On generalization of theorems of Pestryakov

TL;DR

This paper examines the extent to which Pestryakov's theorems on cardinal functions of Baire class function spaces can be generalized to larger spaces, using selection principles to establish new results.

Contribution

It identifies which of Pestryakov's theorems extend to spaces containing finite linear combinations of characteristic functions of zero-sets, and introduces new propositions via selection principles.

Findings

Certain Pestryakov theorems are valid for broader function spaces.

New propositions for function spaces are established using selection principles.

The paper clarifies the scope of generalizations of Pestryakov's results.

Abstract

In 1987, A.V. Pestryakov proved a series of theorems for cardinal functions of the space $B_{\alpha}(X)$ of all real-valued functions of Baire class $\alpha$ $(\alpha> 0)$, and he conjectured that most of these theorems are true for spaces containing all finite linear combinations of characteristic functions of zero-sets in $X$. In this paper we investigate for which theorems of Pestriakov generalizations are valid. Also we prove some additional propositions for function spaces applying the theory of selection principles.

Equations

$$f_{V}(x)=\left\{\begin{array}[]{rcl}i,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,x\in V_{i},\\ 0,\,\,\,\,if\,\,x\in X\setminus\bigcup\limits_{i=1}^{n}V_{i},\\ \end{array}\right.$$

Full text

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1 Introduction

In this paper by a space we shall always mean a Tychonoff space. Let $C_{p}(X)$ denote the space of continuous real-valued functions $C(X)$ on a space $X$ with the topology of pointwise convergence. Let $B_{0}(X)=C(X)$ and inductively define $B_{\alpha}(X)$ for each ordinal $\alpha\leq\omega_{1}$ to be the space of pointwise limits of sequences of functions in $\bigcup\limits_{\beta<\alpha}B_{\beta}(X)$. So $B_{\alpha}(X)$ a set of all functions of Baire class $\alpha$, defined on a Tychonoff space $X$, provided with the pointwise convergence topology.

The family of Baire sets of a space $X$ is the smallest family of sets containing the zero sets of continuous real-valued functions (i.e. of the form $Z(f)=\{x\in X:f(x)=0\}$), and closed under countable unions and countable intersections. The Baire sets of $X$ of multiplicative class [math], denoted $Z(X)$, are the zero-sets of continuous real-valued functions. The sets of additive class [math], denoted $CZ(X)$, are the complements of the sets in $Z(X)$.

Let $\mathcal{A}$ and $\mathcal{B}$ be sets whose elements are families of subsets of an infinite set $X$. Then $S_{1}(\mathcal{A},\mathcal{B})$ denotes the selection principle:

For each sequence $(A_{n}:n\in\mathbb{N})$ of elements of $\mathcal{A}$ there is a sequence $(b_{n}:n\in\mathbb{N})$ such that for each $n$, $b_{n}\in A_{n}$, and $\{b_{n}:n\in\mathbb{N}\}$ is an element of $\mathcal{B}$.

The following prototype of many classical properties is called ”$\mathcal{A}$ choose $\mathcal{B}$” in [22].

${\mathcal{A}\choose\mathcal{B}}$ : For each $A\in\mathcal{A}$ there exists $B\subset A$ such that $B\in\mathcal{B}$.

Clearly that $S_{1}(\mathcal{A},\mathcal{B})$ implies ${\mathcal{A}\choose\mathcal{B}}$.

In this paper, by a cover we mean a nontrivial one, that is, $\mathcal{U}$ is a cover of $X$ if $X=\bigcup\mathcal{U}$ and $X\notin\mathcal{U}$.

A cover $\mathcal{U}$ of a space $X$ is:

$\bullet$ an $\omega$-cover if every finite subset of $X$ is contained in a member of $\mathcal{U}$.

$\bullet$ a $\gamma$-cover if it is infinite and each $x\in X$ belongs to all but finitely many elements of $\mathcal{U}$. Note that every $\gamma$-cover contains a countably $\gamma$-cover.

For a topological space $X$ we denote:

$\bullet$ $\Omega$ — the family of all open $\omega$-covers of $X$;

$\bullet$ $\Gamma$ — the family of all open $\gamma$-covers of $X$;

$\bullet$ $Z_{\Omega}$ — the family of all countable $\omega$-covers of $X$ by zero-sets in $X$;

$\bullet$ $Z_{\Gamma}$ — the family of all countable $\gamma$-covers of $X$ by zero-sets in $X$.

Let $(X,\tau)$ be a topological space. The Baire topology $\tau_{b}$ on $X$ is the topology on the underlying set $X$ having for a basis the family of all zero-sets of $X$. Since the countable intersection of zero-sets is also a zero-set, it follows that the space $X$ endowed with the Baire topology and denoted by $X_{\aleph_{0}}$ is a $P$-space. Recall that a topological space is called a $P$-space if the intersection of a countable family of open sets is open. Let us recall also that the family of $G_{\delta}$-sets in $X$ forms a base of the topology $\tau_{\delta}$ on $X$, and the space $X$ with the topology $\tau_{\delta}$ is called the $P$-modification of $X$ and is denoted by $PX$ or $X_{\delta}$ (see [4, 6, 9]). Clearly, $PX$ is a $P$-space and $\tau_{\delta}$ is finer than the Baire topology $\tau_{b}$. If $X$ is a Tychonoff space, then $X_{\aleph_{0}}=PX$ and $X_{\aleph_{0}}$ is a Tychonoff space. Note that the topology $X_{\aleph_{0}}$ coincides with the weak topology generated by $B_{\alpha}(X)$ for each $\alpha>0$ ([16]).

Further, we consider the spaces $C(X_{\aleph_{0}})$ and $B_{\alpha}(X)$ with the topology of pointwise convergence.

We will use the standard notation for usual cardinal invariants, so $c$, $\chi$, $\pi\chi$, $\psi$, $w$, $\pi w$, $\psi w$, $nw$, $d$, $t$, $l$, $s$, denote cellularity, character, $\pi$-character, pseudocharacter, weight, $\pi$-weight, pseudoweight, network weight, density, tightness, the Lindel$\ddot{o}$f number, spread, respectively, see [7, 8]. For a cardinal function $\epsilon$ denoted by $h\epsilon(Y)=\{\epsilon(Z):Z\subseteq Y\}$, $i\epsilon(Y)=\min\{\epsilon(Z):$ $Y$ admits a one-to-one continuous mapping onto a space $Z\}$ and $\epsilon^{*}(Y)=\sup\{\epsilon(Y^{n}):n\in\mathbb{N}\}$.

