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

Alexander V. Osipov

doi:10.1515/math-2018-0070

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

Alexander V. Osipov

Published/Copyright: July 17, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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(B_Γ, B_Γ);
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 : 2^κ is not selective sequentially separable}. This is a partial answer to the question of A. Bella, M. Bonanzinga and M. Matveev.

Keywords: S1(𝓓, 𝓓); S1(𝓢,𝓢); Sfin(𝓢, 𝓢); Function spaces; Selection principles; Borel function; σ-set; S1(BΩ, BΩ); S1(BΓ, BΓ); S1(BΩ, BΓ); Sequentially separable; Selectively separable; Selective sequentially separable; Countably selective sequentially separable

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 (A_n : n ∈ ℕ) of elements of 𝓐 there is a sequence (b_n : n ∈ ℕ) such that for each n, b_n ∈ A_n, and {b_n : n ∈ ℕ} is an element of 𝓑.

S_fin(𝓐, 𝓑) is the selection hypothesis: for each sequence (A_n : n ∈ ℕ) of elements of 𝓐 there is a sequence (B_n: n ∈ ℕ) of finite sets such that for each n, B_n ⊆ A_n, and ⋃_n∈ℕB_n ∈ 𝓑.

U_fin(𝓐, 𝓑) is the selection hypothesis: whenever 𝓤₁, 𝓤₂, … ∈ 𝓐 and none contains a finite subcover, there are finite sets 𝓕_n ⊆ 𝓤_n, n ∈ ℕ, such that {⋃ 𝓕_n : 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 x ∈ X 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;
B_Ω — the family of all countable Borel ω-covers of X;
B_Γ — the family of all countable Borel γ-covers of X;
F_Γ — 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 γ_F-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 Γ_F, the family of all countable γ_F-shrinkable γ-covers of X.

We will use the following notations.

C_p(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 A ⊆ X, then the sequential closure of A, denoted by [A]_seq, is the set of all limits of sequences from A. A set D ⊆ X 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 Xstrongly 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 (D_n : n ∈ ℕ) of (countable) dense subsets of X one can pick finite F_n ⊂ D_n, n ∈ ℕ, so that ⋃ {F_n : 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 (D_n : n ∈ ℕ) of (countable) sequentially dense subsets of X, one can pick finite F_n ⊂ D_n, n ∈ ℕ, so that ⋃{F_n : n ∈ ℕ} is sequentially dense in X.

In Scheeper’s terminology [16], countably selectively separability equivalently to the selection principle S_fin(𝓓, 𝓓), and countably selective sequentially separability equivalently to the S_fin(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^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^ℕ. A set B ⊂ [ℕ]^∞ is unbounded if the set of all increasing enumerations of elements of B is unbounded in ℕ^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 G_δ. 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 realsX, the following statements are equivalent:

B(X) satisfiesS₁(𝓢, 𝓢) andB(X) is sequentially separable;
XsatisfiesS₁(B_Γ, B_Γ);
B(X) ∈ S_fin(𝓢, 𝓢) andB(X) is sequentially separable;
XsatisfiesS_fin(B_Γ, B_Γ);
B₁(X) satisfiesS₁(𝓢, 𝓢);
XsatisfiesS₁(F_Γ, F_Γ);
B₁(X) satisfiesS_fin(𝓢, 𝓢).

Proof

It is obvious that (1) ⇒ (3).

(2) ⇔ (4). By Theorem 1 in [15], U_fin(B_Γ, B_Γ) = S₁(B_Γ, B_Γ) = S_fin(B_Γ, B_Γ).

(3) ⇒ (2). Let {𝓕_i} ⊂ B_Γ and 𝓢 = {h_m}_m∈ℕ be a countable sequentially dense subset of B(X). For each i ∈ ℕ we consider a countable sequentially dense subset 𝓢_i of B(X) and 𝓕_i = {Fim}m∈N where

Si={fim}:={fim∈B(X):fim↾Fim=hmandfim↾(X∖Fim)=1form∈N}.

