$G_\delta$-refinements

TL;DR

This paper investigates how certain topological properties are preserved under $G_\delta$-refinements in specific classes of scattered spaces, addressing open questions about tightness in $G_\delta$-refinements of $\sigma$-products.

Contribution

It establishes preservation results for metacompactness, paralindel"ofness, metalindel"ofness, and linear lindel"ofness in $ ext{SP}$-scattered spaces under $G_\delta$-refinements, and explores related generalizations.

Findings

Preservation of several covering properties in $ ext{SP}$-scattered spaces.

Analysis of $G_\delta$-refinements in $ ext{N}$-scattered and $ ext{ω}$-scattered spaces.

Addressing the question of tightness in $G_\delta$-refinements of $\sigma$-products.

Abstract

In this work we deal with the preservation by $G_\delta$-refinements. We prove that for $\mathrm{SP}$-scattered spaces the metacompactness, paralindel\"ofness, metalindel\"ofness and linear lindel\"ofness are preserved by $G_\delta$-refinements. In this context we also consider some other generalizations of discrete spaces like $\omega$-scattered and $N$-scattered. In the final part of this paper we look at a question of Juh\'asz, Soukup, Szentmikl\'ossy and Weiss concerning the tighness of the $G_\delta$-refinement of a $\sigma$-product.

Equations

$$\{W_s\cap C:s\in S\text{ and }C\in {\mathcal{C}}_{x_s}\}$$

$${\mathcal{D}}_n=\{W_s\cap C:s\in S\text{ e }C\in {\mathcal{C}}_{x_s}\}.$$

$$Z=Z_x\cap \bigcap \{O_t:t\in T\}.$$

$${\mathcal{D}}_n(Z)$$

$$=\{W_i\cap C:s\in S,C\in {\mathcal{C}}_{x_s}\text{ e }{W}_s\cap C\cap Z\ne \varnothing \}$$

$$\subseteq \{W_t\cap C:t\in T\text{ e }C\in {\mathcal{C}}_{x_t}(Z)\},$$

$${\mathcal{C}}^{\prime }=\{C_y\}\cup \bigcup \{{\mathcal{D}}_n:n\in \omega \}$$

$${\mathcal{D}}_n=\{W_i\cap C:i\in I\text{ e }C\in {\mathcal{C}}_{x_i}\}.$$

$${\mathcal{C}}^{\prime }=\{C_y\}\cup \bigcup \{{\mathcal{D}}_n:n\in \omega \}$$

$${\mathcal{D}}_n=\{W_i\cap C\cap (U_n\setminus {\mathrm{cl}}_{\tau }(U_{n+2}):i\in I\text{ and }C\in {\mathcal{C}}_{x_i}\}.$$

$${\mathcal{C}}^{\prime }=\{C_y\}\cup \bigcup \{{\mathcal{D}}_n:n\in \omega \}$$

$$V_{\alpha }={\mathrm{int}}_{\tau }\left(\bigcup \{C_{\beta }:\beta \le \alpha \}\right).$$

$$O=\left\{x\in X:\text{ there exists α(x)<κ such that x∈Vα(x)}\right\}.$$

$$y\in U_y\subseteq C_{{\alpha }_y}\cup \bigcup \{F_n:n\in \omega \}\subseteq \bigcup \{C_{\beta }:\beta \le \alpha \}.$$

$$\Phi (E)=\left\{x\in \mathbb{R}:\underset{h\to 0}{\lim }\frac{m\left(E\cap ]x-h,x+h[\right)}{2h}=1\right\}.$$

$${\tau }_d=\left\{E\in \mathcal{M}:E\subseteq \Phi (E)\right\}$$

$$\sigma =\sigma (X)=\sigma (X,x^{\ast })=\left\{x\in \prod _{i\in I}{X}_i:\mathrm{supp}(x)\text{ is finite}\right\},$$

$$t(\sigma (X^{\kappa }{)}_{\delta })={\aleph }_1$$

$$t\left(\prod \{X_i:i\le n\}\right)=\max \{t(X_i):i\le n\}.$$

$$t\left(X_{\delta }\right)=\sup \{t(X_n):n\in \omega \}.$$

$$Y_n=\{y\in Y:\mathrm{supp}(y)\subseteq n\}.$$

$$q\in U=\prod \{U_{\alpha }:\alpha \in I\}\times \prod \{X_{\alpha }:\alpha \in \kappa \setminus I\},$$

$$t\left(\sigma {\left(X\right)}_{\delta }\right)\le \lambda .$$

$$t\left(\sigma {\left(X^{\kappa }\right)}_{\delta }\right)\le t(X{)}^{{\aleph }_0}.$$

$$t\left(\sigma {\left(X^{\kappa }\right)}_{\delta }\right)>\lambda ?$$

$$t\left(\sigma {\left(X^{\kappa }\right)}_{\delta }\right)={\aleph }_{\omega +1}?$$

$$t\left(X_{\delta }\right)\le {\mathrm{Cov}}_{\omega }(\lambda )\cdot \sup \{t(X_{\alpha }):\alpha <\lambda \}.$$

$$C=\bigcup \{C_J:J\in \mathcal{J}\}.$$

$$t(\sigma (X^{{\aleph }_{\omega }},x^{\ast }{)}_{\delta })=t(X).$$

$$Y_n=\{y\in Y:\mathrm{supp}(y)\subseteq {\omega }_n\}.$$

Full text

1. Preliminaries

Definition 1.1**.**

For any space $\langle X,\tau\rangle$, the topology $\tau_{\delta}$ obtained by letting every $G_{\delta}$ subset of $X$ be open is called the $G_{\delta}$-topology and the space so obtained is denoted by $X_{\delta}$.

