Guessing clubs for aD, non D-spaces

The notion of a D-space was probably first introduced by van Douwen and since than, many work had been done in this topic. Investigating the properties of Dspaces and the connections between other covering properties led to the definition of aD-spaces, defined by Arhangel'skii in [2]. As it turned out, property aD is much more docile then property D. In [3] Arhangel'skii asked the following: Problem 4.6. Is there a Tychonoff aD-space which is not a D-space?

A negative answer to this question would settle almost all of the questions about the relationship of classical covering properties to property D. Quite similarly, Guo and Junnila in [6] asked the following about a weakening of property D: Problem 2.12. Is every aD-space linearly D?

In G. Gruenhage’s survey on D-spaces [5], another version of this question is stated (besides the original Arhangel’skii), namely: Question 3.6(2) Is every scattered, aD-space a D-space?

The main result of this paper is the following answer to the questions above.

Theorem 1.1. There exists a 0-dimensional T 2 space X such that X is scattered, aD and non linearly D.

In [9] the author showed that the existence of a locally countable, locally compact space X of size ω 1 which is aD and non linearly D is independent of ZFC. Here we refine those methods and using Shelah’s club guessing theory we answer the above questions in ZFC.

The paper has the following structure. Sections 2, 3 and 4 gather all the necessary facts about D-spaces, MAD families and club guessing. In Section 5 we define spaces X[λ, µ, M, C], where λ and µ = cf (µ) are cardinals, M is a MAD family on µ and C is a guessing sequence. It is shown in Claim 5.2 that (0) X[λ, µ, M, C] is always T 2 , 0-dimensional and scattered. Section 6 contains two important results:

is aD under certain assumptions (see Corollary 6.9). Finally in Section 7 we show how to produce such spaces X[λ, µ, M, C] depending on the cardinal arithmetic and using Shelah’s club guessing.

The author would like to thank Assaf Rinot for his ideas and advices to look deeper into the theory of club guessing in ZFC.

An open neighborhood assignment (ONA, in short) on a space (X, τ ) is a map U : X → τ such that x ∈ U (x) for every x ∈ X. X is said to be a D-space if for every neighborhood assignment U , one can find a closed discrete D ⊆ X such that X = d∈D U (d) = U [D] (such a set D is called a kernel for U ). In [2] the authors introduced property aD :

It is clear that D-spaces are aD. Proving that a space is aD, the notion of an irreducible space will play a key role. A space X is irreducible iff every open cover U has a minimal open refinement U 0 ; meaning that no proper subfamily of U 0 covers X. In [3] Arhangel’skii showed the following equivalence.

Theorem 2.2 ([3, Theorem 1.8]). A T 1 -space X is an aD-space if and only if every closed subspace of X is irreducible.

Another generalization of property D is due to Guo and Junnila [6]. For a space X a cover U is monotone iff it is linearly ordered by inclusion. Definition 2.3. A space (X, τ ) is said to be linearly D iff for any ONA U : X → τ for which {U (x) : x ∈ X} is monotone, one can find a closed discrete set D ⊆ X such that X = U [D].

We will use the following characterization of linear D property. A set D ⊆ X is said to be U-big for a cover U iff there is no U ∈ U such that D ⊆ U . Theorem 2.4 ([6, Theorem 2.2]). The following are equivalent for a T 1 -space X:

(1) X is linearly D.

(2) For every non-trivial monotone open cover U of X, there exists a closed discrete U-big set in X.

As MAD families will play an essential part in our constructions we observe some easy facts about them. Let µ be any infinite cardinal. We call

We will use the following rather trivial combinatorial fact.

From our point of view the sizes of MAD families are important. Clearly there is a MAD family on ω of size 2 ω . The analogue of this does not always hold for ω 1 . Baumgartner in [4] proves that it is consistent with ZFC that there is no almost disjoint family on ω 1 of size 2 ω1 . However, we have the following fact.

In Section 7 we use nonstationary MAD families M N S ⊆ [µ] µ meaning that M N S is a MAD family such that every M ∈ M N S is nonstationary in µ. Observe, that using Zorn’s lemma to almost disjoint families of nonstationary sets of µ we can get nonstationary MAD families.

The constructions of the upcoming sections will use the following amazing results of Shelah. For a cardinal λ and a regular cardinal µ let S λ µ denote the ordinals in λ with cofinality µ. For an S ⊆ S λ µ an S-club sequence is a sequence C = C δ : δ ∈ S such that C δ ⊆ δ is a club in δ of order type µ. µ } denote an S λ µ -club sequence. We define a topological space X = X[λ, µ, M, C] as follows. The underlying set of our topology will be a subset of the product λ × κ. Let

) is a disjoint union. Define the topology on X by neighborhood bases as follows;

and let B(α) = {U (α, β) : β < α} be a base for the poin

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