Since 𝓕_i = {Fim}m∈N is a Borel γ-cover of X and 𝓢 is a countable sequentially dense subset of B(X), we have that 𝓢_i is a countable sequentially dense subset of B(X) for each i ∈ ℕ. Indeed, let h ∈ B(X), there is a sequence {h_s}_s∈ℕ ⊂ 𝓢 such that {h_s}_s∈ℕ converges to h. We claim that {fis}s∈N converges to h. Let K = {x₁, …, x_k} be a finite subset of X, ϵ > 0 and let W = 〈h, K, ϵ〉 := {g ∈ B(X) : |g(x_j) – h(x_j)| < ϵ for j = 1, …, k} be a base neighborhood of h, then there is m₀ ∈ ℕ such that K ⊂ Fim for each m > m₀ and h_s ∈ W for each s > m₀. Since fis ↾ K = h_s ↾ K for every s > m₀, fis ∈ W for every s > m₀. It follows that { fis}_s∈ℕ converges to h.

Since B(X) satisfies S_fin(𝓢, 𝓢), there is a sequence (Fi={fim1,...,fims(i)}:i∈N) such that for each i, F_i ⊂ 𝓢_i, and ⋃_i∈ℕF_i is a countable sequentially dense subset of B(X).

For 0 ∈ B(X) there is a sequence {fijms(ij)}j∈N⊂⋃i∈NFisuch that{fijms(ij)}j∈N converges to 0. Consider a sequence ( Fijms(ij) : j ∈ ℕ). Then

Fijms(ij) ∈ 𝓕_{i_j};
{Fijms(ij) : j ∈ ℕ} is a γ-cover of X.

Indeed, let K be a finite subset of X and U = 〈0, K, 12〉 be a base neighborhood of 0, then there is j₀ ∈ ℕ such that fijms(ij) ∈ U for every j > j₀. It follows that K ⊂ Fijms(ij) for every j > j₀. We thus get that X satisfies U_fin(B_Γ, B_Γ), and, hence, by Theorem 1 in [15], X satisfies S₁(B_Γ, B_Γ).

(2) ⇒ (1). Let {S_i} ⊂ 𝓢 and S = {d_n : n ∈ ℕ} ∈ 𝓢. Consider the topology τ generated by the family 𝓟 = {f^–1(G) : G is an open set of ℝ and f ∈ S∪⋃i∈NSi}. Since P = S∪⋃i∈NSi 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 f ∈ P, considered as mapping from Y to ℝ, is a continuous function i.e. f ∈ C(Y) for each f ∈ P. 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(B_Γ, B_Γ). By Corollary 21 in [17], B(Y) is an α₂ space.

Let q : ℕ ↦ ℕ × ℕ be a bijection. Then we enumerate {S_i}_i∈ℕ as {S_q(i)}_{q(i)∈ℕ×ℕ}. For each d_n ∈ S there are sequences s_n,m ⊂ S_n,m such that s_n,m converges to d_n for each m ∈ ℕ. Since B(Y) is an α₂ space, there is {b_n,m : m ∈ ℕ} such that for each m, b_n,m ∈ s_n,m, and, b_n,m → d_n (m → ∞). Let B = {b_n,m : n, m ∈ ℕ}. Note that S ⊂ [B]_seq.

Since X is a σ-set (that is, each Borel subset of X is F_σ)(see [17]), B₁(X) = B(X) and φ(B(Y)) = φ(B₁(Y)) ⊆ B(X) where φ(B(Y)) := {p ∘ φ : p ∈ B(Y)} and φ(B₁(Y)) := {p ∘ φ : p ∈ B₁(Y)}.

Since S is a countable, sequentially dense subset of B(X), for any g ∈ B(X) there is a sequence {g_n}_n∈ℕ ⊂ S such that {g_n}_n∈ℕ converges to g. But g we can consider as a mapping from Y into ℝ and a set {g_n : n ∈ ℕ} as subset of C(Y). It follows that g ∈ B₁(Y). We get that φ(B(Y)) = B(X).

We claim that B ∈ 𝓢, i.e. that [B]_seq = B(X). Let f ∈ B(Y) and {f_k : k ∈ ℕ} ⊂ S such that f_k → f (k → ∞). For each k ∈ ℕ there is { fkn : n ∈ ℕ} ⊂ B such that fkn → f_k (n → ∞). Since Y is a QN-space (Theorem 16 in [6]), there exists an unbounded β ∈ ℕ^ℕ such that { fkβ(k)} converges to f on Y. It follows that { fkβ(k) : 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 {𝓕_i} ⊂ F_Γ and 𝓢 = {h_m}_m∈ℕ be a countable sequentially dense subset of B₁(X).