Definition 1.2**.**

Let $X$ be a set and let $\mu$ be a cardinal such that $\mu\leq|X|$. We say that $\mathscr{C}\subseteq[X]^{\mu}$ is cofinal in $[X]^{\mu}$ if, for all $x\in[X]^{\mu}$, there exists $y\in\mathscr{C}$ such that $x\subseteq y$. For cardinals $\mu\leq\kappa$, we define $\mathrm{cf}\left([\kappa]^{\mu},\subseteq\right)$ as the least cardinality of a cofinal family in $[\kappa]^{\mu}$. Given a infinite cardinal $\kappa$, we define $\mathrm{Cov}_{\omega}(\kappa)=\mathrm{cf}\left([\kappa]^{\aleph_{0}},\subseteq\right)$.

Theorem 1.3** (Passos [Pas2007]).**

*Let $\kappa$ be a infinite cardinal such that $\mathrm{Cov}_{\omega}(\kappa)=\kappa$. Given a set $X$ of cardinality $\kappa$, there exists an $\omega$-covering elementary submodel $M$ such that $X\subseteq M$ and $|M|=\kappa$. *

Recall that a subset $F$ of $X$ is $\kappa$-closed, where $\kappa$ is an infinite cardinal iff whenever $S\subseteq F$ and $|S|\leq\kappa$ then $S\subseteq F$. It is well known that $t(X)\leq\kappa$ iff every $\kappa$-closed set in $X$ is closed.

2. $\mathrm{SP}$-scattered spaces

Definition 2.1**.**

A point $p$ in a topological space $X$ is called a strong $P$-point if it has a neighborhood consisting of $P$-points. The set of all strong $P$-points of $X$ is denoted by $\mathrm{SP}(X)$.

Observe that $\mathrm{SP}(X)=\mathrm{int}_{X}P(X)$.

Definition 2.2** ([HRW2007]).**

Recursively, define:

• •

$S_{0}(X)=X$ and $S_{1}(X)=X\setminus\mathrm{SP}(X)$;

• •

$S_{\alpha+1}(X)=S_{1}(S_{\alpha}(X))$ for any ordinal $\alpha\geq 1$;

• •

$S_{\lambda}(X)=\bigcap\{\,S_{\alpha}(X):\alpha<\lambda\,\}$, if $\lambda$ is a limit ordinal.

In [HRW2007], Henriksen, Raphael and Woods proved the following generalizations of the well known theorems 5.1 and 5.2 of [LR1981], respectively:

Theorem 2.3**.**

If $X$ is a Lindelöf $\mathrm{SP}$-scattered regular space, then $X_{\delta}$ is Lindelöf.

Theorem 2.4**.**

If $X$ is a paracompact $\mathrm{SP}$-scattered Hausdorff space, then $X_{\delta}$ is paracompact.

In the same article they asked:

Question 2.5**.**

If $X$ is a metacompact $\mathrm{SP}$-scattered regular space, so is $X_{\delta}$ a metacompact space?

In this section, we will see that not just the metacompactness, but also the paralindelöfness, the metalindelöfness and the linear lindelöfness are preserved by $G_{\delta}$-refinements on the class of $\mathrm{SP}$-scattered regular spaces.

Theorem 2.6** ([Mis1972]).**

A regular $P$-space $X$ is paralindelöf if, and only if, it is paracompact.

Proposition 2.7** ([HRW2007]).**

If $X$ is a regular space, then the following are equivalent:

1. (1)

$X$* is $\mathrm{SP}$-scattered;* 2. (2)

if $A\subseteq X$ is nonempty, then \mathrm{int}_{A}\{\,a:\text{a$is a$P$-point of$A}\,\}\neq\emptyset.

Theorem 2.8**.**

If $X$ is a regular $\mathrm{SP}$-scattered paralindelöf space, then $X_{\delta}$ is paracompact.

Proof.

By the theorem 2.6, it is enough to show that $X_{\delta}$ is paralindelöf. Let $\mathscr{C}$ be an open cover of $X_{\delta}$. Let $O$ be the set of all points $x\in X$ such that $x\in\mathrm{int}_{\tau}\bigcup\mathscr{C}^{\prime}$ for some locally countable open partial refinement $\mathscr{C}^{\prime}$ of $\mathscr{C}$ in $X_{\delta}$.