Since $B_{\alpha}(X)$ is dense in $\mathbb{R}^{X}$ ($0\leq\alpha\leq\omega_{1}$), $c(B_{\alpha}(X))=\omega_{0}$, $\pi\chi(B_{\alpha}(X))=\chi(B_{\alpha}(X))=\pi w(B_{\alpha}(X))=w(B_{\alpha}(X))=|X|$.

In 1987 A.V. Pestryakov proved the following theorems (P1-P9) for a space $B_{\alpha}(X)$ ($0<\alpha\leq\omega_{1}$)([17, 18]).

Theorem 1.1**.**

$({\bf P1})$* $t(B_{\alpha}(X))=l^{*}(X_{\aleph_{0}})$.*

Theorem 1.2**.**

$({\bf P2})$* $hd(B_{\alpha}(X))=hl^{*}(X_{\aleph_{0}})$.*

Theorem 1.3**.**

$({\bf P3})$* $hl(B_{\alpha}(X))=hd^{*}(X_{\aleph_{0}})$.*

Theorem 1.4**.**

$({\bf P4})$* $s(B_{\alpha}(X))=s^{*}(X_{\aleph_{0}})$.*

Theorem 1.5**.**

$({\bf P5})$* The followings statements are equivalent.*

$B_{\alpha}(X)$* is Fr*$\acute{e}$chet-Urysohn; 2. 2.

$B_{\alpha}(X)$* is sequential;* 3. 3.

$B_{\alpha}(X)$* is a $k$-space;* 4. 4.

$B_{\alpha}(X)$* is a $\omega_{1}$-$k$-space;* 5. 5.

$B_{\alpha}(X)$* has countable tightness;* 6. 6.

$X_{\aleph_{0}}$* satisfies ${\Omega\choose\Gamma}$;* 7. 7.

$X_{\aleph_{0}}$* is Lindel*$\ddot{o}$f.

Theorem 1.6**.**

$({\bf P6})$* $d(B_{\alpha}(X))=iw(X)$.*

Theorem 1.7**.**

$({\bf P7})$* For $0<\alpha\leq\omega_{1}$, $\psi(B_{\alpha}(X))=\psi w(B_{\alpha}(X))=i\chi(B_{\alpha}(X))=iw(B_{\alpha}(X))=d(X_{\aleph_{0}})$.*

Theorem 1.8**.**

$({\bf P8})$* $nw(B_{\alpha}(X))=nw(X_{\aleph_{0}})$.*

Theorem 1.9**.**

$({\bf P9})$* $l(B_{\alpha}(X))\geq t^{*}(X_{\aleph_{0}})$.*

Put $L(X)=\{n_{1}\cdot f_{Z_{1}}+...+n_{k}\cdot f_{Z_{k}}:$ $f_{Z_{i}}$ is the characteristic function of $Z_{i}$, $Z_{i}\in Z(X)$, $k,i,n_{k}\in\mathbb{N}\}$. For a space $X$, define $\mathbb{B}=\{Y:L(X)\subseteq Y\subseteq C(X_{\aleph_{0}})\}$. For example, $B_{\alpha}(X)\in\mathbb{B}$ for each $\alpha>0$, $C(X_{\aleph_{0}})\in\mathbb{B}$ and $[C_{p}(X)]^{\prime}_{\omega_{0}}=\bigcup\{cl_{\mathbb{R}^{X}}B:B\subset C_{p}(X),|B|\leq\omega_{0}\}\in\mathbb{B}$.

Pestryakov conjectured that most of theorems (P1-P9) are true for any $\mathbb{B}(X)\in\mathbb{B}$. In this paper we check for which Pestryakov’s theorems generalizations are valid. Also we prove some additional propositions for spaces in the class $\mathbb{B}$.

2 Tightness

The following result is well known [2].

Theorem 2.1**.**

(Arhangel’skii-Pytkeev) $t(C_{p}(X))=l^{*}(X)$.

We prove an analogue of Theorem P1 and the Arhangel’skii-Pytkeev Theorem for a space $\mathbb{B}(X)\in\mathbb{B}$.

Theorem 2.2**.**

$t(\mathbb{B}(X))=l^{*}(X_{\aleph_{0}})$.

Proof.

Since $t(C_{p}(Y))=l^{*}(Y)$ for a space $Y$ (the Arhangel’skii-Pytkeev Theorem) and $\mathbb{B}(X)\subseteq C(X_{\aleph_{0}})$, then $t(\mathbb{B}(X))\leq l^{*}(X_{\aleph_{0}})$.

Fix $n\in\mathbb{N}$. Assume that $\eta$ is an open cover of $X^{n}_{\aleph_{0}}$. Clearly that, whenever $V\in\eta$ and $x=(x_{1},...,x_{n})\in V$ there exists $W_{x}=\prod\limits_{i=1}^{n}\{V_{x_{i}}$ : $V_{x_{i}}$ is an open in $X_{\aleph_{0}}$ and $x_{i}\in V_{x_{i}}\}$ such that $x\in W_{x}\subset V$. Then we can consider the cover $\mu=\{W_{x}:x\in X^{n}\}$ of $X_{\aleph_{0}}^{n}$ such that $\mu$ is a refinement of $\eta$.

For each $x=(x_{1},...,x_{n})\in X^{n}$ denote $\widetilde{x}=\{x_{1},...,x_{n}\}\subset X$.

Let $m\in\mathbb{N}$, $z=(z_{1},...,z_{m})\in X^{m}$.

Fix $F(z_{i})\in Z(X)$ such that $z_{i}\in F(z_{i})$ ($1\leq i\leq m$) and if $\widetilde{x}\subset\widetilde{z}$ (i.e. $x=(z_{i_{1}},...,z_{i_{n}})$) then $F(z_{i_{k}})\subset V_{z_{i_{k}}}$ ($k=1,...,n$).

Let $f_{z}$ be the characteristic function of $\bigcup\{F(z_{i}):1\leq i\leq m\}$. The symbol $\bf{1}$ stands for the constant function to $1$. Note that $F=\{f_{z}:z\in\bigcup\{X^{m}:1\leq m<\omega\}\}\subset\mathbb{B}(X)$ and ${\bf 1}\in\overline{F}$. Then there exists $F^{\prime}\subset F$ such that ${\bf 1}\in\overline{F^{\prime}}$, $|F^{\prime}|\leq\tau=t(\mathbb{B}(X))$. Then there is $A\subset X^{m}$ such that $|A|\leq\tau$ and $F^{\prime}=\{f_{z}\in F:z\in A\}$. Let $Y=\{y\in X^{n}:\widetilde{y}\subset\widetilde{z},z\in A\}$. Clearly that $|Y|\leq\tau$.

