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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# On the different kinds of separability of the space of Borel functions

Alexander V. Osipov
• Corresponding author
• Krasovskii Institute of Mathematics and Mechanics, Ural Federal University, Ural State University of Economics, 620219, Yekaterinburg, Russia
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Published Online: 2018-07-17 | DOI: https://doi.org/10.1515/math-2018-0070

## Abstract

In paper we prove that:

• a space of Borel functions B(X) on a set of reals X, with pointwise topology, to be countably selective sequentially separable if and only if X has the property S1(BΓ, BΓ);

• there exists a consistent example of sequentially separable selectively separable space which is not selective sequentially separable. This is an answer to the question of A. Bella, M. Bonanzinga and M. Matveev;

• there is a consistent example of a compact T2 sequentially separable space which is not selective sequentially separable. This is an answer to the question of A. Bella and C. Costantini;

• min{𝔟, 𝔮} = {κ : 2κ is not selective sequentially separable}. This is a partial answer to the question of A. Bella, M. Bonanzinga and M. Matveev.

MSC 2010: 54C35; 54C05; 54C65; 54A20

## 1 Introduction

In [12], Osipov and Pytkeev gave necessary and sufficient conditions for the space B1(X) of the Baire class 1 functions on a Tychonoff space X, with pointwise topology, to be (strongly) sequentially separable. In this paper, we consider some properties of a space B(X) of Borel functions on a set of reals X, with pointwise topology, that are stronger than (sequential) separability.

## 2 Main definitions and notation

Many topological properties are defined or characterized in terms of the following classical selection principles. Let 𝓐 and 𝓑 be sets consisting of families of subsets of an infinite set X. Then:

S1(𝓐, 𝓑) is the selection hypothesis: for each sequence (An : n ∈ ℕ) of elements of 𝓐 there is a sequence (bn : n ∈ ℕ) such that for each n, bnAn, and {bn : n ∈ ℕ} is an element of 𝓑.

Sfin(𝓐, 𝓑) is the selection hypothesis: for each sequence (An : n ∈ ℕ) of elements of 𝓐 there is a sequence (Bn: n ∈ ℕ) of finite sets such that for each n, BnAn, and ⋃n∈ℕ Bn ∈ 𝓑.

Ufin(𝓐, 𝓑) is the selection hypothesis: whenever 𝓤1, 𝓤2, … ∈ 𝓐 and none contains a finite subcover, there are finite sets 𝓕n ⊆ 𝓤n, n ∈ ℕ, such that {⋃ 𝓕n : n ∈ ℕ} ∈ 𝓑.

An open cover 𝓤 of a space X is:

• an ω-cover if X does not belong to 𝓤 and every finite subset of X is contained in a member of 𝓤;

• a γ-cover if it is infinite and each xX belongs to all but finitely many elements of 𝓤.

For a topological space X we denote:

• Ω — the family of all countable open ω-covers of X;

• Γ — the family of all countable open γ-covers of X;

• BΩ — the family of all countable Borel ω-covers of X;

• BΓ — the family of all countable Borel γ-covers of X;

• FΓ — the family of all countable closed γ-covers of X;

• 𝓓 — the family of all countable dense subsets of X;

• 𝓢 — the family of all countable sequentially dense subsets of X.

A γ-cover 𝓤 of co-zero sets of X is γF-shrinkable if there exists a γ-cover {F(U) : U ∈ 𝓤} of zero-sets of X with F(U) ⊂ U for every U ∈ 𝓤.

For a topological space X we denote ΓF, the family of all countable γF-shrinkable γ-covers of X.

We will use the following notations.

• Cp(X) is the set of all real-valued continuous functions C(X) defined on a space X, with pointwise topology.

• B1(X) is the set of all first Baire class 1 functions B1(X) i.e., pointwise limits of continuous functions, defined on a space X, with pointwise topology.

• B(X) is the set of all Borel functions, defined on a space X, with pointwise topology.

If X is a space and AX, then the sequential closure of A, denoted by [A]seq, is the set of all limits of sequences from A. A set DX is said to be sequentially dense if X = [D]seq. If D is a countable, sequentially dense subset of X then X call sequentially separable space.

Call a space X strongly sequentially separable if X is separable and every countable dense subset of X is sequentially dense.