If $O=X$ then, for each $x\in X$, there exists a locally countable open partial refinement $\mathscr{C}_{x}$ of $\mathscr{C}$ in $X_{\delta}$ such that $x\in V_{x}=\mathrm{int}_{\tau}\bigcup\mathscr{C}_{x}$. Since $\langle X,\tau\rangle$ is paralindelöf, the open cover $\mathscr{V}=\{\,V_{x}:x\in X\,\}$ admits a locally countable open refinement $\{\,W_{s}:s\in S\,\}$, with $W_{s}\neq W_{s^{\prime}}$ whenever $s\neq s^{\prime}$. For each $s\in S$, take $x_{s}\in X$ such that $W_{s}\subseteq V_{x_{s}}$. So

[TABLE]

is locally countable open refinement of $\mathscr{C}$ in $X_{\delta}$.

Now it remains to show that in fact $O=X$. Suppose this is not the case. Since $\langle X,\tau\rangle$ is $\mathrm{SP}$-scattered, by the proposition 2.7, there exists a $y\in X\setminus O$ and an open neighborhood $U$ of $y$ in $X$ such that $(X\setminus O)\cap U$ is a $P$-subspace of $X$. Take $C_{y}\in\mathscr{C}$ such that $y\in C_{y}$. We can suppose that $C_{y}=\bigcap\{\,U_{n}:n\in\omega\,\}$, where, for each $n\in\omega$, $U_{n}\in\tau$ and $\mathrm{cl}_{\tau}(U_{n})\subseteq U$. So $C_{y}\cup O$ is an open subset of $X$, for $(X\setminus O)\cap C_{y}$ is a subspace of the $P$-space $U$. As $X$ is regular, $y$ has an open neighborhood $U_{y}$ in $X$ such that $\mathrm{cl}_{\tau}(U_{y})\subseteq C_{y}\cup O$.

Fix $n\in\omega$. Let $F=\mathrm{cl}_{\tau}(U_{y})\setminus U_{n}$. Since $F\subseteq O$, for each $x\in F$, there exists a locally countable open partial refinement $\mathscr{C}_{x}$ of $\mathscr{C}$ in $X_{\delta}$ such that $x\in V_{x}=\mathrm{int}_{\tau}\bigcup\mathscr{C}_{x}$. As $\langle X,\tau\rangle$ is paralindelöf and $F$ is closed, $\mathscr{V}=\{\,V_{x}:x\in F\,\}$ admits a locally countable open partial refinement $\mathscr{W}=\{\,W_{s}:s\in S\,\}$ that covers $F$, where $W_{s}\neq W_{s^{\prime}}$ whenever $s\neq s^{\prime}$. For each $s\in S$, choose $x_{s}\in F$ such that $W_{s}\subseteq V_{x_{s}}$. Consider the family

[TABLE]

The family $\mathscr{D}_{n}$ is a locally countable open cover of $F$ in $X_{\delta}$. Indeed, let $x\in X$. Since $\mathscr{W}$ is a locally countable open family in $X$, there exists an open neighborhood $Z_{x}$ of $x$ in $X$ such that $T=\{\,s\in S:W_{s}\cap Z_{x}\neq\emptyset\,\}$ is countable. For each $t\in T$, take an open neighborhood $O_{t}$ of $x$ in $X_{\delta}$ such that $\{\,C\in\mathscr{C}_{x_{t}}:C\cap O_{t}\neq\emptyset\,\}$ is countable. Consider the following open neighborhood of $x$ em $X_{\delta}$:

[TABLE]

Note that for each $t\in T$, $\mathscr{C}_{x_{t}}(Z)=\{\,C\in\mathscr{C}_{x_{t}}:C\cap Z\neq\emptyset\,\}$ is countable. Seeing that

[TABLE]

the family $\mathscr{D}_{n}({Z})$ is countable.

Thus,

[TABLE]

is a locally countable open partial refinement of $\mathscr{C}$ in $X_{\delta}$ such that $y\in U_{y}\subseteq\mathrm{int}_{\tau}\bigcup\mathscr{C}^{\prime}$, contradicting the fact that $y\notin O$. ∎

Theorem 2.9**.**

If $X$ is a regular $\mathrm{SP}$-scattered metalindelöf space, then $X_{\delta}$ is metalindelöf.

Proof.

Let $\mathscr{C}$ be an open cover of $X_{\delta}$ and consider the set $O$ whose elements are all those $x\in X$ such that $x\in\mathrm{int}_{\tau}\bigcup\mathscr{C}^{\prime}$ for some pointwise countable open partial refinement $\mathscr{C}^{\prime}$ of $\mathscr{C}$ in $X_{\delta}$.

If $O=X$ then, proceeding in the same way as in the proof of theorem 2.8, we can obtain a pointwise countable open refinement of $\mathscr{C}$ in $X_{\delta}$.