We claim that $\{W_{y}:y\in Y\}\subset\mu$ is a cover of $X^{n}$. Let $x\in X^{n}$. Then $W=\{f\in\mathbb{B}(X):f(t)>0$ for all $t\in\widetilde{x}\}$ is a neighborhood of ${\bf 1}$. There is an $f_{z}\in F^{\prime}\bigcap W$. We have $\widetilde{x}\subset\bigcup\limits_{i=1}^{m}F(z_{i})$. Let $x_{k}\in F(z_{i_{k}})$ for $1\leq k\leq n$, $y=(z_{i_{1}},...,z_{i_{n}})$. Then $y\in Y$ and $x\in\prod\{F(z_{i_{k}}):1\leq k\leq n\}\subset\prod\{V_{y_{k}}:1\leq k\leq n\}=W_{y}$.

∎

Corollary 2.3**.**

$t(B_{\alpha}(X))=t([C(X)]^{\prime}_{\omega_{0}})=t(C_{p}(X_{\aleph_{0}}))=l^{*}(X_{\aleph_{0}})$ ($\alpha>0$).

Recall that a space is said to be scattered if every nonempty subspace of it has an isolated point.

Note that if $X$ is scattered, then $l(X)=l(X_{\aleph_{0}})$ [9]. Then we have the following result.

Corollary 2.4**.**

If $X$ is scattered, then $t(\mathbb{B}(X))=l^{*}(X)$.

Note that $l(X_{\aleph_{0}})=\omega_{0}$ implies that $l(X^{n}_{\aleph_{0}})=\omega_{0}$ [10].

Corollary 2.5**.**

$t(\mathbb{B}(X))=t(C_{p}(X_{\aleph_{0}}))=\omega_{0}$ if and only if $X_{\aleph_{0}}$ is Lindel$\ddot{o}$f.

3 Hereditary density

The following result is well known in $C_{p}$-theory [2] (for $hd(C_{p}(X))=\omega_{0}$ see [23]).

Theorem 3.1**.**

$hd(C_{p}(X))=hl^{*}(X)$.

We prove an analogue of this theorem and Theorem P2 for a space in class $\mathbb{B}$.

Theorem 3.2**.**

$hd(\mathbb{B}(X))=hl^{*}(X_{\aleph_{0}})$.

Proof.

Since $hd(C_{p}(Y))=hl^{*}(Y)$ for a space $Y$ and $\mathbb{B}(X)\subseteq C_{p}(X_{\aleph_{0}})$, then $hd(\mathbb{B}(X))\leq hl^{*}(X_{\aleph_{0}})$.

Let $Y$ be a subspace of $X^{n}_{\aleph_{0}}$, we consider the family $\mu=\{W_{y}:y\in Y\}$ of sets in $Y$ where $W_{y}=\prod\limits_{i=1}^{n}\{Z_{y_{i}}$ : $Z_{y_{i}}$ is a zero-set in $X$, $y_{i}\in Z_{y_{i}}\}$ such that $Y\subseteq\bigcup\mu$ and $|\mu|=l(Y)$. Then $\gamma=\{W_{y}\cap Y:W_{y}\in\mu\}$ is an open cover of $Y$.

Suppose $A=\{f_{Z_{y}}:Z_{y}=\bigcup\limits_{i=1}^{n}Z_{y_{i}},W_{y}\cap Y\in\gamma$ and $f_{Z_{y}}$ is the characteristic function of $Z_{y}\}$. Note that $A\subset\mathbb{B}(X)$. Let $D\subset A$ such that $|D|<|A|\leq|\gamma|$. Then the family $\eta=\{W_{y}\cap Y:f_{Z_{y}}\in D\}$ is not a cover of $Y$. Hence, there are $W_{y}\in\mu$ and $y\in Y$ such that $W_{y}\cap Y\notin\eta$ and $y=(y_{1},...,y_{n})\in W_{y}\cap(Y\setminus\bigcup\eta)$. Then $\{h:|h(y_{i})-f_{Z_{y}}(y_{i})|<1,1\leq i\leq n\}\cap D=\emptyset$. It follows that $D$ is not dense in $A$, and $d(A)\geq l(Y)$.

∎

Corollary 3.3**.**

$hd(B_{\alpha}(X))=hd([C(X)]^{\prime}_{\omega_{0}})=hd(C_{p}(X_{\aleph_{0}}))=hl^{*}(X_{\aleph_{0}})$ ($\alpha>0$).

Note that if $X$ is scattered, then $hl(X)=hl(X_{\aleph_{0}})$ [9]. Then we have the following result.

Corollary 3.4**.**

If $X$ is scattered, then $hd(\mathbb{B}(X))=hd(C_{p}(X_{\aleph_{0}}))=hl^{*}(X)$.

4 Hereditary Lindel$\ddot{o}$f degree

The following result is well known in $C_{p}$-theory [2, 23]

Theorem 4.1**.**

If $ind(X)=0$, then $hl(C_{p}(X))=hd^{*}(X)$.

Note that $ind(X_{\aleph_{0}})=0$ for any space $X$. Then we have the following theorem.

Theorem 4.2**.**

$hl(\mathbb{B}(X))=hd^{*}(X_{\aleph_{0}})$.

Proof.

Since $ind(X_{\aleph_{0}})=0$ and $\mathbb{B}(X)\subseteq C_{p}(X_{\aleph_{0}})$, then, by Theorem 4.1, $hl(\mathbb{B}(X))\leq hd^{*}(X_{\aleph_{0}})$.

First we prove an auxiliary proposition.

Proposition 4.3**.**

If $Y\subset X^{n}$ is such that whenever $y=(y_{1},...,y_{n})\in Y$ and $y_{i}\neq y_{j}$ for $i\neq j$, then $hl(\mathbb{B}(X))\geq d(Y_{\aleph_{0}})$.

Proof.