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

(6) ⇒ (5). By Corollary 20 in [17], X satisfies S₁(B_Γ, B_Γ). 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). ForXa separable metric space, the following are equivalent:

C_p(X) satisfiesS₁(𝓓, 𝓓);
XsatisfiesS₁(Ω, Ω).

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

Theorem 3.3

For a set of realsX, the following are equivalent:

B(X) satisfiesS₁(𝓓, 𝓓);
XsatisfiesS₁(B_Ω, B_Ω).

Proof

(1) ⇒ (2). Let X be a set of reals satisfying the hypotheses and β be a countable base of X. Consider a sequence {𝓑_i}_i∈ℕ of countable Borel ω-covers of X where 𝓑_i = {Wij}j∈N for each i ∈ ℕ.

Consider a topology τ generated by the family 𝓟 = { Wij ∩ A : i, j ∈ ℕ and A ∈ β} ⋃ {(X ∖ Wij) ∩ A : i, j ∈ ℕ and A ∈ β}.

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

Note that {𝓑_i} 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 C_p(Z) is a dense subset of B(Z), C_p(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 {Wij(i)}i∈N such that Wij(i) ∈ 𝓑_i and { Wij(i) : i ∈ ℕ} is an open ω-cover of Z. It follows that { Wij(i) : i ∈ ℕ} is Borel ω-cover of X.

(2) ⇒ (1). Assume that X has the property S₁(B_Ω, B_Ω). Let {D_k}_k∈ℕ be a sequence countable dense subsets of B(X) and D_k = { fik : i ∈ ℕ} for each k ∈ ℕ. We claim that for any f ∈ B(X) there is a sequence {f_k} ⊂ B(X) such that f_k ∈ D_k for each k ∈ ℕ and f ∈ {f_k : k ∈ ℕ}. Without loss of generality we can assume f = 0. For each fik ∈ D_k let Wik={x∈X:−1k<fik(x)<1k}.

If for each j ∈ ℕ there is k(j) such that Wi(j)k(j)=X,, then a sequence fk(j)=fi(j)k(j) uniformly converges to f and, hence, f ∈ {f_k(j)} : j ∈ ℕ}.

We can assume that Wik ≠ X for any k, i ∈ ℕ.

{ Wik}_i∈ℕ a sequence of Borel sets of X.
For each k ∈ ℕ, { Wik : i ∈ ℕ} is a ω-cover of X.

By (2), X has the property S₁(B_Ω, B_Ω), hence, there is a sequence {Wi(k)k}k∈Nsuch thatWi(k)k∈{Wik}i∈N for each k ∈ ℕ and {Wi(k)k}k∈N is a ω-cover of X.

Consider {fi(k)k}. We claim that f∈{fi(k)k:k∈N}¯. 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 1k0<ϵandK⊂Wi(k0)k0. It follows that fi(k0)k0∈U.

Let D = {d_n : n ∈ ℕ} be a dense subspace of B(X). Given a sequence {D_i}_i∈ℕ of dense subspace of B(X), enumerate it as {D_n,m : n, m ∈ ℕ}. For each n ∈ ℕ, pick d_n,m ∈ D_n,m so that d_n ∈ {d_n,m : m ∈ ℕ}. Then {d_n,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). ForXa separable metric space, the following are equivalent:

C_p(X) satisfiesS_fin(𝓓, 𝓓);
XsatisfiesS_fin(Ω, Ω).

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

Theorem 3.5

For a set of realsX, the following are equivalent:

B(X) satisfiesS_fin(𝓓, 𝓓);
XsatisfiesS_fin(B_Ω, B_Ω).

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 spaceZ, such thatZis 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(B_Ω, B_Ω), but X does not have the property U_fin(Γ, Γ).

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

Z is sequentially separable. Since X is Lindelöf and X satisfies S₁(B_Ω, B_Ω), X has the property S₁(Ω, Ω).
By Theorem 3.2, C_p(X) satisfies S₁(𝓓, 𝓓), and, hence, C_p(X) satisfies S_fin(𝓓, 𝓓).
Z is selectively separable. By Theorem 4.1 in [11], U_fin(Γ, Γ) = U_fin(Γ_F, Γ) for Lindelöf spaces.
Since X does not have the property U_fin(Γ, Γ), X does not have the property S_fin(Γ_F, Γ). By Theorem 8.11 in [9], C_p(X) does not have the property S_fin(𝓢, 𝓢).
Z is not selective sequentially separable.□

Theorem 4.2

(CH). There is a consistent example of a spaceZ, such thatZis 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(B_Ω, B_Ω), but X does not have the property U_fin(Γ, Γ) and, hence, X does not have the property S_fin(B_Γ, B_Γ).