In order to complete the proof, it is enough to show that $O=X$. Suppose on the contrary that $O\neq X$. As $X$ is regular and $\mathrm{SP}$-scattered, by the proposition 2.7, there exist a point $y\in X\setminus O$ and an open neighborhood $U$ of $y$ in $X$ such that $U\cap(X\setminus O)$ is a $P$-subspace of $X$. Choose a $C_{y}\in\mathscr{C}$ such that $y\in C_{y}$. We can suppose that $C_{y}=\bigcap\{\,U_{n}:n\in\omega\,\}$, where for each $n\in\omega$, $U_{n}\in\tau$ and $\mathrm{cl}_{\tau}(U_{n})\subseteq U$. Note that $C_{y}\cup O$ is an open subset of $X$. Hence, from the regularity of $X$ it follows that there exist an open neighborhood $U_{y}$ of $y$ in $X$ such that $\mathrm{cl}_{\tau}(U_{y})\subseteq C_{y}\cup O$.

Fix $n\in\omega$. Note that $F=\mathrm{cl}_{\tau}(U_{y})\setminus U_{n}\subseteq O$. Then, for each $x\in F$, there is a pointwise countable open partial refinement $\mathscr{C}_{x}$ of $\mathscr{C}$ in $X_{\delta}$ such that $x\in V_{x}=\mathrm{int}_{\tau}(\bigcup\mathscr{C}_{x})$. Because $X$ is metalindelöf and $F$ is closed then $\{\,V_{x}:x\in F\,\}$ has a pointwise countable open partial refinement $\mathcal{W}=\{\,W_{i}:i\in I\,\}$, where $W_{i}\neq W_{j}$ whenever $i\neq j$. For each $i\in I$, choose $x_{i}\in F$ such that $W_{i}\subseteq V_{x_{i}}$. Consider the family

[TABLE]

It is easily checked that each $\mathscr{D}_{n}$ is a pointwise countable open partial refinement of $\mathscr{C}$ which covers $\mathrm{cl}_{\tau}(U_{y})\setminus U_{n}$. So,

[TABLE]

is a pointwise countable open partial refinement of $\mathscr{C}$ in $X_{\delta}$ such that $y\in U_{y}\subseteq\mathrm{int}_{\tau}(\bigcup\mathscr{C}^{\prime})$, contradicting the fact that $y\notin O$. Thus, $O=X$. ∎

Theorem 2.10**.**

If $X$ is a regular $\mathrm{SP}$-scattered metacompact space, then $X_{\delta}$ is metacompact.

Proof.

Let $\mathscr{C}$ be an open cover of $X_{\delta}$ and consider the set $O$ whose elements are all $x\in X$ such that $x\in\mathrm{int}_{\tau}(\bigcup\mathscr{C}^{\prime})$ for some pointwise finite open partial refinement $\mathscr{C}^{\prime}$ of $\mathscr{C}$ in $X_{\delta}$.

Similarly to what it has been done in theorem 2.8, we can get, from the assumption $O=X$, a pointwise finite open refinement of $\mathscr{C}$ in $X_{\delta}$.

We complete the proof by showing that $O=X$. Suppose that $O\neq X$. Since $\langle X,\tau\rangle$ is $\mathrm{SP}$-scattered and regular, by the proposition 2.7, there are a point $y\in X\setminus O$ and an open neighborhood $U$ of $y$ in $X$ such that $(X\setminus O)\cap U$ is a $P$-subspace of $X$. Take a $C_{y}\in\mathscr{C}$ such that $y\in C_{y}$. We can suppose that $C_{y}=\bigcap\{\,U_{n}:n\in\omega\,\}$, where for each $n\in\omega$, $U_{n}\in\tau$ and $\mathrm{cl}_{\tau}(U_{n+1})\subseteq U_{n}\subseteq\mathrm{cl}_{\tau}(U_{n})\subseteq U$. Note that $C_{y}\cup O$ is an open subset of $X$. Once $X$ is regular, $y$ has an open neighborhood $U_{y}$ in $X$ such that $\mathrm{cl}_{\tau}(U_{y})\subseteq(C_{y}\cup O)\cap U_{0}$.

As $F_{n}\subseteq O$, for each $x\in F_{n}$, there exists a pointwise finite open partial refinement $\mathscr{C}_{x}$ of $\mathscr{C}$ in $X_{\delta}$ such that $x\in V_{x}=\mathrm{int}_{\tau}(\bigcup\mathscr{C}_{x})$. Since $\langle X,\tau\rangle$ is metacompact and $F_{n}$ is closed then $\mathcal{V}=\{\,V_{x}:x\in F_{n}\,\}$ has a pointwise finite open partial refinement $\mathcal{W}$ which covers $F_{n}$. We can suppose that $\mathcal{W}=\{\,W_{i}:i\in I\,\}$, where $W_{i}\neq W_{j}$ whenever $i\neq j$. For each $i\in I$, choose $x_{i}\in F_{n}$ such that $W_{i}\subseteq V_{x_{i}}$. Consider the family

[TABLE]

Note that each $\mathscr{D}_{n}$ is a pointwise finite open partial refinement of $\mathscr{C}$ in $X_{\delta}$ such that $\bigcup\mathscr{D}_{n}=U_{n}\setminus\mathrm{cl}_{\tau}(U_{n+2})$. Therefore,

[TABLE]

is a pointwise finite open partial refinement of $\mathscr{C}$ in $X_{\delta}$ such that $y\in U_{y}\subseteq\mathrm{int}_{\tau}(\bigcup\mathscr{C}^{\prime})$, contradicting the fact that $y\notin O$. ∎

Theorem 2.11**.**

If $X$ is a regular $\mathrm{SP}$-scattered linearly Lindelöf space, then $X_{\delta}$ is linearly Lindelöf.