For each $y=(y_{1},...,y_{n})\in Y$ we fix a local base $\beta(y)$ at $y$ in $X^{n}_{\aleph_{0}}$ the following of the form $\beta(y)=\{V=\prod\limits_{i=1}^{n}V_{i}:y_{i}\in V_{i}\in Z(X),V_{i}\cap V_{j}=\emptyset$ for $i\neq j\}$. Let

[TABLE]

and $A=\{f_{V}:V\in\beta(y),y\in Y\}$. Clearly that $A\subset\mathbb{B}(X)$. Let $U(y)=\{f\in\mathbb{B}(X):|f(y_{i})-i|<1\}$ and $\gamma=\{U(y):y\in Y\}$. Then $\gamma$ is a cover of $A$. There is a subcover $\gamma^{\prime}\subseteq\gamma$ such that $|\gamma^{\prime}|=l(A)\leq hl(\mathbb{B}(X))$. Consider $S=\{y\in Y:U(y)\in\gamma^{\prime}\}$. Note that $|S|\leq|\gamma^{\prime}|$. We claim that $S$ is dense in $Y_{\aleph_{0}}$. Fix $z=(z_{1},...,z_{n})\in Y$, $V\in\beta(z)$. Then $f_{V}\in A$ and there is $U(y)\in\gamma^{\prime}$ such that $f_{V}\in U(y)$. It follows that $y\in V\cap S$ and $S$ is dense in $Y_{\aleph_{0}}$. ∎

Now, by indiction on $n$, we claim that $hl(\mathbb{B}(X))\geq hd(X^{n}_{\aleph_{0}})$.

For $n=1$ by Proposition 4.3.

Suppose that $hl(\mathbb{B}(X))\geq hd(X^{k}_{\aleph_{0}})$ for $k<n$. Note that $X^{n}_{ij}=\{(x_{1},...,x_{n}):x_{i}=x_{j}\}$ for $i\neq j$ is homeomorphic to the space $X^{n-1}$. Set $D=\bigcup\{X^{n}_{ij}:1\leq i\neq j\leq n\}$. Let $Z\subseteq X^{n}$. Then, by Proposition 4.3, $d(Z_{\aleph_{0}}\setminus D)\leq hl(\mathbb{B}(X))$ and, by the inductive hypothesis, $d(Z_{\aleph_{0}})\leq d(Z_{\aleph_{0}}\setminus D)+\sum\limits_{1\leq i\neq j\leq n}d(Z_{\aleph_{0}}\cap X^{n}_{ij})\leq hl(\mathbb{B}(X))$.

∎

Corollary 4.4**.**

$hl(B_{\alpha}(X))=hl([C(X)]^{\prime}_{\omega_{0}})=hl(C_{p}(X_{\aleph_{0}}))=hd^{*}(X_{\aleph_{0}})$

($\alpha>0$).

5 Spread

The well-known the following result in $C_{p}$-theory [2].

Theorem 5.1**.**

If $ind(X)=0$, then $s(C_{p}(X))=s^{*}(X)$.

Since $ind(X_{\aleph_{0}})=0$ for any space $X$, we have the following theorem for the spread $s(\mathbb{B}(X))$ of a space $\mathbb{B}(X)$ from the class $\mathbb{B}$.

Theorem 5.2**.**

$s(\mathbb{B}(X))=s^{*}(X_{\aleph_{0}})$.

Proof.

Since $ind(X_{\aleph_{0}})=0$ and $\mathbb{B}(X)\subseteq C_{p}(X_{\aleph_{0}})$, then, by Theorem 5.1, $s(\mathbb{B}(X))\leq s^{*}(X_{\aleph_{0}})$.

First we prove an auxiliary proposition.

Proposition 5.3**.**

Assume that $Y\subset X^{n}$ is such that whenever $y=(y_{1},...,y_{n})\in Y$ and $y_{i}\neq y_{j}$ for $i\neq j$. If $Y$ is discrete in $X^{n}_{\aleph_{0}}$, then $|Y|\leq s(\mathbb{B}(X))$.

Proof.

For each $y=(y_{1},...,y_{n})\in Y$ we fix $V=\prod\limits_{i=1}^{n}V_{i}$ such that $V\cap Y=\{y\}$, $y_{i}\in V_{i}\in Z(X),V_{i}\cap V_{j}=\emptyset$ for $i\neq j$. Let

[TABLE]

and $A=\{f_{V}:y\in Y\}$. Clearly that $A\subset\mathbb{B}(X)$ and $|A|=|Y|$. We claim that $A$ is discrete. If $f_{U}\in(f_{V},y,1)\cap A$, then $|f_{U}(y_{i})-i|<1$ for $1\leq i\leq n$. Hence, $y=(y_{1},...,y_{n})\in U$. It follows that $U=V$ and $f_{U}=f_{V}$. ∎

Now, by indiction on $n$, we claim that $s(\mathbb{B}(X))\geq s(X^{n}_{\aleph_{0}})$.

For $n=1$ by Proposition 5.3.

Suppose that $s(\mathbb{B}(X))\geq s(X^{k}_{\aleph_{0}})$ for $k<n$. Note that $X^{n}=\widetilde{X}^{n}\cup D$, where $D=\bigcup\{X^{n}_{ij}:1\leq i\neq j\leq n\}$, $X^{n}_{ij}=\{(x_{1},...,x_{n}):x_{i}=x_{j}\}$, $\widetilde{X}^{n}=X^{n}\setminus D$. Let $Y$ be discrete in $X^{n}_{\aleph_{0}}$. Put $Y_{1}=Y\cap\widetilde{X}^{n}$, $Y_{2}=Y\cap D$, then $Y=Y_{1}\cap Y_{2}$. By Proposition 5.3, $|Y_{1}|\leq s(\mathbb{B}(X))$. If $i\neq j$, then $X^{n}_{ij}$ is homeomorphic to the space $X^{n-1}$ and, hence, $|Y_{2}|\leq s(X^{n-1}_{\aleph_{0}})$. By the inductive hypothesis, $|Y_{2}|\leq s(\mathbb{B}(X))$. Note that $|Y|=|Y_{1}|+|Y_{2}|$. It follows that $s(\mathbb{B}(X))\geq s^{*}(X_{\aleph_{0}})$.

∎

Corollary 5.4**.**

$s(B_{\alpha}(X))=s([C(X)]^{\prime}_{\omega_{0}})=s(C_{p}(X_{\aleph_{0}}))=s^{*}(X_{\aleph_{0}})$ ($\alpha>0$).

6 Modification of Theorem P5.

Let $\kappa$ be an infinite cardinal. A space $C$ is said to be $\kappa$-initially compact (see [12]) if every open cover $\mathcal{V}$ of $C$ with $|\mathcal{V}|\leq\kappa$ has a finite subcover. A space $E$ is a $\kappa$-$k$ ($\kappa$-$k$-space), if whenever the subspace $A$ is non-closed in $E$, there is a $\kappa$-initially compact subspace $C$ of $E$ with $C\cap A$ non-closed in $C$ ([15]).