A space X is (countably) selectively separable (or M-separable, [3]) if for every sequence (Dn : n ∈ ℕ) of (countable) dense subsets of X one can pick finite FnDn, n ∈ ℕ, so that ⋃ {Fn : n ∈ ℕ} is dense in X.

In [3], the authors started to investigate a selective version of sequential separability.

A space X is (countably) selectively sequentially separable (or M-sequentially separable, [3]) if for every sequence (Dn : n ∈ ℕ) of (countable) sequentially dense subsets of X, one can pick finite FnDn, n ∈ ℕ, so that ⋃{Fn : n ∈ ℕ} is sequentially dense in X.

In Scheeper’s terminology [16], countably selectively separability equivalently to the selection principle Sfin(𝓓, 𝓓), and countably selective sequentially separability equivalently to the Sfin(𝓢, 𝓢).

Recall that the cardinal 𝔭 is the smallest cardinal so that there is a collection of 𝔭 many subsets of the natural numbers with the strong finite intersection property but no infinite pseudo-intersection. Note that ω1 ≤ 𝔭 ≤ 𝔠.

For f, g ∈ ℕ, let f* g if f(n) ≤ g(n) for all but finitely many n. 𝔟 is the minimal cardinality of a ≤*-unbounded subset of ℕ. A set B ⊂ [ℕ] is unbounded if the set of all increasing enumerations of elements of B is unbounded in ℕ, with respect to ≤*. It follows that |B| ≥ 𝔟. A subset S of the real line is called a Q-set if each one of its subsets is a Gδ. The cardinal 𝔮 is the smallest cardinal so that for any κ < 𝔮 there is a Q-set of size κ. (See [7] for more on small cardinals including 𝔭).

## 3 Properties of a space of Borel functions

#### Theorem 3.1

For a set of reals X, the following statements are equivalent:

1. B(X) satisfies S1(𝓢, 𝓢) and B(X) is sequentially separable;

2. X satisfies S1(BΓ, BΓ);

3. B(X) ∈ Sfin(𝓢, 𝓢) and B(X) is sequentially separable;

4. X satisfies Sfin(BΓ, BΓ);

5. B1(X) satisfies S1(𝓢, 𝓢);

6. X satisfies S1(FΓ, FΓ);

7. B1(X) satisfies Sfin(𝓢, 𝓢).

#### Proof

It is obvious that (1) ⇒ (3).

(2) ⇔ (4). By Theorem 1 in [15], Ufin(BΓ, BΓ) = S1(BΓ, BΓ) = Sfin(BΓ, BΓ).

(3) ⇒ (2). Let {𝓕i} ⊂ BΓ and 𝓢 = {hm}m∈ℕ be a countable sequentially dense subset of B(X). For each i ∈ ℕ we consider a countable sequentially dense subset 𝓢i of B(X) and 𝓕i = $\begin{array}{}\left\{{F}_{i}^{m}{\right\}}_{m\in \mathbb{N}}\end{array}$ where

$\begin{array}{}{\mathcal{S}}_{i}=\left\{{f}_{i}^{m}\right\}:=\left\{{f}_{i}^{m}\in B\left(X\right):{f}_{i}^{m}↾{F}_{i}^{m}={h}_{m}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{f}_{i}^{m}↾\left(X\setminus {F}_{i}^{m}\right)=1\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}m\in \mathbb{N}\right\}.\end{array}$

Since 𝓕i = $\begin{array}{}\left\{{F}_{i}^{m}{\right\}}_{m\in \mathbb{N}}\end{array}$ is a Borel γ-cover of X and 𝓢 is a countable sequentially dense subset of B(X), we have that 𝓢i is a countable sequentially dense subset of B(X) for each i ∈ ℕ. Indeed, let hB(X), there is a sequence {hs}s∈ℕ ⊂ 𝓢 such that {hs}s∈ℕ converges to h. We claim that $\begin{array}{}\left\{{f}_{i}^{s}{\right\}}_{s\in \mathbb{N}}\end{array}$ converges to h. Let K = {x1, …, xk} be a finite subset of X, ϵ > 0 and let W = 〈h, K, ϵ〉 := {gB(X) : |g(xj) – h(xj)| < ϵ for j = 1, …, k} be a base neighborhood of h, then there is m0 ∈ ℕ such that K$\begin{array}{}{F}_{i}^{m}\end{array}$ for each m > m0 and hsW for each s > m0. Since $\begin{array}{}{f}_{i}^{s}\end{array}$K = hsK for every s > m0, $\begin{array}{}{f}_{i}^{s}\end{array}$W for every s > m0. It follows that { $\begin{array}{}{f}_{i}^{s}\end{array}$}s∈ℕ converges to h.