Proof.

Let $\mathscr{C}=\{\,C_{\alpha}:\alpha<\kappa\,\}$ be an open cover of $X_{\delta}$, where $\kappa$ is an uncountable regular cardinal. For each $\alpha<\kappa$, let

[TABLE]

Define

[TABLE]

Claim**.**

$O=X$**

Proof of claim

Suppose on the contrary that $O\neq X$. As $\langle X,\tau\rangle$ is a $\mathrm{SP}$-scattered regular space, by the proposition 2.7, there are $y\in X\setminus O$ and an open neighborhood $U$ of $y$ in $X$ such that $(X\setminus O)\cap U$ is a $P$-subspace of $X$. Choose $\alpha_{y}<\kappa$ such that $y\in C_{\alpha_{y}}$. We can suppose that $C_{\alpha_{y}}=\bigcap\{\,U_{n}:n\in\omega\,\}$, where, for each $n\in\omega$, $U_{n}\in\tau$ and $\mathrm{cl}_{\tau}(U_{n})\subseteq U$. Note that $C_{y}\cup O$ is an open subset of $X$. Once $X$ is regular, $y$ has an open neighborhood $U_{y}$ in $X$ such that $\mathrm{cl}_{\tau}(U_{y})\subseteq C_{\alpha_{y}}\cup O$.

Fix $n\in\omega$. Let $F_{n}=\mathrm{cl}_{\tau}(U_{y})\setminus U_{n}$. Note that $F_{n}$ is a closed subset of $X$ and so it is linearly Lindelöf. Moreover, $F_{n}\subseteq O$; this implies that, for each $x\in F_{n}$, we can take $\alpha(x)<\kappa$ such that $x\in V_{\alpha(x)}$. Then, $\left\{\,V_{\alpha(x)}:x\in F_{n}\,\right\}$ is a family of open subsets of $X$ which covers $F_{n}$ and it is linearly ordered by inclusion. Therefore, there is a countable subset $E_{n}\subseteq F_{n}$ such that $\left\{\,V_{\alpha(x)}:x\in E_{n}\,\right\}$ covers $F_{n}$. So, $\alpha=\sup(\{\alpha_{y}\}\cup\bigcup\{\,E_{n}:n\in\omega\,\})<\kappa$ e

[TABLE]

Then $y\in V_{\alpha}$ and, thus, $y\in O$. This is a contradiction. $\triangle$

By the claim above, $\mathcal{V}=\{\,V_{\alpha}:\alpha<\kappa\,\}$ is an open cover of $\langle X,\tau\rangle$. Since $\langle X,\tau\rangle$ is a linearly Lindelöf space, $\mathcal{V}$ has a subcover whose cardinality is less than $\kappa$. Because $\kappa$ is regular, $V_{\alpha}=X$ for some $\alpha<\kappa$. Thus, $\{\,C_{\beta}:\beta<\alpha\,\}$ is a subcover of $\mathscr{C}$ whose cardinality is less than $\kappa$. ∎

3. Other generalizations of scattered

Clearly, if a regular Lindelöf space $X$ is a countable union of scattered closed subspaces, then $X_{\delta}$ is Lindelöf. As we shall see, at least consistently, this is not the case when it is not required that the subspaces are closed.

A space is $\sigma$-scattered if it is an union of a countable family of scattered subspaces.

Example 1**.**

Assuming $\mathsf{CH}$, there exists a regular $\sigma$-scattered Lindelöf space whose $G_{\delta}$-refinement is not Lindelöf.

Proof.

It is enough to take a Luzin subset of the real line containing the rational numbers and consider it as a subspace of the Michael line. ∎

Question 3.1**.**

Is there a regular $\sigma$-scattered Lindelöf space whose $G_{\delta}$-refinement is not Lindelöf?

Question 3.2**.**

Is there a regular $\sigma$-scattered paracompact space whose $G_{\delta}$-refinement is not paracompact?

Hdeib and Pareek introduced in [HP1989] the following natural generalization of scattered spaces: a space $X$ is $\omega$-scattered if, for each non-empty subset $A$ of $X$, there exist a point $x\in A$ and an open neighborhood $U_{x}$ of $x$ such that $U_{x}\cap A$ is countable.

Every scattered space is $\omega$-scattered, but the reverse is not true: the set of rational numbers with the usual topology is $\omega$-scattered and non-scattered.

