On the Menger covering property and $D$-spaces

The main results of this note are: It is consistent that every subparacompact space $X$ of size $\omega_1$ is a $D$-space; If there exists a Michael space, then all productively Lindel\"of spaces have the Menger property, and, therefore, are $D$-spaces; and Every locally $D$-space which admits a $\sigma$-locally finite cover by Lindel\"of spaces is a $D$-space.


Introduction
A neighbourhood assignment for a topological space X is a function N from X to the topology of X such that x ∈ N(x) for all x. A topological space X is said to be a D-space [6], if for every neighbourhood assignment N for X there exists a closed and discrete subset A ⊂ X such that N(A) = x∈A N(x) = X.
It is unknown whether paracompact (even Lindelöf) spaces are D-spaces. Our first result in this note answers [7,Problem 3.8] in the affirmative and may be thought of as a very partial solution to this problem 1 .
Our second result shows that the affirmative answer to [19,Problem 2.6], which asks whether all productively Lindelöf spaces are D-spaces, is consistent. It is worth mentioning that our premises (i.e., the existence of a Michael space) are not known to be inconsistent.
Our third result is a common generalization of two theorems from [10]. Most of our proofs use either the recent important result of Aurichi [2] asserting that every topological space with the Menger property is a Dspace, or the ideas from its proof. We consider only regular topological spaces. For the definitions of small cardinals d and cov(M) used in this paper we refer the reader to [22].
The first author was supported by SRA grants P1-0292-0101 and J1-2057-0101. The second author acknowledges the support of FWF grant P19898-N18. We would also like to thank Leandro Aurichi, Franklin Tall, and Hang Zhang for kindly making their recent papers available to us. 1 While completing this manuscript we have learned that this result has been independently obtained by Hang Zhang and Wei-Xue Shi, see [15].

Subparacompact spaces of size ω 1
Following [4] we say that a topological space X has the property E * ω if for every sequence u n : n ∈ ω of countable open covers of X there exists a sequence v n : n ∈ ω such that v n ∈ [u n ] <ω and n∈ω ∪v n = X. In the realm of Lindelöf spaces the property E * ω is usually called the Menger property or fin (O, O), see [21] and references therein.
We say that a topological space X has property D ω , if for every neighbourhood assignment N there exists a countable collection {A n : n ∈ ω} of closed discrete subsets of X such that X = n∈ω N(A n ). Observe that the property D ω is inherited by all closed subsets.
The following theorem is the main result of this section.
Theorem 2.1. Suppose that a topological space X has properties D ω and E * ω . Then X is a D-space. The proof of Theorem 2.1 is analogous to the proof of [2, Proposition 2.6]. In particular, it uses the following game of length ω on a topological space X: On the nth move player I chooses a countable open cover u n = {U n,k : k ∈ ω} such that U n,k ⊂ U n,k+1 for all k ∈ ω, and player II responds by choosing a natural number k n . Player II wins the game if n∈ω U n,kn = X. Otherwise, player I wins. We shall call this game an E * ω -game. In the realm of Lindelöf spaces this game is known under the name Menger game. It is well-known that a Lindelöf space X has the property E * ω if and only if the first player has no winning strategy in the E * ω -game on X, see [8,14]. The proof of [14,Theorem 13] also works without any change for non-Lindelöf spaces.
We are in a position now to present the proof of Theorem 2.1.
Proof. We shall define a strategy Υ : Suppose that for some m ∈ ω and all s ∈ ω ≤m we have already defined a closed subset F s of X, an increasing sequence A s,k : k ∈ ω of closed discrete subsets of F s , and a countable open cover Υ(s) = u s of X such that Fix s ∈ ω m+1 . Since X has the property D ω , so does its closed subspace F s := X \ i<m+1 N(A s↾i,s(i) ), and hence there exists an increasing sequence A s,k : k ∈ ω} of closed discrete subsets of F s such that F s ⊂ k∈ω N(A s,k ).
This completes the definition of Υ.
Since X has the property E * ω , Υ is not winning. Thus there exists z ∈ ω ω such that X = n∈ω (X \ F z↾n ) ∪N(A z↾n,z(n) ). By the inductive construction, X \ F ∅ = ∅ and X \ F z↾n = i<n N(A z↾i,z(i) ) for all n > 0. It follows from above that X = n∈ω N(A z↾n,z(n) ). In addition, A z↾n,z(n) ⊂ F z↾n = X \ i<n N(A z↾i,z(i) ) for all n > 0, which implies that A := n∈ω A z↾n,z(n) is a closed discrete subset of X. It suffices to note that N(A) = X.
We recall from [5] that a topological space X is called subparacompact, if every open cover of X has a σ-locally finite closed refinement.
Suppose that X is a subparacompact topological space which can be covered by ω 1 -many of its Lindelöf subspaces. Then X has the property D ω . 2 In particular, every subparacompact space of size ω 1 has the property D ω .
Proof. Let L = {L ξ : ξ < ω 1 } be an increasing cover of X by Lindelöf subspaces, τ be the topology of X, and N : X → τ be a neighbourhood assignment. Construct by induction a sequence C α : α < ω 1 of (possibly empty) countable subsets of X such that The subparacompactness of X yields a closed cover F = n∈ω F n of X which refines U = {N(x) : x ∈ C} and such that each F n is locally-finite. Since every element of U contains at most countably many elements of C, so do elements of F . Therefore for every F ∈ F n such that C ∩ F = ∅ we can write this intersection in the form {x n,F,m : m ∈ ω}. Now it is easy to see that A n,m := {x n,F,m : F ∈ F n , C ∩ F = ∅} is a closed discrete subset of X and n,m∈ω A n,m = C.
Remark 2.4. What we have actually used in the proof of Lemma 2.3 is the following weakening of subparacompactness: every open cover U which is closed under unions of its countable subsets admits a σ-locally finite closed refinement. We do not know whether this property is strictly weaker than subparacompactness.
Corollary 2.5. Let X be a countably tight subparacompact topological space of density ω 1 . Then X has the property D ω .
It is well-known [9, Theorem 4.4] (and it easily follows from corresponding definitions) that any Lindelöf space of size < d has the Menger property. The same argument shows that every topological space of size < d has the property E * ω . Combining this with Theorem 2.1 and Lemma 2.3 we get the following corollary, which implies the first of the results mentioned in our abstract.
Corollary 2.6. Suppose that X is a subparacompact topological space of size |X| < d which can be covered by ω 1 -many of its Lindelöf subspaces. Then X is a D-space.