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.

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 S_fin(𝓓, 𝓓).
Z is countably selectively separable. Since X does not have the property S_fin(B_Γ, B_Γ), by Theorem 3.1, B₁(X) does not have the property S_fin(𝓢, 𝓢).
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 compactT₂sequentially separable space which is not selective sequentially separable.

Proof

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

We claim that 2^𝔟 is not selective sequentially separable.

On the contrary, suppose that 2^𝔟 is selective sequentially separable. Since non(S_fin(B_Γ, B_Γ)) = 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 S_fin(B_Γ, B_Γ). Hence there exists sequence (A_n : n ∈ ℕ) of elements of B_Γ that for any sequence (B_n : n ∈ ℕ) of finite sets such that for each n, B_n ⊆ A_n, we have that ⋃_n∈ℕB_n ∉ B_Γ.

Consider an identity mapping id : D ↦ X from the space D onto the space X. Denote Cni = id^–1( Ani) for each Ani ∈ A_n and n, i ∈ ℕ. Let C_n = { Cni}_i∈ℕ (i.e. C_n = id^–1(A_n)) and let 𝓢 = {h_i}_i∈ℕ be a countable sequentially dense subset of B(D, {0, 1}) = 2^𝔟.

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

𝓢_n = { fni} := { fni ∈ B(D, 2) : fni ↾ Cni = h_i and fni ↾ (X ∖ Cni) = 1 for i ∈ ℕ}.

Since C_n = { Cni}_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 𝓢_n is a countable sequentially dense subset of B(D, {0, 1}) for each n ∈ ℕ.

Indeed, let h ∈ B(D, {0, 1}), there is a sequence {h_s}_s∈ℕ ⊂ 𝓢 such that {h_s}_s∈ℕ converges to h. We claim that {fns}s∈N converges to h. Let K = {x₁, …, x_k} be a finite subset of D, ϵ = {ϵ₁, …, ϵ_k} where ϵ_j ∈ {0, 1} for j = 1, …, k, and W = 〈h, K, ϵ〉 := {g ∈ B(D, {0, 1}) : |g(x_j) – h(x_j)| ∈ ϵ_j for j = 1, …, k} be a base neighborhood of h, then there is a number m₀ such that K ⊂ Cni for i > m₀ and h_s ∈ W for s > m₀. Since fns ↾ K = h_s ↾ K for each s > m₀, fns ∈ W for each s > m₀. It follows that a sequence { fns}_s∈ℕ converges to h.

Since B(D, {0, 1}) is selective sequentially separable, there is a sequence {Fn={fni1,...,fnis(n)}:n∈N} such that for each n, F_n ⊂ 𝓢_n, and ⋃_n∈ℕF_n is a countable sequentially dense subset of B(D, {0, 1}).

For 0 ∈ B(D, {0, 1}) there is a sequence {fnjij}j∈N⊂⋃n∈NFnsuch that{fnjij}j∈N converges to 0. Consider a sequence { Cnjij : j ∈ ℕ}. Then

Cnjij ∈ C_{n_j};
{Cnjij : 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 fnjij ∈ U for every j > j₀. It follows that K ⊂ Cnjij for every j > j₀. Hence, { Anjij = id( Cnjij) : j ∈ ℕ} ∈ B_Γ in the space X, a contradiction.□

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

Theorem 5.2

μ = min{𝔟, 𝔮}.

Proof

Let κ < min{𝔟, 𝔮}. Then, by Proposition 3 in [13], 2^κ 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}) = 2^X = 2^κ 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, 2^𝔟 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]):

μ = 𝔭 = 𝔟 < 𝔮;
𝔭 < μ = 𝔟 = 𝔮;
and
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?

References

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Received: 2017-06-18

Accepted: 2018-05-24

Published Online: 2018-07-17

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0070

Keywords for this article

S1(𝓓, 𝓓); S1(𝓢,𝓢); Sfin(𝓢, 𝓢); Function spaces; Selection principles; Borel function; σ-set; S1(BΩ, BΩ); S1(BΓ, BΓ); S1(BΩ, BΓ); Sequentially separable; Selectively separable; Selective sequentially separable; Countably selective sequentially separable

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