D-spaces, irreducibility and trees

Leandro F. Aurichi  Lúcia R. Junqueira  Paul B. Larson^{∗}

August 14, 2008

Abstract

We show that the removal of one point from 2^{ω_{1}} gives a counterexample to a conjecture of Ishiu on D-spaces. We also show that Martin’s Axiom implies that there are no Lindelöf non-D-spaces that can be written as union of less than continuum many compact subspaces. Finally we show that the property of being a D-space is preserved by forcing with trees of height ω.

An open neighborhood assignment (ONA) on a topological space X is a function N which assigns to each point x ∈ X an open set N(x) containing x. Given an ONA N on a space X, and subset Y of X, we let N[Y] denote ⋃{N(x) | x ∈ Y }. A space X is a D-space [11] if for every ONA N on X there is a closed discrete C ⊆ X such that N[C] = X. These spaces were introduced by van Dowen in 1979[11], and while they have attracted a lot of attention in recent years[2, 3, 4, 6, 7, 8, 9, 10, 13, 14], many basic questions remain open [12]. Probably the best known is whether every regular Lindelöf space is a D-space (see [16]).

In the first section we prove that removing one point from 2^{ω_{1}} gives a counterexample to a conjecture of Tetsuya Ishiu, as the resulting space is irreducible but not a D-space. In the second section, we prove that, assuming Martin’s Axiom, there are no “small” Lindelöf non-D-spaces, where “small” means a union of less than continuum many compact subspaces. Finally, in the third section we consider the effects of forcing with trees of height ω. For instance, we show that if T is such a tree and X is a D-space, then X remains a D-space after forcing with T.

1 Irreducibility and the Revised Range Conjecture

Tetsuya Ishiu proposed what he called the Revised Range Conjecture, asserting that every topological space X has a basis B such that for any two ONA’s N_{0}, N_{1} (on X) with the same range R ⊆ B, there is a closed discrete set C_{0} such that N_{0}[C_{0}] = X if and only if there is a closed discrete set C_{1} such that N_{1}[C_{1}] = X. We will see in this section that this conjecture is false.

∗The work in this paper was supported in part by NSF Grant DMS-0401603 (third author), FAPESP Grant 2005/60183-5 (second and third authors) and CNPq (first author)

A topological space X is said to be irreducible [1] if for every open cover O of X there is an open cover O′ such that each element of O′ is contained in a member of O and contains a point not in any other member of O′ (such a O′ is said to be a minimal open refinement of O).

Lemma 1.1. Let X be an irreducible space in which every open set has the same cardinality. If the Revised Range Conjecture holds for X, then X is a D-space.

Proof. Let N be an ONA on X. We may assume that the range of N is contained in a basis B witnessing the Revised Range Conjecture for X. Since X is irreducible, there exists a minimal open refinement O′ of the range of N covering X. For each O ∈ O′ pick a point in O not in any other member of O′, and let Y be the set of picked points. Then Y is a closed discrete set, and we can define a partial ONA N′ on Y by letting N′(y) be any member of the range of N containing the member of O′ containing y, for each y ∈ Y. It suffices now to extend N′ to an ONA on all of X with the same range as N. Since Y is closed discrete, each open set has intersection of size |X| with the complement of Y. The range of N has cardinality κ ≤ |X|. Let ⟨B_{α} : α < κ⟩ be a wellordering of the range of N, and choose points ⟨x_{α} : α < κ⟩ such that each x_{α} ∈ B_{α} \ (Y ∪ {x_{β} : β < α}), and define N′(x_{α}) = B_{α} for each α < κ. For each x ∈ X \ (Y ∪ {x_{α} : α < κ}), let N′(x) = N(x).

In [18] it was shown that the removal of one point from 2^{ω_{1}} gives an irreducible space. However, this space is not a D-space, as shown by the following lemma. Note that every open subset of this space has the same cardinality.

Lemma 1.2. The space 2^{ω_{1}} with one point removed contains a closed copy of ω_{1}.

Proof. For simplicity, let the removed point be the constant 0 function. For each α < ω_{1}, let x_{α} be (α × {0}) ∪ ((ω_{1} \ α) × {1}). The subspace {x_{α} : α < ω_{1}} is closed. Furthermore, if for each β < ω_{1}, we let O_{β} = {z ∈ 2^{ω_{1}} | z(β) = 0} and I_{β} = {z ∈ 2^{ω_{1}} | z(β) = 1}, then the O_{β}’s and I_{β}’s generate the ω_{1}-topology on {x_{α} : α < ω_{1}}.