Since B(X) satisfies Sfin(𝓢, 𝓢), there is a sequence $\begin{array}{}\left({F}_{i}=\left\{{f}_{i}^{{m}_{1}},...,{f}_{i}^{{m}_{s\left(i\right)}}\right\}:i\in \mathbb{N}\right)\end{array}$ such that for each i, Fi ⊂ 𝓢i, and ⋃i∈ℕ Fi is a countable sequentially dense subset of B(X).

For 0 ∈ B(X) there is a sequence $\begin{array}{}\left\{{f}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}{\right\}}_{j\in \mathbb{N}}\subset \bigcup _{i\in \mathbb{N}}{F}_{i}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{such that}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left\{{f}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}{\right\}}_{j\in \mathbb{N}}\end{array}$ converges to 0. Consider a sequence ( $\begin{array}{}{F}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}\end{array}$ : j ∈ ℕ). Then

1. $\begin{array}{}{F}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}\end{array}$ ∈ 𝓕ij;

2. {$\begin{array}{}{F}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}\end{array}$ : j ∈ ℕ} is a γ-cover of X.

Indeed, let K be a finite subset of X and U = 〈0, K, $\begin{array}{}\frac{1}{2}\end{array}$〉 be a base neighborhood of 0, then there is j0 ∈ ℕ such that $\begin{array}{}{f}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}\end{array}$U for every j > j0. It follows that K$\begin{array}{}{F}_{{i}_{j}}^{{m}_{s\left({i}_{j}\right)}}\end{array}$ for every j > j0. We thus get that X satisfies Ufin(BΓ, BΓ), and, hence, by Theorem 1 in [15], X satisfies S1(BΓ, BΓ).

(2) ⇒ (1). Let {Si} ⊂ 𝓢 and S = {dn : n ∈ ℕ} ∈ 𝓢. Consider the topology τ generated by the family 𝓟 = {f–1(G) : G is an open set of ℝ and f$\begin{array}{}S\cup \bigcup _{i\in \mathbb{N}}{S}_{i}\end{array}$}. Since P = $\begin{array}{}S\cup \bigcup _{i\in \mathbb{N}}{S}_{i}\end{array}$ is a countable dense subset of B(X) and X is Tychonoff, we have that the space Y = (X, τ) is a separable metrizable space. Note that a function fP, considered as mapping from Y to ℝ, is a continuous function i.e. fC(Y) for each fP. Note also that an identity map φ from X on Y, is a Borel bijection. By Corollary 12 in [6], Y is a QN-space and, hence, by Corollary 20 in [17], Y has the property S1(BΓ, BΓ). By Corollary 21 in [17], B(Y) is an α2 space.

Let q : ℕ ↦ ℕ × ℕ be a bijection. Then we enumerate {Si}i∈ℕ as {Sq(i)}q(i)∈ℕ×ℕ. For each dnS there are sequences sn,mSn,m such that sn,m converges to dn for each m ∈ ℕ. Since B(Y) is an α2 space, there is {bn,m : m ∈ ℕ} such that for each m, bn,msn,m, and, bn,mdn (m → ∞). Let B = {bn,m : n, m ∈ ℕ}. Note that S ⊂ [B]seq.

Since X is a σ-set (that is, each Borel subset of X is Fσ)(see [17]), B1(X) = B(X) and φ(B(Y)) = φ(B1(Y)) ⊆ B(X) where φ(B(Y)) := {pφ : pB(Y)} and φ(B1(Y)) := {pφ : pB1(Y)}.

Since S is a countable, sequentially dense subset of B(X), for any gB(X) there is a sequence {gn}n∈ℕS such that {gn}n∈ℕ converges to g. But g we can consider as a mapping from Y into ℝ and a set {gn : n ∈ ℕ} as subset of C(Y). It follows that gB1(Y). We get that φ(B(Y)) = B(X).