In 1982, Pytkeev [19] and Gerlits [15] independently proved the following result.

Theorem 6.1**.**

(Pytkeev-Gerlits) For a space $X$, the following are equivalent:

$C_{p}(X)$* is Fr*$\acute{e}$chet-Urysohn; 2. 2.

$C_{p}(X)$* is sequential;* 3. 3.

$C_{p}(X)$* is a $k$-space.*

Gerlits and Nagy defined three properties [14, 15]:

$\bullet$ the property $(\gamma)$: for every open $\omega$-cover $\mathcal{V}$ of $X$ there exists a sequence $G_{n}\in\mathcal{V}$ such that $\{G_{n}:n\in\omega\}$ is a $\gamma$-cover of $X$ (${\Omega\choose\Gamma}$ in terminology of selection principles).

$\bullet$ the property $(\epsilon)$ is one of the following equivalent properties:

(a) $X^{n}$ is Lindel$\ddot{o}$f for all $n\in\omega$ ($l^{*}(X)=\omega_{0}$);

(b) Every open $\omega$-cover of $X$ contains a countable $\omega$-subcover;

(c) $t(C_{p}(X))=\omega_{0}$.

$\bullet$ the property $(\varphi)$: whenever $\mathcal{U}=\bigcup\{\mathcal{U}_{n}:n\in\mathbb{N}\}$ is an open $\omega$-cover of $X$, $\mathcal{U}_{n}\subset\mathcal{U}_{n+1}$ ($n\in\mathbb{N}$), there exists a sequence $X_{n}\subset X$ such that $\underline{\lim}X_{n}=X$ and $X_{n}$ is $\omega$-covered by $\mathcal{U}_{n}$.

Gerlits proved that a space $X$ has the property $(\gamma)$ if and only if it has both $(\varphi)$ and $(\epsilon)$ (Theorem 1 in [15]).

Let $X$ be a topological space, and $x\in X$. A subset $A$ of $X$ converges to $x$, $x=\lim A$, if $A$ is infinite, $x\notin A$, and for each neighborhood $U$ of $x$, $A\setminus U$ is finite. Consider the following collection:

$\bullet$ $\Omega_{x}=\{A\subseteq X:x\in\overline{A}\setminus A\}$;

$\bullet$ $\Gamma_{x}=\{A\subseteq X:x=\lim A\}$.

In 1982 Gerlits and Nagy [14] proved

Theorem 6.2**.**

(Gerlits-Nagy) For a space $X$, the following are equivalent:

$C_{p}(X)$* satisfies $S_{1}(\Omega_{\bf 0},\Gamma_{\bf 0})$;* 2. 2.

$C_{p}(X)$* is Fr*$\acute{e}$chet-Urysohn; 3. 3.

$X$* satisfies $S_{1}(\Omega,\Gamma)$;* 4. 4.

$X$* has the property ($\gamma$), i.e. $X$ satisfies ${\Omega\choose\Gamma}$.*

In 1984, A.V. Arhangel’skii [1] proved the following theorem in the class of $P$-spaces.

Theorem 6.3**.**

(Arhangel’skii) For a $P$-space $X$, the following are equivalent:

$C_{p}(X)$* has countable tightness;* 2. 2.

$C_{p}(X)$* is Fr*$\acute{e}$chet-Urysohn; 3. 3.

$X$* is Lindel*$\ddot{o}$f.

Similar to Theorem 3 in [15], we get the next result.

Theorem 6.4**.**

If $\mathbb{B}(X)$ is a $\omega$-$k$-space, then $X_{\aleph_{0}}$ has the property $(\varphi)$.

Proof.

Otherwise $\mathbb{B}(X)$ is a $\omega$-$k$-space, yet $X$ has not the property $(\varphi)$, and let $\mathcal{U}=\bigcup\{\mathcal{U}_{n}:n\in\mathbb{N}\}$ witness this. Put for $n\in\mathbb{N}$, $n\geq 1$ and $A_{n}=\{f\in\mathbb{B}(X):f^{-1}(-\infty,n)$ is $\omega$-covered by $\mathcal{U}_{n}\}$, $A=\bigcup\{A_{n}:n\in\mathbb{N}\}$. Then $A_{n}$ is closed in $\mathbb{B}(X)$ for any $n$. On the other hand, $A$ is not closed in $\mathbb{B}(X)$, because ${\bf 0}\in\overline{A}\setminus A$. As $\mathbb{B}(X)$ is a $\omega$-$k$-space, there is a countably compact subset $C$ of $\mathbb{B}(X)$ such that $C\cap A$ is non-closed in $C$. As $C$ is countably compact, so also is each of its projections on the real line: for each $x\in X$ there is an $n(x)\in\mathbb{N}$ such that for each $f\in C$, $f(x)\leq n(x)$. Put $X_{n}=\{x\in X:n(x)\leq n\}$. As the sets $X_{n}$ monotonically increase and their union is $X$, we have $\underline{\lim}X_{n}=X$. Using now that $\{\mathcal{U}_{n}\}$ witnesses that $X$ has not $(\varphi)$, we get an $m\in\omega$ such that no $\mathcal{U}_{k}$ $\omega$-covers $X_{m}$.

Note that $C\cap A_{k}=\emptyset$ if $m<k<\omega$. Indeed, let $f\in A_{k}$, $m<k<\omega$. $f^{-1}(-\infty,k)$ is $\omega$-covered by $\mathcal{U}_{k}$, but $X_{m}$ is not, so $X_{m}\setminus f^{-1}(-\infty,k)\neq\emptyset$, hence, there is a point $x\in X_{m}$ such that $f(x)\geq k>m$ and $n(x)\leq m$. The definition of $X_{m}$ implies now that $f\notin C$.

However, this is impossible because then $C\cap A=\bigcup\{C\cap A_{k}:k\leq m\}$ would be closed in $C$, contrary to the choice of $C$.

∎

The following theorem is proved similarly to Theorem 4 in [15]; therefore, we omit the proof of this theorem.

Theorem 6.5**.**

If $\mathbb{B}(X)$ is a $\omega_{1}$-$k$-space, then $X_{\aleph_{0}}$ has the property $S_{1}(\Omega,\Gamma)$.

Now we can consider a modification of Theorem P5.

Theorem 6.6**.**

Let $X$ be a Tychonoff space and $\mathbb{B}(X)\in\mathbb{B}$. Then the following are equivalent.

$\mathbb{B}(X)$* is Fr*$\acute{e}$chet-Urysohn; 2. 2.