2 Lindelöfness and Martin’s Axiom

Our second section concerns Lindelöf non-D-spaces and Martin’s Axiom (MA). Recall that Martin’s Axiom is the statement that if P is a partial order without uncountable antichains, and D is a collection of dense subsets of P such that |D| < 𝔠 (where 𝔠 denotes the cardinality of the continuum), then there is a filter G ⊆ P intersecting each element of D (see [17], for instance). The covering number for the meager ideal (cov(M)) is the smallest cardinality of a family of meager sets of reals whose union is all of ℝ (see [5], for instance); restated, it is the smallest cardinality of a collection D consisting of dense subsets of the partial order (ω^{ω}, ⊆) with the property that no filter intersects every member of D. The Baire Category Theorem implies that cov(M) ≥ ℵ_{1}. Martin’s Axiom (indeed, its restriction to Cohen forcing) implies that cov(M) = 𝔠.

We first prove that there are no Lindelöf non-D-spaces of cardinality less than cov(M). One easy consequence is that one can not prove, only assuming ZFC, that there is such space of cardinality ℵ_{1}.

We begin with the following.

Lemma 2.1. If X is a T_{1} Lindelöf space and N is an open neighborhood assignment on X, then there is a countable Y ⊆ X such that for every finite a ⊆ Y and every x ∈ X \ N[a] there is a y ∈ Y \ N[a] such that x ∈ N[y].

Proof. We find countable sets Y_{i} ⊆ X (i < ω), and let ⟨a_{i} : i < ω⟩ be a listing of all the finite subsets of ⋃_{i<ω} Y_{i}, such that each a_{i} ⊆ ⋃_{j≤i} Y_{j}. Let Y_{0} be any countable subset of X such that N[Y_{0}] = X. Given a_{i}, consider the open cover of X given by the restriction of N to a_{i} ∪ (X \ N[a_{i}]), and let Y_{i+1} be the set of x such that N(x) is in some fixed countable subcover. Then Y = ⋃_{i<ω} Y_{i} is as desired.

Theorem 2.2. If X is a T_{1} Lindelöf space and |X| < cov(M), then X is a D-space.

Proof. Let Y = ⟨y_{i} : i < ω⟩ be as in Lemma 2.1, and consider the set A of a ∈ 2^{ω} such that for each i ∈ a^{−1}(1), y_{i} ∉ N[{y_{j} : j ∈ i ∩ a^{−1}(1)}]. Then A is a perfect subset of 2^{ω}, and for each x ∈ X the set of a ∈ A with x ∉ N[{y_{i} : i ∈ a}] is nowhere dense in A. Since |X| < cov(M), there is an a ∈ A such that N[a] = X.

The assumptions of the Theorem 2.2 are implied by MA(Cohen forcing) when |X| < 𝔠 (see [5]).

Corollary 2.3 (MA(Cohen forcing)). If X is a Lindelöf space such that |X| < 2^{ω}, then X is a D-space.

Corollary 2.4 (MA(Cohen forcing)). If X is a hereditary Lindelöf space such that it is not a D-space, then |X| = 2^{ω}.

Proof. This is immediate, since if Y is a hereditary Lindelöf space then |Y | ≤ 2^{ω}, by a result of de Groot (see [15]).

Modifying the proof of Theorem 2.2, we can obtain a stronger result. First we note the following consequence of Lemma 2.1.

Lemma 2.5. Let X be a Lindelöf space and N be an open neighborhood assignment on X. Then there is a countable Y ⊆ X such that for every finite a ⊆ Y, there is b ⊆ Y ∖ N[a] such that X = N[a] ∪ N[b].

Let (X, τ) be a topological space. Let f : ω^{<ω} ⟶ X × τ be a function. If s ∈ ω^{<ω} and f(s) = (x, V), then we denote by f_{X}(s) = x and by f_{τ} (s) = V.

The idea for the next lemma is the following: we will construct an ω-tree using the Y given by the previous lemma. The successors of every element of the tree will be all the points of Y that are “not yet covered” by our construction. At the same time we will assure that every finite subset of Y that is not yet covered can be added to the tree in finitely many steps.

Lemma 2.6. Let (X, τ) be a Lindelöf space and N be an open neighborhood assignment on X. Then there is f : ω^{<ω} ∖ {∅} ⟶ X × τ such that:

(i) if s ∈ ω^{<ω} ∖ {∅} then f_{τ} (s) ⊆ N(f_{X}(s));

(ii) if r is a branch of ω^{<ω}, then {f_{X}(s) : s ∈ r} is closed discrete in ⋃{f_{τ} (s) : s ∈ r};

(iii) if C ⊆ X is compact, then D_{C} = {s ∈ ω^{<ω} : C ⊆ ⋃_{t≤s} f_{τ} (t)} is dense in ω^{<ω}.