Concerning the existence of a Michael space
A topological space X is said to be productively Lindelöf, if X × Y is Lindelöf for all Lindelöf spaces Y . It was asked in [19] whether productively Lindelöf spaces are D-spaces. The positive answer to the above question has been proved consistent and in a stream of recent papers (see the list of references in [19]) several sufficient set-theoretical conditions were established. The following statement gives a uniform proof for some of these results. In particular, it implies [16,Theorems 5 and 7], [1,Corollary 4.5], and answers [17,Question 15] in the affirmative.
A Lindelöf space Y is called a Michael space, if ω ω × Y is not Lindelöf. We refer the reader to [11] where the existence of a Michael space was reformulated in a combinatorial language and a number of set-theoretic conditions guaranteeing the existence of Michael spaces were established.
In the proof of Proposition 3.1 we shall use set-valued maps, see [13]. By a set-valued map Φ from a set X into a set Y we understand a map from X into P(Y ) and write Φ : X ⇒ Y (here P(Y ) denotes the set of all subsets of Y ). For a subset A of X we set Φ(A) = x∈A Φ(x) ⊂ Y . A set-valued map Φ from a topological spaces X to a topological space Y is said to be The proof of the following claim is straightforward.
(1) Suppose that X, Y are topological spaces, X is Lindelöf, and Φ : X ⇒ Y is a compact-valued upper semicontinuous map such that Y = Φ(X). Then Y is Lindelöf.
Proof of Proposition 3.1. Suppose, contrary to our claim, that X is a productively Lindelöf space which does not have the Menger property and Y is a Michael space. It suffices to show that X × Y is not Lindelöf. Indeed, by [23,Theorem 8] there exists a compact-valued upper semicontinuous map Φ : X → ω ω such that Φ(X) = ω ω . By Claim 3.2(2) the product ω ω × Y is the image of X × Y under a compact-valued upper semicontinuous map. By the definition of a Michael space, ω ω × Y is not Lindelöf. By applying Claim 3.2(1) we can conclude that X × Y is not Lindelöf neither. 2 By a result of Tall [16] the existence of a Michael space implies that all productively Lindelöf analytic metrizable spaces are σ-compact. Combining recent results obtained in [1] and [12] we can consistently extend this result to all Σ 1 2 definable subsets of 2 ω . Theorem 3.3. Suppose that cov(M) > ω 1 and there exists a Michael space. Then every productively Lindelöf Σ 1 2 definable subset of 2 ω is σ-compact. Proof. Let X be a productively Lindelöf Σ 1 2 definable subset of 2 ω . If X cannot be written as a union of ω 1 -many of its compact subspaces, then it contains a closed copy of ω ω [12], and hence the existence of the Michael space implies that X is not productively Lindelöf, a contradiction.
Thus X can be written as a union of ω 1 -many of its compact subspaces, and therefore it is σ-compact by [1,Corollary 4.15].
We do not know whether the assumption cov(M) > ω 1 can be dropped from Theorem 3.3.
Question 3.4. Suppose that there exists a Michael space. Is every coanalytic productively Lindelöf space σ-compact?
By [18,Proposition 31] the affirmative answer to the question above follows from the Axiom of Projective Determinacy.

Locally finite unions
Theorem 4.1. Suppose that X is a locally D-space which admits a σ-locally finite cover by Lindelöf spaces. Then X is a D-space.
Proof. Let F = n∈ω F n be a cover of X by Lindelöf subspaces such that F n is locally finite. Fix F ∈ F n . For every x ∈ F there exists an open neighbourhood U x of x such thatŪ x is a D-space. Let C F be a countable subset of F such that F ⊂ x∈C F U x . Then Z F = {F ∩ U x : x ∈ C F } is a countable cover of F consisting of closed D-subspaces of X such that F ∩ Z is dense in Z for all Z ∈ Z F . It follows from the above that X admits a σ-locally finite cover consisting of closed D-subspaces. Since a union of a locally finite family of closed D-subspaces is easily seen to be a closed Dsubspace, X is a union of an increasing sequence of its closed D-subspaces. Therefore it is a D-space by results of [3].
Corollary 4.2. If a topological space X admits a σ-locally finite locally countable cover by topological spaces with the Menger property, then it is a D-space.
In particular, a locally Lindelöf space admitting a σ-locally finite cover by topological spaces with the Menger property is a D-space.
Proof. The second part is a direct consequence of the first one since every σ-locally countable family of subspaces of a locally Lindelöf space is locally countable.
To prove the first assertion, note that by local countability every point x ∈ X has a closed neighbourhood which is a countable union of its subspaces with the Menger property, and hence it has the Menger property itself. Therefore X is a locally D-space. It now suffices to apply Theorem 4.1.
It is known that every Lindelöf C-scattered space is C-like, and that Clike spaces have the Menger property, see [20, p. 247