The theorem 3.12 of [HP1989] states that in the class of regular $\omega$-scattered spaces the Lindelöf property is preserved by $G_{\delta}$-refinements. However, this is not true once the space of the example 1 is $\omega$-scattered.

Question 3.3**.**

Is there a Hausdorff $\omega$-scattered paracompact space $X$ such that $X_{\delta}$ is not paracompact?

A space $X$ is $N$-scattered if every nowhere dense subset of $X$ is a scattered subspace of $X$. The next example was noticed by Santi Spadaro.

Example 2**.**

Assuming $\mathsf{CH}$, there exists a $N$-scattered Lindelöf space whose $G_{\delta}$-refinement is not Lindelöf.

Proof.

Let $\mathcal{M}$ the family of all Lebesgue measurable subsets of the real line. For each $E\in\mathcal{M}$, define

[TABLE]

Then

[TABLE]

is a topology on $\mathbb{R}$ stronger than that usual, well known as density topology. Denote by $\mathbb{R}_{d}$ the topological space $\langle\mathbb{R},\tau_{d}\rangle$. By corollary 4.3 of [Tal1976], $\mathsf{CH}$ implies that $\langle\mathbb{R},\tau_{d}\rangle$ has a hereditarily Lindelöf, non-separable, regular and Baire subspace $X$. By theorem 2.7 of [Tal1976], every nowhere dense subset of $X$ is discrete (and closed). Therefore, $X$ is $N$-scattered. On the other side, the pseudocharacter of $X$ is countable, for $X$ is Hausdorff and hereditarily Lindelöf. Then $X_{\delta}$ is discrete and uncountable and, thus, it is not Lindelöf. ∎

Question 3.4**.**

Is there a Hausdorff paracompact $N$-scattered space whose $G_{\delta}$-refinement is not a paracompact space?

4. The tightness of $G_{\delta}$-refinement of $\sigma$-products

Given a family of topological spaces $\{\,X_{i}:i\in I\,\}$ and a point $x^{\ast}\in X=\prod\{\,X_{i}:{i\in I}\,\}$, define

[TABLE]

where $\mathrm{supp}(x)=\{\,i\in I:x(i)\neq x^{\ast}(i)\,\}$. The $\sigma$-product of $X$ at $x^{\ast}$ is the set $\sigma$ equipped with the topology induced by the Tychonoff product $\prod\{\,X_{i}:{i\in I}\,\}$.

In [JSSW], Juhász, Soukup, Szentmiklóssy and Weiss proved:

Theorem 4.1**.**

Let $\kappa$ and $\lambda$ be cardinals, with $\kappa\leq\aleph_{1}$. Let $X$ be the one point lindelöfication of a discrete space of cardinality $\kappa$ by a point $p$ and let $x^{\ast}\in X^{\kappa}$, where $x^{\ast}(\alpha)=p$ for all $\alpha<\kappa$. Then $\left(\sigma(X^{\kappa},x^{\ast})\right)_{\delta}$ has tighness $\aleph_{1}$.

In the same article, it was asked:

Question 4.2**.**

Assume that $X$ is a Lindelöf $P$-space such that $t(X)=\aleph_{1}$. Is it true that

[TABLE]

for all cardinal $\kappa$?

We will see that the answer is positive.

Lemma 4.3** ([DW1986]).**

If $\{\,X_{i}:i\leq n\,\}$ is a finite family of regular locally Lindelöf $P$-spaces, then

[TABLE]

Lemma 4.4**.**

If $X=\sigma\{\,X_{n}:n\in\omega\,\}$ is a $\sigma$-product of regular locally Lindelöf $P$-spaces, then

[TABLE]

Proof.

Let $\lambda=\sup\{\,t(X_{n}):n\in\omega\,\}$. Let $Y$ be a non-closed subset of $\sigma_{\delta}=\sigma\left(X,x^{\ast}\right)_{\delta}$ and let $q\in\mathrm{cl}(Y)\setminus Y$. For each $n\in\omega$, let

[TABLE]

Since $Y=\bigcup\{\,Y_{n}:n\in\omega\,\}$ and $\sigma(X,x^{\ast})_{\delta}$ is a $P$-space, there exists a $m\in\omega$ such that $q\in\mathrm{cl}(Y_{m})$. Now, $\pi_{m}(q)\in\mathrm{cl}\left(\pi_{m}[Y_{m}]\right)$, where $\pi_{m}$ is the natural projection from $\prod\{\,X_{n}:n\in\omega\,\}$ in $\prod\{\,X_{i}:i\in m\,\}$. Since by the corollary 4.3 the tightness of $\prod\{\,X_{i}:i\in m\,\}$ is $\leq\lambda$, there exists $Z^{\prime}\subseteq\pi_{m}[Y_{m}]$ of cardinality $\leq\lambda$ such that $\pi_{m}(q)\in\mathrm{cl}(Z^{\prime})$. Then $Z=Z^{\prime}\times\prod\{\,\{x^{\ast}(n)\}:n\geq m\,\}\subseteq Y_{m}\subseteq Y$ and since $\mathrm{supp}(q)\subseteq m$, we have $q\in\mathrm{cl}(Z)$. ∎

Lemma 4.5**.**

Let $\kappa$ be a infinite cardinal. Let $X=\prod\{\,X_{\alpha}:\alpha<\kappa\,\}$. If for each countable subset $I\subseteq\kappa$, $\sigma\{\,X_{\alpha}:\alpha\in I\,\}_{\delta}$ has tightness $\leq\lambda$, with $\mathrm{Cov}_{\omega}(\lambda)=\lambda$, then $\sigma(X)_{\delta}$ has tightness $\leq\lambda$.

