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

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 separability.


Introduction
In [12], Osipov and Pytkeev gave necessary and su cient conditions for the space B (X) of the Baire class 1 functions on a Tychono 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.

Main de nitions and notation
Many topological properties are de ned or characterized in terms of the following classical selection principles. Let A and B be sets consisting of families of subsets of an in nite set X. Then: S (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 n (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 nite sets such that for each n, B n ⊆ A n , and ⋃ n∈N B n ∈ B.
U n (A, B) is the selection hypothesis: whenever U , U , ... ∈ A and none contains a nite subcover, there are nite 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 nite subset of X is contained in a member of U; • a γ-cover if it is in nite and each x ∈ X belongs to all but nitely 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.
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) de ned on a space X, with pointwise topology.
• B (X) is the set of all rst Baire class functions B (X) i.e., pointwise limits of continuous functions, de ned on a space X, with pointwise topology.
• B(X) is the set of all Borel functions, de ned 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 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 (D n ∶ n ∈ N) of (countable) dense subsets of X one can pick nite F n ⊂ D n , 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 (D n ∶ n ∈ N) of (countable) sequentially dense subsets of X, one can pick nite F n ⊂ D n , 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 S n (D, D), and countably selective sequentially separability equivalently to the S n (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 nite intersection property but no in nite pseudo-intersection. Note that ω ≤ p ≤ c.
For f , g ∈ N N , let f ≤ * g if f (n) ≤ g(n) for all but nitely 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).

Properties of a space of Borel functions
Theorem 3.1. For a set of reals X, the following statements are equivalent: 1
( ) ⇒ ( ). Let {S i } ⊂ S and S = {d n ∶ n ∈ N} ∈ S. Consider the topology τ generated by the family and X is Tychono , 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 S (B Γ , B Γ ). By Corollary 21 in [17], Since X is a σ-set (that is, each Borel subset of X is F σ )(see [17]), B (X) = B(X) and 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 ∈ B (Y). We get that ϕ(B(Y)) = B(X).
We claim the theorem for a space B(X) of Borel functions.

Consider a topology τ generated by the family P = {W
Note that if χ P is a characteristic function of P for each P ∈ P, then a diagonal mapping ϕ = ∆ P∈P χ P ∶ X ↦ 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 S (D, D). Since C p (Z) is a dense subset of B(Z), C p (Z) has the property S (D, D), too.
By Theorem 3.2, the space Z has the property S (Ω, Ω). It follows that there is a sequence {W Assume that X has the property S (B Ω , B Ω ). Let {D k } k∈N be a sequence countable dense subsets of B(X) and D k = {f k i ∶ 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 We can assume that W k i ≠ X for any k, i ∈ N. (a). {W k i } i∈N a sequence of Borel sets of X.
We claim that f ∈ {f k i(k) ∶ k ∈ N}. Let K be a nite subset of X, > and U = ⟨f , K, ⟩ be a base neighborhood of f , then there is k ∈ N such that k < and K ⊂ W k i (k ) . It follows that f k i(k ) ∈ 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).
Then for the space B(X) we have an analogous result. Proof. It is proved similarly to the proof of Theorem 3.3.

Question of A. Bella, M. Bonanzinga and M. Matveev
In [3], Question 4.3, it is asked to nd 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 S (B Ω , B Ω ), but X does not have the property U n (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.
(a). Z is sequentially separable. Since X is Lindelöf and X satis es S (B Ω , B Ω ), X has the property S (Ω, Ω).
By Theorem 3.2, C p (X) satis es S (D, D), and, hence, C p (X) satis es S n (D, D). (b). Z is selectively separable. By Theorem 4.1 in [11], U n (Γ , Γ ) = U n (Γ F , Γ ) for Lindelöf spaces. Since X does not have the property U n (Γ , Γ ), X does not have the property S n (Γ F , Γ ). By Theorem 8.11 in [9], C p (X) does not have the property S n (S , S).
(c). 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 S (B Ω , B Ω ), but X does not have the property U n (Γ , Γ ) and, hence, X does not have the property S n (B Γ , B Γ ). Consider a space Z = B (X). By Velichko's Theorem in [18], a space B (X) is sequentially separable for any separable metric space X.
(a). Z is sequentially separable. By Theorem 3.3, B(X) satis es S (D, D). Since Z is dense subset of B(X) we have that Z satis es S (D, D) and, hence, Z satis es S n (D, D).
(b). Z is countably selectively separable. Since X does not have the property S n (B Γ , B Γ ), by Theorem 3.1, B (X) does not have the property S n (S , S).
(c). Z is not countably selective sequentially separable.

Question of A. Bella and C. Costantini
In [5], Question 2.7, it is asked to nd a compact T sequentially separable space which is not selective sequentially separable.
The following theorem answers this question.
There is a consistent example of a compact T sequentially separable space which is not selective sequentially separable.
Proof. Let D be a discrete space of size b. Since b < q, a space b is sequentially separable (see Proposition 3 in [13]). We claim that b is not selective sequentially separable.
On the contrary, suppose that b is selective sequentially separable. Since non(S n (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 n (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 nite sets such that for each n, B n ⊆ A n , we have that ⋃ n∈N B n ∉ B Γ .
Consider an identity mapping id ∶ D ↦ X from the space D onto the space X.
For each n ∈ N we consider a countable sequentially dense subset S n of B(D, { , }) where (2) {C i j n j ∶ j ∈ N} is a γ-cover of D. Indeed, let K be a nite subset of D and U = ⟨ , K, { }⟩ be a base neighborhood of , then there is a number j such that f i j n j ∈ U for every j > j . It follows that K ⊂ C i j n j for every j > j . Hence, {A i j n j = id(C i j n j ) ∶ j ∈ N} ∈ B Γ in the space X, a contradiction.
Let X be a set of reals such that X = κ and X be a Q-set.
Since µ ≤ q, we suppose that µ > b and b < q. Then, by Theorem 5.1, b is not selective sequentially separable. It follows that µ = min{b, q}.