$\mathbb{B}(X)$* is sequential;* 3. 3.

$\mathbb{B}(X)$* is a $k$-space;* 4. 4.

$\mathbb{B}(X)$* is a $\omega_{1}$-$k$-space;* 5. 5.

$\mathbb{B}(X)$* has countable tightness;* 6. 6.

$X_{\aleph_{0}}$* satisfies $S_{1}(\Omega,\Gamma)$;* 7. 7.

$X_{\aleph_{0}}$* is Lindel*$\ddot{o}$f.

Proof.

$(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)$, $(1)\Rightarrow(5)$ are immediate. Since $\mathbb{B}(X)\subseteq C(X_{\aleph_{0}})$, then, by Theorem 6.2, we have that $(6)\Rightarrow(1)$ holds.

By Lemma 2.2, we have that $(5)\Rightarrow(7)$.

By Theorem 6.5, if $\mathbb{B}(X)$ is a $\omega_{1}$-$k$-space, then $X_{\aleph_{0}}$ satisfies $S_{1}(\Omega,\Gamma)$, i.e. $(4)\Rightarrow(6)$ holds.

$(7)\Rightarrow(1)$. Since $X_{\aleph_{0}}$ is a $P$-space, then, by Theorem 6.3, we have that $C_{p}(X_{\aleph_{0}})$ is Fr$\acute{e}$chet-Urysohn. But $\mathbb{B}(X)\subseteq C_{p}(X_{\aleph_{0}})$, hence $\mathbb{B}(X)$ is Fr$\acute{e}$chet-Urysohn, too.

∎

Corollary 6.7**.**

Let $X$ be a Tychonoff space. Then the following are equivalent.

$C_{p}(X_{\aleph_{0}})$ is Fr$\acute{e}$chet-Urysohn; 2. 2.

$C_{p}(X_{\aleph_{0}})$ is sequential; 3. 3.

$C_{p}(X_{\aleph_{0}})$ is a $k$-space; 4. 4.

$C_{p}(X_{\aleph_{0}})$ is a $\omega_{1}$-$k$-space; 5. 5.

$C_{p}(X_{\aleph_{0}})$ has countable tightness; 6. 6.

$X_{\aleph_{0}}$ satisfies $S_{1}(\Omega,\Gamma)$; 7. 7.

$X_{\aleph_{0}}$ is Lindel$\ddot{o}$f.

Corollary 6.8**.**

Let $\mathbb{B}(X)$ be a $\omega_{1}$-$k$-space and $B_{1}(X)\subseteq\mathbb{B}(X)$. Then $B_{1}(X)=\mathbb{B}(X)=C(X_{\aleph_{0}})$.

Corollary 6.9**.**

Assume that $X_{\aleph_{0}}$ satisfies $S_{1}(\Omega,\Gamma)$. Then $B_{1}(X)=C(X_{\aleph_{0}})$.

Corollary 6.10**.**

Assume that $X$ is a perfectly normal space and $B_{\alpha}(X)$ is $k$-space for some $1\leq\alpha\leq\omega_{1}$. Then $X$ is countable.

Proposition 6.11**.**

There exists a space $X$ such that $B_{\alpha}(X)$ is a $\omega$-$k$-space, but not a $\omega_{1}$-$k$-space.

Proof.

Let $X$ be the space $\omega_{2}\setminus L$, where $L$ denotes the set of $\omega$-limits in $\omega_{2}$; then $X=X_{\aleph_{0}}$, $C_{p}(X)=B_{\alpha}(X)$ ($0\leq\alpha\leq\omega_{1}$) and $B_{\alpha}(X)$ is $\omega$-$k$ but not $\omega_{1}$-$k$ (see Example in [15]). ∎

Proposition 6.12**.**

There exists a space $X$ such that $\omega_{0}=t(C_{p}(X))<t(B_{\alpha}(X))$ for $\alpha>0$.

Proof.

The space $C_{p}([0,1])$ is not sequential (Fr$\acute{e}$chet-Urysohn, $k$-space), but $\omega_{0}=t(C_{p}([0,1]))<t(B_{\alpha}([0,1]))=\mathfrak{c}$ for any $\alpha>0$. ∎

Proposition 6.13**.**

(MA+$\neg$ CH) There exists a set of reals $X$ such that $C_{p}(X)$ is sequential, but $t(\mathbb{B}(X))>\omega_{0}$ for any $\mathbb{B}(X)\in\mathbb{B}$.

Proof.

By Theorem 1 in [13], assuming Martin’s axiom, there exists a set of reals $X$ of cardinality the continuum such that $X$ has the property $S_{1}(\Omega,\Gamma)$. Then $X_{\aleph_{0}}$ is not Lindel$\ddot{o}$f and, hence, by Theorem 6.6, $t(\mathbb{B}(X))>\omega_{0}$ for any $\mathbb{B}(X)\in\mathbb{B}$. ∎

Theorem 6.14**.**

If $\mathbb{B}(X)$ is a $\omega$-$k$-space, then $X$ satisfies $S_{1}(Z_{\Omega},Z_{\Gamma})$.

Proof.

Let $\alpha=\{F_{i}:i\in\mathbb{N}\}$ be a $\omega$-cover of $X$ by zero-sets of $X$. Consider $A=\{h_{n}:h_{n}=n\cdot f_{n}$, $f_{n}$ is the characteristic function of $X\setminus F_{n}$, $F_{n}\in\alpha$, $n\in\mathbb{N}\}$. Note that ${\bf 0}\in\overline{A}\setminus A$. Hence, there exists a countably compact set $C$ such that $A\bigcap C$ is not a closed subset of $C$. Since $C$ is a countably compact set, whenever $x\in X$ there is $n(x)\in\mathbb{N}$ such that $f(x)<n(x)$ for each $f\in C$. Let $X_{n}=\{x\in X:f(x)<n$ for each $f\in C\}$. Then $X_{n+1}\supseteq X_{n}$ and $X=\bigcup\limits_{n}X_{n}$.

If for every $n$ there exists $i(n)$ such that $X_{n}\subseteq F_{i(n)}$, then $\{F_{i(n)}:n\in\mathbb{N}\}$ is a $\gamma$-cover of $X$. Otherwise, there is an $n^{\prime}$ such that $X_{n^{\prime}}\setminus F_{i}\neq\emptyset$ for each $i\in\mathbb{N}$. Fix an $n\in\mathbb{N}$ such that $n>n^{\prime}$. There is an $x\in X_{n^{\prime}}\setminus F_{n}$ such that $h_{n}(x)=n>n^{\prime}$. It follows that $h_{n}\notin C$. Thus, we have that $A\bigcap C=\{h_{i}:i<n^{\prime}+1\}\bigcap C$ is not a closed subset of $C$, a contradiction.