Proof.

Let $Y$ be a non-closed subset of $\sigma_{\delta}=\sigma(X,x^{\ast})_{\delta}$ and let $q\in\mathrm{cl}(Y)\setminus Y$. By the theorem 1.3, there exists a $\omega$-covering elementary submodel $M$ of cardinality $\lambda$ such that $\{X,x^{\ast},\kappa,\lambda,q,Y\}\cup\lambda\subseteq M$. We are going to show that $q\in\mathrm{cl}(Y\cap M)$. Suppose that

[TABLE]

where $I$ is a countable subset of $\kappa$ and each $U_{\alpha}$ is an open subset of $X_{\alpha}$. Since $M$ is $\omega$-covering, there exists $J\in M$, a countable subset of $\kappa$ such that $I\cap M\subseteq J$. Now note that $\pi_{J}(q)\in\mathrm{cl}(\pi_{J}[Y])$ and $\pi_{J}[Y]\in M$; besides $\pi_{J}[\sigma(X,x^{\ast})_{\delta}]$ belongs to $M$ and, by hypothesis, its tightness is $\leq\lambda$. So by the elementarity there exists $Z\in M$, a subset of $\pi_{J}[Y]$ whose cardinality is at most $\lambda$, such that $\pi_{J}(q)\in\mathrm{cl}(Z)$. Let $z\in\pi_{J}[U]\cap Z$. Note that $z\in M$, because, since $Z\in M$ and $Z$ has cardinality at most $\lambda$ and $\lambda\subseteq M$, $Z\subseteq M$. So by the elementarity there exists $y\in Y\cap M$ such that $\pi_{J}(y)=z$.

We claim that $y\in U$. Indeed, since $\mathrm{supp}(y)$, $\mathrm{supp}(q)\subseteq M$, it follows that if $\alpha\in I\setminus M$ so $y(\alpha)=x^{\ast}(\alpha)=q(\alpha)\in U_{\alpha}$. On the other side, if $\alpha\in I\cap M$ so $\alpha\in J$ and thus $y(\alpha)=z(\alpha)\in U_{\alpha}\cap M$. Therefore, $y\in\prod\{\,U_{\alpha}:{\alpha\in I}\,\}\times\prod\{\,X_{\alpha}:\alpha\in\kappa\setminus I\,\}$. ∎

Theorem 4.6**.**

If $X=\prod\{\,X_{\alpha}:\alpha<\kappa\,\}$, with each $X_{\alpha}$ being a Lindelöf $P$-space such that $t(X_{\alpha})\leq\lambda$, with $\mathrm{Cov}_{\omega}(\lambda)=\lambda$, then

[TABLE]

In particular, if $X$ is a Lindelöf $P$-space whose tightness is $\aleph_{n}$ then the tightness of $\sigma\left(X^{\kappa},x^{\ast}\right)_{\delta}$ is $\aleph_{n}$.

As a corollary of the previous theorem we have that, for a regular Lindelöf $P$-space,

[TABLE]

It remains to be seen whether:

Question 4.7**.**

Is there a Lindelöf $P$-space $X$ such that $t(X)=\lambda$, with $\mathrm{Cov}_{\omega}(\lambda)>\lambda$, and

[TABLE]

Question 4.8**.**

Assuming that $\mathrm{Cov}_{\omega}(\aleph_{\omega})=\aleph_{\omega+1}$, is there a Lindelöf $P$-space $X$ such that $t(X)=\aleph_{\omega}$ and

[TABLE]

Based on the theorem 3.1 from [DW1986], we have:

Lemma 4.9**.**

Let $\lambda$ be a infinite cardinal. If $X=\sigma\{\,X_{\alpha}:\alpha<\lambda\,\}$ is a $\sigma$-product of regular Lindelöf $P$-spaces, then

[TABLE]

Proof.

Let $\kappa=\mathrm{Cov}_{\omega}(\lambda)\cdot\sup\{\,t(X_{\alpha}):\alpha<\lambda\,\}$. For each $I\subseteq\lambda$, let $\sigma_{I}=\sigma\{\,X_{\alpha}:\alpha\in I\,\}_{\delta}$. Suppose that $A\subseteq\sigma_{\delta}$ is $\kappa$-closed, and $a\in\mathrm{cl}_{\sigma_{\delta}}(A)$.