We claim that B ∈ 𝓢, i.e. that [B]seq = B(X). Let fB(Y) and {fk : k ∈ ℕ} ⊂ S such that fkf (k → ∞). For each k ∈ ℕ there is { $\begin{array}{}{f}_{k}^{n}\end{array}$ : n ∈ ℕ} ⊂ B such that $\begin{array}{}{f}_{k}^{n}\end{array}$fk (n → ∞). Since Y is a QN-space (Theorem 16 in [6]), there exists an unbounded β ∈ ℕ such that { $\begin{array}{}{f}_{k}^{\beta \left(k\right)}\end{array}$} converges to f on Y. It follows that { $\begin{array}{}{f}_{k}^{\beta \left(k\right)}\end{array}$ : k ∈ ℕ} converge to f on X and [B]seq = B(X).

(5) ⇒ (6). By Velichko’s Theorem ([18]), a space B1(X) is sequentially separable for any separable metric space X.

Let {𝓕i} ⊂ FΓ and 𝓢 = {hm}m∈ℕ be a countable sequentially dense subset of B1(X).

Similarly implication (3) ⇒ (2) we get X satisfies Ufin(FΓ, FΓ), and, hence, by Lemma 13 in [17], X satisfies S1(FΓ, FΓ).

(6) ⇒ (5). By Corollary 20 in [17], X satisfies S1(BΓ, BΓ). Since X is a σ-set (see [17]), B1(X) = B(X) and, by implication (2) ⇒ (1), we get B1(X) satisfies S1(𝓢, 𝓢).□

In [16], (Theorem 13) M. Scheepers proved the following result.

#### Theorem 3.2

(Scheepers). For X a separable metric space, the following are equivalent:

1. Cp(X) satisfies S1(𝓓, 𝓓);

2. X satisfies S1(Ω, Ω).

We claim the theorem for a space B(X) of Borel functions.

#### Theorem 3.3

For a set of reals X, the following are equivalent:

1. B(X) satisfies S1(𝓓, 𝓓);

2. X satisfies S1(BΩ, BΩ).

#### Proof

(1) ⇒ (2). Let X be a set of reals satisfying the hypotheses and β be a countable base of X. Consider a sequence {𝓑i}i∈ℕ of countable Borel ω-covers of X where 𝓑i = $\begin{array}{}\left\{{W}_{i}^{j}{\right\}}_{j\in \mathbb{N}}\end{array}$ for each i ∈ ℕ.

Consider a topology τ generated by the family 𝓟 = { $\begin{array}{}{W}_{i}^{j}\end{array}$A : i, j ∈ ℕ and Aβ} ⋃ {(X$\begin{array}{}{W}_{i}^{j}\end{array}$) ∩ A : i, j ∈ ℕ and Aβ}.

Note that if χP is a characteristic function of P for each P ∈ 𝓟, then a diagonal mapping φ = ΔP∈𝓟χP : X ↦ 2ω is a Borel bijection. Let Z = φ(X).

Note that {𝓑i} is countable open ω-cover of Z for each i ∈ ℕ. Since B(Z) is a dense subset of B(X), then B(Z) also has the property S1(𝓓, 𝓓). Since Cp(Z) is a dense subset of B(Z), Cp(Z) has the property S1(𝓓, 𝓓), too.

By Theorem 3.2, the space Z has the property S1(Ω, Ω). It follows that there is a sequence $\begin{array}{}\left\{{W}_{i}^{j\left(i\right)}{\right\}}_{i\in \mathbb{N}}\end{array}$ such that $\begin{array}{}{W}_{i}^{j\left(i\right)}\end{array}$ ∈ 𝓑i and { $\begin{array}{}{W}_{i}^{j\left(i\right)}\end{array}$ : i ∈ ℕ} is an open ω-cover of Z. It follows that { $\begin{array}{}{W}_{i}^{j\left(i\right)}\end{array}$ : i ∈ ℕ} is Borel ω-cover of X.

(2) ⇒ (1). Assume that X has the property S1(BΩ, BΩ). Let {Dk}k∈ℕ be a sequence countable dense subsets of B(X) and Dk = { $\begin{array}{}{f}_{i}^{k}\end{array}$ : i ∈ ℕ} for each k ∈ ℕ. We claim that for any fB(X) there is a sequence {fk} ⊂ B(X) such that fkDk for each k ∈ ℕ and f ∈ {fk : k ∈ ℕ}. Without loss of generality we can assume f = 0. For each $\begin{array}{}{f}_{i}^{k}\end{array}$Dk let $\begin{array}{}{W}_{i}^{k}=\left\{x\in X:-\frac{1}{k}<{f}_{i}^{k}\left(x\right)<\frac{1}{k}\right\}.\end{array}$

If for each j ∈ ℕ there is k(j) such that $\begin{array}{}{W}_{i\left(j\right)}^{k\left(j\right)}=X,\end{array}$, then a sequence $\begin{array}{}{f}_{k\left(j\right)}={f}_{i\left(j\right)}^{k\left(j\right)}\end{array}$ uniformly converges to f and, hence, f ∈ {fk(j)} : j ∈ ℕ}.