∎

Recall that a space $X$ is called proper analytic if it admits a perfect map onto an analytic subset of a complete separable metric space. A space $X$ is disjoint analytic if and only if it is a one-to-one continuous image of a proper analytic space [16]. Note that any $K$-Lusin space is a disjoint analytic space.

Theorem 6.15**.**

Let $X$ be a disjoint analytic space and $B_{1}(X)\subseteq\mathbb{B}(X)$. Then the following are equivalent:

$X$* is scattered;* 2. 2.

$\mathbb{B}(X)$* is Fr*$\acute{e}$chet-Urysohn.

Proof.

If $X$ is scattered, then $l(X)=l(X_{\aleph_{0}})$ [9]. By Theorem 6.6, $\mathbb{B}(X)$ is Fr$\acute{e}$chet-Urysohn.

If $\mathbb{B}(X)$ is Fr$\acute{e}$chet-Urysohn, then, by Theorem 6.6 and Corollary 6.8, $B_{1}(X)=\mathbb{B}(X)=C_{p}(X_{\aleph_{0}})$. Then, by Theorem 6 in [16], $X$ is scattered. ∎

It is well-known that for a compact space $X$, $C_{p}(X)$ is Fr$\acute{e}$chet-Urysohn if and only if $C_{p}(X)$ is a $k$-space if and only if $X$ is scattered [15, 19].

Corollary 6.16**.**

For a compact space $X$ and $\alpha>0$, $B_{\alpha}(X)$ is Fr$\acute{e}$chet-Urysohn if and only if $B_{\alpha}(X)$ is a $k$-space if and only if $X$ is scattered.

Thus we have that if a compact space $X$ is not scattered, then $t(B_{\alpha}(X))\geq l(X_{\aleph_{0}})\geq\mathfrak{c}$.

Note that there exists a scattered space $Z$ such that $t(B_{1}(Z))>\omega_{0}$.

Example 6.17**.**

Let $Z$ be the set of all countable ordinals endowed with the interval topology. Then $Z$ is scattered pseudocompact and $t(B_{1}(Z))>\omega_{0}$.

A.V. Arhangel’skii [3] (see also [24]) asked the question: For what compact spaces $X$ does the inequality $l(X_{\aleph_{0}})\leq\mathfrak{c}$ hold ?

It is well-known that the answer is positive in the following cases:

1. $X$ is a finite product of ordered compact spaces [24].

2. $X$ is a compact space of countable tightness [20].

3. $X$ is a weakly Corson compact space [21].

This implies, in particular, $t(B_{\alpha}(X))\leq\mathfrak{c}$ for any space $X$ in these classes of spaces.

In [3, 24], it was shown that the Lindel$\ddot{o}$f number of $X_{\aleph_{0}}$ for a compact space $X$ can be arbitrary large (for example, the Stone-$\check{C}$ech compactification $\beta(D)$ of a discrete space $D$). Therefore, the tightness of $B_{\alpha}(X)$ for compact spaces $X$ is not bounded. E.G. Pytkeev proved the following remarkable result (Theorem 1.1. in [21]).

Theorem 6.18**.**

(Pytkeev) Let $X$ be a Tychonoff space. Then

$t(C_{p}(X))\leq t(B_{\alpha}(X))\leq exp(t(C_{p}(X))\cdot t(X))$.

7 Density

Recall that the $i$-weight $iw(X)$ of a space $X$ is the smallest infinite cardinal number $\tau$ such that $X$ can be mapped by a one-to-one continuous mapping onto a Tychonoff space of the weight not greater than $\tau$.

Theorem 7.1**.**

(Noble [11]) $d(C_{p}(X))=iw(X)$.

Let $A\subset Y$. Put $[A]^{\prime}_{\tau}=\bigcup\{\overline{B}:B\subset A,|B|\leq\tau\}$, $T(x,A,Y)=\min\{\tau:x\in[A]^{\prime}_{\tau}\}$, $T(A,Y)=\sup\{T(x,A,Y):x\in\overline{A}\}$. Then $T(C_{p}(X),B_{\alpha}(X))=\omega_{0}$. Since $C_{p}(X)$ is dense in $B_{\alpha}(X)$, $d(B_{\alpha}(X))\leq d(C_{p}(X))=iw(X)$.

Let $\mu=d(B_{\alpha}(X))$. Then there is $D\subset B_{\alpha}(X)$ such that $|D|=\mu$ and $\overline{D}=B_{\alpha}(X)$. The equality $T(C_{p}(X),B_{\alpha}(X))=\omega_{0}$ means that $[C_{p}(X)]^{\prime}_{\omega_{0}}=B_{\alpha}(X)$. For each $d\in D$, fix a set $C_{d}\subset C_{p}(X)$ such that $|C_{d}|\leq\omega_{0}$ and $d\in\overline{C_{d}}$. Then the set $S=\bigcup\{C_{d}:d\in D\}$ is dense in $C_{p}(X)$ and $|S|\leq\mu$. Hence, $d(B_{\alpha}(X))\geq d(C_{p}(X))$. Thus, we have the Theorem P6 of Pestryakov that $d(B_{\alpha}(X))=iw(X)$ ($0<\alpha\leq\omega_{1}$).

Example 7.2**.**

Let $X$ be a first-countable space such that $|X|\leq\mathfrak{c}$ and $iw(X)>\omega_{0}$. Then $d(B_{\alpha}(X))=iw(X)>iw(X_{\aleph_{0}})=d(C_{p}(X_{\aleph_{0}}))$.

For example, if $Z$ is the set of all countable ordinals endowed with the interval topology, then $d(B_{\alpha}(Z))>d(C_{p}(Z_{\aleph_{0}}))$.

Note also that if $\mathfrak{c}<2^{\omega_{1}}$ then $|B_{\omega_{1}}(Z)|=\mathfrak{c}<2^{\omega_{1}}=|C_{p}(Z_{\aleph_{0}})|$, otherwise $|B_{\omega_{1}}(Z)|=|C_{p}(Z_{\aleph_{0}})|$.

8 Pseudocharacter, pseudoweight

It is well-known that $\psi(C_{p}(X))=iw(C_{p}(X))=d(X)$ [2].

Theorem 8.1**.**

$\psi(\mathbb{B}(X))=\psi w(\mathbb{B}(X))=i\chi(\mathbb{B}(X))=iw(\mathbb{B}(X))=d(X_{\aleph_{0}})$.