Note that, for each countable subset $J\subseteq\lambda$, $\pi_{J}[\mathrm{cl}_{\sigma(\lambda)}(A)]\subseteq\mathrm{cl}_{\sigma_{J}}(\pi_{J}[A])$. Indeed, let $x\in\pi_{J}[\mathrm{cl}_{\sigma(\lambda)}(A)]$. Then there exists $z\in\mathrm{cl}_{\sigma(\lambda)}(A)$ such that $\pi_{J}(z)=x$. If $\prod\{\,U_{j}:j\in J\,\}$ is an basic neighborhood of $x$ in $\sigma_{J}$, then $\prod\{\,U_{j}:j\in J\,\}\times\prod\{\,X_{\alpha}:\alpha\in{\lambda\setminus J}\,\}$ is an open neighborhood of $z$. So $\left(\prod\{\,U_{j}:j\in J\,\}\times\prod\{\,X_{\alpha}:\alpha\in{\lambda\setminus J}\,\}\right)\cap A\neq\emptyset$ and, thus, $\left(\prod\{\,U_{j}:j\in J\,\}\right)\cap\pi_{J}[A]\neq\emptyset$. Therefore, $x\in\mathrm{cl}_{\sigma_{J}}(\pi_{J}[A])$.

Then, for each countable subset $J\subseteq\lambda$, $\pi_{J}(a)\in\mathrm{cl}_{\sigma_{J}}(\pi_{J}[A])$. By lemma 4.4, $t(\sigma_{J})\leq\kappa$, then we can take $B_{J}\subseteq\pi_{J}[A]$ of cardinality $\leq\kappa$ such that $\pi_{J}(a)\in\mathrm{cl}_{\sigma_{J}}(B_{J})$. For each $b\in B_{J}$ choose $x_{b}\in A$ such that $\pi_{J}(x_{b})=b$, and let $C_{J}=\{\,x_{b}:b\in B_{J}\,\}$.

Now, let $\mathcal{J}$ be a cofinal family in $[\lambda]^{\aleph_{0}}$ and let

[TABLE]

Note that $|C|\leq\mathrm{Cov}_{\omega}(\lambda)\cdot t(X)=\kappa$. Then $\mathrm{cl}_{\sigma(\lambda)}(C)\subseteq A$. So, it remains to be proved that $a\in\mathrm{cl}_{\sigma(\lambda)}(C)$. Let $U=\prod\{\,U_{j}:{j\in J^{\prime}}\,\}\times\prod\{\,X_{\alpha}:\alpha\in{\lambda\setminus J^{\prime}}\,\}$ be an basic neighborhood of $a$ in $\sigma(\lambda)$. Let $J\in\mathcal{J}$ such that $J^{\prime}\subseteq J$. Since $\pi_{J}(a)\in\mathrm{cl}_{\sigma_{J}}(B_{J})=\mathrm{cl}_{\sigma_{J}}\left(\pi_{J}[C_{J}]\right)$, then $\pi_{J}[U]\cap\pi_{J}[C_{J}]\neq\emptyset$; so $U\cap C_{J}\neq\emptyset$. Therefore, $U\cap C\neq\emptyset$. ∎

In the same way we have proved the lemma 4.4, we can show the following result for the cases in which $\mathrm{Cov}_{\omega}(t(X))>t(X)$:

Lemma 4.10**.**

If $X$ is a Lindelöf $P$-space then

[TABLE]

Proof.

Let $\kappa=t(X)$. Suppose that $\mathrm{Cov}_{\omega}(\kappa)>\kappa$. Note that $\kappa\geq\aleph_{\omega}$. Let $Y$ be a non-closed subset of $\sigma(X^{\omega},x^{\ast})_{\delta}$ and let $q\in\mathrm{cl}(Y)\setminus Y$. For each $n\in\omega$, let

[TABLE]

Since $Y=\bigcup\{\,Y_{n}:n\in\omega\,\}$ and $\sigma(X^{\omega},x^{\ast})_{\delta}$ is a $P$-space, there exists a $m\in\omega$ such that $q\in\mathrm{cl}(Y_{m})$. Now, $\pi_{m}(q)\in\mathrm{cl}\left(\pi_{m}[Y_{m}]\right)$, where $\pi_{m}$ is the natural projection from $X^{\aleph_{\omega}}$ in $X^{\aleph_{m}}$. Since by the theorem 4.9 the tightness of $X^{\aleph_{m}}$ is $\kappa$, there exists $Z^{\prime}\subseteq\pi_{m}[Y_{m}]$ of cardinality $\leq\kappa$ such that $\pi_{m}(q)\in\mathrm{cl}(Z^{\prime})$. Then $Z=Z^{\prime}\times\prod\{\,\{x^{\ast}(n)\}:n\geq m\,\}\subseteq Y_{m}\subseteq Y$ and since $\mathrm{supp}(q)\subseteq\omega_{m}$, we have $q\in\mathrm{cl}(Z)$. ∎