space $Y$ in which $X$ is embedded.

Given a space $X$ and an ultrafilter $\mathcal{U}$ on it, we put $a{\cal U}=\bigcap\{\overline{U}:U\in\mathcal{U}\}$.

For an $n$-Hausdorff space $X$, with $n\geq 2$, an open ultrafilter $\mathcal{U}$ on $X$ is said to be full if $|a\mathcal{U}|$ $=n-1$.

Theorem 4.3**.**

[4] Let $n\geq 2$, and $X$ be a space. The following are equivalent:

• (a)

$X$ is $n$-Hausdorff

• (b)

if $\mathcal{U}$ is an open ultrafilter of $X$, then $|a\mathcal{U}|$ $\leq n-1$.

Theorem 4.4**.**

[4] Let $n\geq 2$, and $X$ be an $n$-Hausdorff space. The following are equivalent:

• (a)

$X$ is $n$-H-closed

• (b)

every open ultrafilter on $X$ is full.

Recall the following construction, made in [4]. Let $n\geq 2$, $X$ be an $n$-Hausdorff space and ${\mathfrak{U}}=\{{\cal U}:{\cal U}\hbox{ is an open ultrafilter such that }|a{\cal U}|<n-1\}$. We index ${\mathfrak{U}}$ by ${\mathfrak{U}}=\{{\cal U}_{\alpha}:\alpha\in|{\mathfrak{U}}|\}$. For each $\alpha\in|{\mathfrak{U}}|$, let $k\alpha=(n-1)-|a{\cal U}_{\alpha}|$ and $\{p_{\alpha i}:1\leq i\leq k\alpha\}$ be a set of distinct points disjoint from $X$. Let $Y=X\cup\{p_{\alpha i}:1\leq i\leq k\alpha,\alpha\in|{\mathfrak{U}}|\}$. A set $V$ is defined to be open in $Y$ if $V\cap X$ is open in $X$ and if $p_{\alpha i}\in V$ for $1\leq i\leq k\alpha$, $V\cap X\in{\cal U}_{\alpha}$. The space $Y$ is an $n$-Hausdorff space.

In the following results we use the notation of the previous contruction.

Proposition 4.5**.**

[4] For every $\alpha\in|{\mathfrak{U}}|$,

[TABLE]

where $\tau(Y)$ is the topology on $Y$.

By the previous proposition the space $Y$ has the property that every open ultrafilter on $Y$ is full. Indeed the points $p_{\alpha i}$, $1\leq i\leq k\alpha$, added to the space $X$, are in the closure of each element of ${\cal U}_{\alpha}$. Therefore the space $Y$ is $n$-H-closed.

Definition 4.6**.**

[4] Let $n\geq 2$, $S$ and $T$ be $n$-H-closed extensions of an $n$-Hausdorff space $X$. We say $S$ is projectively larger than $T$ if there is a continuous surjection $f:S\rightarrow T$ such that $f(x)=x$ for $x\in X$.

This projectively larger function may not be unique [4].

Theorem 4.7**.**

[4]

Let $n\geq 2$, $X$ be $n$-Hausdorff space and $Y$ be the $n$-H-closed extension of $X$ constructed above. If $Z$ is an $n$-H-closed extension of $X$, there is a continuous surjection $f:Y\rightarrow Z$ such that $f(x)=x$ for all $x\in X$.

Theorem 4.7 shows that the $n$-H-closed extension Y of X is projectively larger than every $n$-H-closed extension of X. Moreover, the space Y has an interesting unique property as it is noted in the next result.

Theorem 4.8**.**

[4]

Let $n\geq 2$, $X$ be an $n$-Hausdorff space and $Y$ be the $n$-H-closed extension of $X$ described above. Let $f:Y\rightarrow Y$ be a continuous surjection such that $f(x)=x$ for all $x\in X$. Then $f$ is a homeomorphism.

Remark 4.9**.**

In the class of Hausdorff spaces the function in Definition 4.6 is unique [4]. Sometimes this is a problem in non-Hausdorff spaces. The $n$-H-closed space $Y$ constructed before for an $n$-Hausdorff space $X$ is a projective maximum, that is $Y$ is projectively larger than every $n$-H-closed extention and given a continuous surjection $f:Y\to Y$ such that $f(x)=x$ for every $x\in X$, then $f$ is a homeomorphism. For the future we denote this $Y$ with $n\hbox{-}kX$ and we call it the n-Katětov extension of $X$.