Proof.

Note that if there exists a condensation (one-to-one continuous map) $f:Y\rightarrow Z$ of a space $Y$ onto a space $Z$ then $\psi(Y)\leq\psi(Z)\leq\chi(Z)\leq w(Z)$ and $\psi(Y)\leq\psi w(Z)\leq w(Z)$. Since the space $Z$ is arbitrary, we get that $\psi(Y)\leq i\chi(Y)\leq iw(Y)$ and $\psi(Y)\leq\psi w(Y)\leq iw(Y)$.

Since $iw(C_{p}(X))=d(X)$ (Theorem 7.1) and $\mathbb{B}(X)\subset C_{p}(X_{\aleph_{0}})$, it is enough to prove that $d(X_{\aleph_{0}})\leq\psi(\mathbb{B}(X))$.

Assume that $d(X_{\aleph_{0}})>\psi(\mathbb{B}(X))$. Let $\{{\bf 0}\}=\bigcap\{U_{\xi}:\xi\in M\}$, $|M|=\psi(\mathbb{B}(X))$. We can assume that $U_{\xi}=(x_{1}(\xi),...,x_{n}(\xi),\epsilon(\xi))=\{f:f\in\mathbb{B}(X),|f(x_{i}(\xi))|<\epsilon(\xi)\}$. Let $A=\{x_{i}(\xi):\xi\in M,1\leq i\leq n(\xi)\}$. Since $|A|<d((X_{\aleph_{0}})$, there exists a zero-set $D$ in $X$ such that $D\cap A=\emptyset$. Note that the characteristic function $\chi_{D}$ of the set $D$ is in $\mathbb{B}(X)$, $\chi_{D}\neq{\bf 0}$ and $\chi_{D}\in\bigcap\{U_{\xi}:\xi\in M\}$, a contradiction. ∎

9 Network weight

Lemma 9.1**.**

Define the function $\varphi:X_{\aleph_{0}}\rightarrow C_{p}(\mathbb{B}(X))$ by the rule: $\varphi(x)(f)=f(x)$ for each $f\in\mathbb{B}(X)$. Then $X_{\aleph_{0}}$ is homeomorphic to $\varphi(X_{\aleph_{0}})\subset C_{p}(\mathbb{B}(X))$.

Proof.

Obviously, $\varphi$ is bijection from $X_{\aleph_{0}}$ onto $\varphi(X_{\aleph_{0}})$.

Note that $\mathbb{B}(X)\subset C_{p}(X_{\aleph_{0}})$. The equality $\varphi^{-1}(\{h:h\in\varphi(X_{\aleph_{0}}),|h(f_{i})-\varphi(x)(f_{i})|<\epsilon,1\leq i\leq n,f_{i}\in\mathbb{B}(X)\})=\bigcap\limits_{i=1}^{n}f^{-1}_{i}(f_{i}(x)-\epsilon,f_{i}(x)+\epsilon)$ implies that $\varphi$ is a continuous map.

The set $\varphi(M)=\{h:h\in\varphi(X),|h(\chi_{M})-1|<1\}$ for a characteristic function $\chi_{M}$ of the zero-set $M$ is an open set in $\varphi(X)$. Thus, $\varphi^{-1}$ is a continuous map.

∎

Theorem 9.2**.**

$nw(\mathbb{B}(X))=nw(X_{\aleph_{0}})$.

Proof.

Since $nw(C_{p}(Y))=nw(Y)$ for a Tychonoff space $Y$ [2] and $\mathbb{B}(X)\subseteq C(X_{\aleph_{0}})$ we get that $nw(\mathbb{B}(X))\leq nw(X_{\aleph_{0}})$. By Lemma 9.1, $nw(X_{\aleph_{0}})\leq nw(C_{p}(\mathbb{B}(X))$. Thus, $nw(X_{\aleph_{0}})\leq nw(\mathbb{B}(X))$. ∎

Note that $nw(X)\leq nw(X_{\aleph_{0}})\leq nw(X)^{\omega_{0}}$. Then we have the following result.

Corollary 9.3**.**

If $\kappa=\kappa^{\omega_{0}}$, then $nw(\mathbb{B}(X))=nw(C_{p}(X_{\aleph_{0}}))=nw(X)=\kappa$.

10 The Lindel$\ddot{o}$f number

The following result is well known in $C_{p}$-theory [5].

Theorem 10.1**.**

(Asanov) $l(C_{p}(X))\geq t^{*}(X)$.

For a space $\mathbb{B}(X)\in\mathbb{B}$, we have the following result.

Theorem 10.2**.**

$l(\mathbb{B}(X))\geq t^{*}(X_{\aleph_{0}})$.

Proof.

Denote as usually $[Y]^{<\omega}$ the set of all non-empty finite subsets of a space $Y$. Consider the topological space $Y_{p}=([Y_{\aleph_{0}}]^{<\omega},\tau)$ where the topology $\tau$ generated by the base $\beta=\{H^{*}:H^{*}=\{F\in[Y_{\aleph_{0}}]^{<\omega}:F\subset H\}$ for any open $H$ in $Y\}$. Since $t(Y^{n})\leq t(Y_{p})$ for every $n\in\omega$ [5] it is enough to prove that $t(X_{\aleph_{0}p})\leq l(\mathbb{B}(X))$.

Let $M\subset X_{\aleph_{0}p}$ and $S\in\overline{M}\setminus M$. Note that the family $\{<p,(-1,1)>:p\in M\}$ is a cover of the set $\{f:f\in\mathbb{B}(X),f(S)=0\}$ where $<p,(-1,1)>=\{f:f\in\mathbb{B}(X),f(p)\subset(-1,1)\}$. Since $\{f:f\in\mathbb{B}(X),f(S)=0\}$ is closed in $\mathbb{B}(X)$, choose $M^{\prime}\subset M$ such that $|M^{\prime}|\leq l(\mathbb{B}(X))$ and $\{<p,(-1,1)>:p\in M^{\prime}\}$ is a cover of $\{f:f\in\mathbb{B}(X),f(S)=0\}$. Then $S\in\overline{M^{\prime}}$.

∎

Note that $l(B_{1}([0,1]))=\mathfrak{c}>\omega_{0}=t^{*}([0,1]_{\aleph_{0}})$.

Question. Is it possible to replace $X_{\aleph_{0}}$ by $X$ in Theorem 10.2 ?

Acknowledgment

The author would like to thank the referee for careful reading and valuable comments and suggestions.