We can assume that $\begin{array}{}{W}_{i}^{k}\end{array}$X for any k, i ∈ ℕ.

1. { $\begin{array}{}{W}_{i}^{k}\end{array}$}i∈ℕ a sequence of Borel sets of X.

2. For each k ∈ ℕ, { $\begin{array}{}{W}_{i}^{k}\end{array}$ : i ∈ ℕ} is a ω-cover of X.

By (2), X has the property S1(BΩ, BΩ), hence, there is a sequence $\begin{array}{}\left\{{W}_{i\left(k\right)}^{k}{\right\}}_{k\in \mathbb{N}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{such that}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{W}_{i\left(k\right)}^{k}\in \left\{{W}_{i}^{k}{\right\}}_{i\in \mathbb{N}}\end{array}$ for each k ∈ ℕ and $\begin{array}{}\left\{{W}_{i\left(k\right)}^{k}{\right\}}_{k\in \mathbb{N}}\end{array}$ is a ω-cover of X.

Consider $\begin{array}{}\left\{{f}_{i\left(k\right)}^{k}\right\}.\end{array}$ We claim that $\begin{array}{}f\in \overline{\left\{{f}_{i\left(k\right)}^{k}:k\in \mathbb{N}\right\}}.\end{array}$ Let K be a finite subset of X, ϵ > 0 and U = 〈f, K, ϵ〉 be a base neighborhood of f, then there is k0 ∈ ℕ such that $\begin{array}{}\frac{1}{{k}_{0}}<ϵ\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}K\subset {W}_{i\left({k}_{0}\right)}^{{k}_{0}}.\end{array}$ It follows that $\begin{array}{}{f}_{i\left({k}_{0}\right)}^{{k}_{0}}\in U.\end{array}$

Let D = {dn : n ∈ ℕ} be a dense subspace of B(X). Given a sequence {Di}i∈ℕ of dense subspace of B(X), enumerate it as {Dn,m : n, m ∈ ℕ}. For each n ∈ ℕ, pick dn,mDn,m so that dn ∈ {dn,m : m ∈ ℕ}. Then {dn,m : m, n ∈ ℕ} is dense in B(X).□

In [16], (Theorem 35) and [4] (Corollary 2.10) proved the following result.

#### Theorem 3.4

(Scheepers). For X a separable metric space, the following are equivalent:

1. Cp(X) satisfies Sfin(𝓓, 𝓓);

2. X satisfies Sfin(Ω, Ω).

Then for the space B(X) we have an analogous result.

#### Theorem 3.5

For a set of reals X, the following are equivalent:

1. B(X) satisfies Sfin(𝓓, 𝓓);

2. X satisfies Sfin(BΩ, BΩ).

#### Proof

It is proved similarly to the proof of Theorem 3.3.□

## 4 Question of A. Bella, M. Bonanzinga and M. Matveev

In [3], Question 4.3, it is asked to find a sequentially separable selectively separable space which is not selective sequentially separable.

The following theorem answers this question.

#### Theorem 4.1

(CH). There is a consistent example of a space Z, such that Z is sequentially separable, selectively separable, not selective sequentially separable.

#### Proof

By Theorem 40 and Corollary 41 in [15], there is a 𝔠-Lusin set X which has the property S1(BΩ, BΩ), but X does not have the property Ufin(Γ, Γ).

Consider a space Z = Cp(X). By Velichko’s Theorem ([18]), a space Cp(X) is sequentially separable for any separable metric space X.

1. Z is sequentially separable. Since X is Lindelöf and X satisfies S1(BΩ, BΩ), X has the property S1(Ω, Ω).

By Theorem 3.2, Cp(X) satisfies S1(𝓓, 𝓓), and, hence, Cp(X) satisfies Sfin(𝓓, 𝓓).