Uspenskiǐ showed in [22] that for any space $X$ there exists a cardinal $\kappa$ and a nonempty subspace $Z\subseteq X^{\kappa}$ such that $X\times Z$ is homogeneous. The space $Z$ is found by selecting a set $A$ such that $\kappa=|A|\geq|X|$ and letting $Z=\{f\in{}^{A}X:\hbox{for each }x\in X,|f^{\leftarrow}(x)|=\kappa\}$, where ${}^{A}X$ is the space of all functions from $A$ to $X$. Both $Z$ and $X\times Z$ are homogeneous and homeomorphic. For our construction we write ${\bf H}(X)=X\times Z$ and consider $X$ as a subspace of ${\bf H}(X)$ [12].

Lemma 4.10**.**

[12]

Let $X$ be a space and $h:X\rightarrow X$ be a homeomorphism and let $id_{Z}$ be the identity function on $Z$ . Then the function $h\times id_{Z}:{\bf H}(X)\rightarrow{\bf H}(X)$ is also a homeomorphism that extends $h$.

Lemma 4.11**.**

Let $n\geq 2$, $X$ be an $n$-Hausdorff space and $h:X\to X$ be a homeomorphism. Then there is a homeomorphism $n$-$kh:n\hbox{-}kX\to n\hbox{-}kX$ that extends $h$.

{proof}

Let $p\in n\hbox{-}kX\setminus X$, then $p=p_{\alpha i}$ for some $\alpha\in|\mathfrak{U}|$ and for some $i=1,...,k\alpha$. The set ${\cal V}=\{h(U):U\in\mathcal{U}_{\alpha}\}$ is an open ultrafilter on $X$ and since $|a{\cal U}_{\alpha}|=|a\mathcal{V}|$, there exists $\beta\in|\mathfrak{U}|$ such that $\mathcal{V}=\mathcal{U}_{\beta}$. Define $n\hbox{-}kh(p_{\alpha i})=p_{\beta i}$ for every $i=1,...,k\alpha=k\beta$. For $x\in X$, define $n\hbox{-}kh(x)=h(x)$. The function $n\hbox{-}kh$ is clearly a homeomorphism that extends $h$.

Theorem 4.12**.**

Let $n\geq 2$, $X$ be an $n$-Hausdorff space. Then $X$ can be embedded in an homogeneous space that is the countable union of $n$-H-closed spaces.

{proof}

Let $H_{1}={\bf H}(n\hbox{-}kX)$. If $H_{m}$ is defined, let’s define $H_{m+1}={\bf H}(n\hbox{-}kH_{m})$ and $H=\bigcup_{m\in\mathbb{N}}H_{m}$. A subset $U\subseteq H$ is open in $H$ if $U\cap H_{m}\in\tau(H_{m})$ for every $m\in\mathbb{N}$. The space $H$ is the countable union of $n$-H-closed spaces. We have to prove that $H$ is homogeneous. Let $p,q\in H$. Since $H_{m}\subseteq H_{m+1}$, there exists $m\in\mathbb{N}$ such that $p,q\in H_{m}$. Each $H_{m}$ is homogeneous, then there exist a homeomorphism $h:H_{m}\to H_{m}$ such that $h(p)=q$. By Lemma 4.11 there exists a homeomorphism $n\hbox{-}kh:n\hbox{-}kH_{m}\to n\hbox{-}kH_{m}$ that extends $h$. By Lemma 4.10 the function $n\hbox{-}kh\times id_{Z}:H_{m+1}\to H_{m+1}$ is a homeomorphism. Put $n\hbox{-}kh=h_{1}$. By induction $h$ can be extended to $h_{k}:H_{m+k}\to H_{m+k}$ for every $k\in{\mathbb{N}}$. The function $g=\bigcup_{k\in{\mathbb{N}}}h_{k}:H\to H$ extends $h$ and it is a homeomorphism on $H$. Then $H$ is homogeneous.

Example 4.13**.**

An example of an $n$-Hausdorff, homogeneous, not $n$-Urysohn space which is the countable union of $n$-H-closed spaces, for every $n\geq 2$.

Let’s take an $n$-Hausdorff, not $n$-Urysohn space $X$ (for example see [3, Example 4]), $n\geq 2$. Then, by Theorem 4.12, $X$ can be embedded in an $n$-Hausdorff, homogeneous space $Y$ which is the countable union of $n$-H-closed spaces. Furthermore $Y$ is not $n$-Urysohn, since $X$ is a non-$n$-Urysohn subset of it. $\triangle$

Acknowledgement: The research was supported by ”National Group for Algebric and Geometric Structures, and their Applications” (GNSAGA-INdAM).