2. Z is selectively separable. By Theorem 4.1 in [11], Ufin(Γ, Γ) = Ufin(ΓF, Γ) for Lindelöf spaces.

Since X does not have the property Ufin(Γ, Γ), X does not have the property Sfin(ΓF, Γ). By Theorem 8.11 in [9], Cp(X) does not have the property Sfin(𝓢, 𝓢).

3. Z is not selective sequentially separable.□

#### Theorem 4.2

(CH). There is a consistent example of a space Z, such that Z is sequentially separable, countably selectively separable, countably selectively separable, not countably selective sequentially separable.

#### Proof

Consider the 𝔠-Lusin set X (see Theorem 40 and Corollary 41 in [15]), then X has the property S1(BΩ, BΩ), but X does not have the property Ufin(Γ, Γ) and, hence, X does not have the property Sfin(BΓ, BΓ).

Consider a space Z = B1(X). By Velichko’s Theorem in [18], a space B1(X) is sequentially separable for any separable metric space X.

1. Z is sequentially separable. By Theorem 3.3, B(X) satisfies S1(𝓓, 𝓓). Since Z is dense subset of B(X) we have that Z satisfies S1(𝓓, 𝓓) and, hence, Z satisfies Sfin(𝓓, 𝓓).

2. Z is countably selectively separable. Since X does not have the property Sfin(BΓ, BΓ), by Theorem 3.1, B1(X) does not have the property Sfin(𝓢, 𝓢).

3. Z is not countably selective sequentially separable.□

## 5 Question of A. Bella and C. Costantini

In [5], Question 2.7, it is asked to find a compact T2 sequentially separable space which is not selective sequentially separable.

The following theorem answers this question.

#### Theorem 5.1

(𝔟 < 𝔮) There is a consistent example of a compact T2 sequentially separable space which is not selective sequentially separable.

#### Proof

Let D be a discrete space of size 𝔟. Since 𝔟 < 𝔮, a space 2𝔟 is sequentially separable (see Proposition 3 in [13]).

We claim that 2𝔟 is not selective sequentially separable.

On the contrary, suppose that 2𝔟 is selective sequentially separable. Since non(Sfin(BΓ, BΓ)) = 𝔟 (see Theorem 1 and Theorem 27 in [15]), there is a set of reals X such that |X| = 𝔟 and X does not have the property Sfin(BΓ, BΓ). Hence there exists sequence (An : n ∈ ℕ) of elements of BΓ that for any sequence (Bn : n ∈ ℕ) of finite sets such that for each n, BnAn, we have that ⋃n∈ℕ BnBΓ.

Consider an identity mapping id : DX from the space D onto the space X. Denote $\begin{array}{}{C}_{n}^{i}\end{array}$ = id–1( $\begin{array}{}{A}_{n}^{i}\end{array}$) for each $\begin{array}{}{A}_{n}^{i}\end{array}$An and n, i ∈ ℕ. Let Cn = { $\begin{array}{}{C}_{n}^{i}\end{array}$}i∈ℕ (i.e. Cn = id–1(An)) and let 𝓢 = {hi}i∈ℕ be a countable sequentially dense subset of B(D, {0, 1}) = 2𝔟.

For each n ∈ ℕ we consider a countable sequentially dense subset 𝓢n of B(D, {0, 1}) where

𝓢n = { $\begin{array}{}{f}_{n}^{i}\end{array}$} := { $\begin{array}{}{f}_{n}^{i}\end{array}$B(D, 2) : $\begin{array}{}{f}_{n}^{i}\end{array}$$\begin{array}{}{C}_{n}^{i}\end{array}$ = hi and $\begin{array}{}{f}_{n}^{i}\end{array}$ ↾ (X$\begin{array}{}{C}_{n}^{i}\end{array}$) = 1 for i ∈ ℕ}.

Since Cn = { $\begin{array}{}{C}_{n}^{i}\end{array}$}i∈ℕ is a Borel γ-cover of D and 𝓢 is a countable sequentially dense subset of B(D, {0, 1}), we have that 𝓢n is a countable sequentially dense subset of B(D, {0, 1}) for each n ∈ ℕ.

Indeed, let hB(D, {0, 1}), there is a sequence {hs}s∈ℕ ⊂ 𝓢 such that {hs}s∈ℕ converges to h. We claim that $\begin{array}{}\left\{{f}_{n}^{s}{\right\}}_{s\in \mathbb{N}}\end{array}$ converges to h. Let K = {x1, …, xk} be a finite subset of D, ϵ = {ϵ1, …, ϵk} where ϵj ∈ {0, 1} for j = 1, …, k, and W = 〈h, K, ϵ〉 := {gB(D, {0, 1}) : |g(xj) – h(xj)| ∈ ϵj for j = 1, …, k} be a base neighborhood of h, then there is a number m0 such that K$\begin{array}{}{C}_{n}^{i}\end{array}$ for i > m0 and hsW for s > m0. Since $\begin{array}{}{f}_{n}^{s}\end{array}$K = hsK for each s > m0, $\begin{array}{}{f}_{n}^{s}\end{array}$W for each s > m0. It follows that a sequence { $\begin{array}{}{f}_{n}^{s}\end{array}$}s∈ℕ converges to h.

Since B(D, {0, 1}) is selective sequentially separable, there is a sequence $\begin{array}{}\left\{{F}_{n}=\left\{{f}_{n}^{{i}_{1}},...,{f}_{n}^{{i}_{s\left(n\right)}}\right\}:n\in \mathbb{N}\right\}\end{array}$ such that for each n, Fn ⊂ 𝓢n, and ⋃n∈ℕ Fn is a countable sequentially dense subset of B(D, {0, 1}).

For 0 ∈ B(D, {0, 1}) there is a sequence $\begin{array}{}\left\{{f}_{{n}_{j}}^{{i}_{j}}{\right\}}_{j\in \mathbb{N}}\subset \bigcup _{n\in \mathbb{N}}{F}_{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{such that}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left\{{f}_{{n}_{j}}^{{i}_{j}}{\right\}}_{j\in \mathbb{N}}\end{array}$ converges to 0. Consider a sequence { $\begin{array}{}{C}_{{n}_{j}}^{{i}_{j}}\end{array}$ : j ∈ ℕ}. Then

1. $\begin{array}{}{C}_{{n}_{j}}^{{i}_{j}}\end{array}$Cnj;

2. {$\begin{array}{}{C}_{{n}_{j}}^{{i}_{j}}\end{array}$ : j ∈ ℕ} is a γ-cover of D.

Indeed, let K be a finite subset of D and U = 〈0, K, {0}〉 be a base neighborhood of 0, then there is a number j0 such that $\begin{array}{}{f}_{{n}_{j}}^{{i}_{j}}\end{array}$U for every j > j0. It follows that K$\begin{array}{}{C}_{{n}_{j}}^{{i}_{j}}\end{array}$ for every j > j0. Hence, { $\begin{array}{}{A}_{{n}_{j}}^{{i}_{j}}\end{array}$ = id( $\begin{array}{}{C}_{{n}_{j}}^{{i}_{j}}\end{array}$) : j ∈ ℕ} ∈ BΓ in the space X, a contradiction.□

Let μ = min{κ : 2κ is not selective sequentially separable}. It is well-known that 𝔭 ≤ μ ≤ 𝔮 (see [3]).

μ = min{𝔟, 𝔮}.

#### Proof

Let κ < min{𝔟, 𝔮}. Then, by Proposition 3 in [13], 2κ is a sequentially separable space.

Let X be a set of reals such that |X| = κ and X be a Q-set.

Analogous to the proof of implication (2) ⇒ (1) in Theorem 3.1, we can claim that B(X, {0, 1}) = 2X = 2κ is selective sequentially separable.

It follows that μ ≥ min{𝔟, 𝔮}.

Since μ ≤ 𝔮, we suppose that μ > 𝔟 and 𝔟 < 𝔮. Then, by Theorem 5.1, 2𝔟 is not selective sequentially separable. It follows that μ = min{𝔟, 𝔮}.□

In [3], Question 4.12 : is it the case μ ∈ {𝔭, 𝔮}?

A partial positive answer to this question is the existence of the following models of set theory (Theorem 8 in [1]):

1. μ = 𝔭 = 𝔟 < 𝔮;

2. 𝔭 < μ = 𝔟 = 𝔮;

and

3. μ = 𝔭 = 𝔮 < 𝔟.

The author does not know whether, in general, the answer can be negative. In this regard, the following question is of interest.

Question. Is there a model of set theory in which 𝔭 < 𝔟 < 𝔮?

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Accepted: 2018-05-24

Published Online: 2018-07-17

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 740–746, ISSN (Online) 2391-5455,

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