Proof. Le Y be as given by Lemma 2.5. We will define f : ω^{<ω} ∖ {∅} ⟶ Y × τ by recursion on the length of s in such a way that:

(a) if s ∈ ω^{<ω} then for every n ∈ ω and every nonzero k ≤ |s|, f_{X}(s^{⌢}n) ∉ f_{τ} (s↾k);

(b) if s ∈ ω^{<ω} then f_{τ} (s) = N(f_{X}(s)) ∖ F where F is a finite subset of Y ∖ {f_{X}(s)};

(c) for every s ∈ ω^{<ω}, if y ∈ Y ∖ ⋃{f_{τ} (s↾k) : 0 < k ≤ |s|}, then there is an n ∈ ω such that y = f_{X}(s^{⌢}n);

(d) if y = f_{X}(s^{⌢}n) for some s ∈ ω^{<ω} and n ∈ ω, then for each finite F ⊆ (Y ∩ N(y)) ∖ {y} there is a k ∈ ω such that f(s^{⌢}k) = (y, N(y) ∖ F);

Note that we can make this construction since Y is countable and so is [Y]^{<ω}. First we will show that if r is a branch of ω^{<ω} ∖ {∅}, then {f_{X}(s) : s ∈ r}

has no accumulation points in ⋃{f_{τ} (s) : s ∈ r}. Let x ∈ ⋃{f_{τ} (s) : s ∈ r}. We will show that it is not an accumulation point of {f_{X}(s) : s ∈ r}. Let s ∈ r such that x ∈ f_{τ} (s). Note that f_{X}(t) ∉ f_{τ} (s) for every t ∈ r, t > s. Then x is separated from these points and, since there are only finitely many points more in r, we have that x is not an accumulation point.

Note that, by Lemma 2.5, we have that, for every s ∈ ω^{<ω} ∖ {∅},

⋃{f_{τ} (s↾k) : 0 < k ≤ |s|} ∪ ⋃_{n∈ω} f_{τ} (s^{⌢}n) = X.

For each C ⊆ X, let D_{C} denote the set of s ∈ ω^{ω} \ {∅} such that C is contained in ⋃{f_{τ} (s↾k) : 0 < k ≤ |s|}. We will show that when C is compact, D_{C} is dense in ω^{ω} \ {∅}. Fix s ∈ ω^{<ω} and suppose s ∉ D_{C}. Then C ∖ ⋃{f_{τ} (s↾k) : 0 < k ≤ |s|} is a compact subset covered by the family (f_{τ} (s^{⌢}n))_{n∈ω}. Then, there are k_{1}, ..., k_{n} ∈ ω such that

C ∖ ⋃{f_{τ} (s↾k) : 0 < k ≤ |s|} ⊆ f_{τ} (s^{⌢}k_{1}) ∪ ⋯ ∪ f_{τ} (s^{⌢}k_{n}).

We can suppose that f_{X}(s^{⌢}k_{i}) ∉ f_{τ} (s^{⌢}k_{j}) for all 1 ≤ i < j ≤ n by property (d). Thus, we can choose p_{1}, ..., p_{n} ∈ ω such that f(s^{⌢}p_{1}) = f(s^{⌢}k_{1}), f(s^{⌢}p_{1}^{⌢}p_{2}) = f(s^{⌢}k_{2}), ..., f(s^{⌢}p_{1}^{⌢}p_{2}^{⌢} ⋯^{⌢} p_{n}) = f(s^{⌢}k_{n}), by property (c). Note that C ⊆ ⋃_{t≤s^{⌢}p_{1}^{⌢}p_{2}^{⌢}···^{⌢}p_{n}} f_{τ} (t).

It was already known that every σ-compact space is a D-space (see [6], for instance). Lemma 2.6 allows us to improve this result.

Theorem 2.7. Every Lindelöf space which is a union of fewer than cov(M) many compact spaces is a D-space.

Proof. Suppose that κ is a cardinal less than cov(M), and that X = ⋃_{ξ<κ} C_{ξ}, where each C_{ξ} is compact. Let f : ω^{<ω} ⟶ X × τ be the function given by Lemma 2.6. Note that for every ξ < κ we have that D_{C_{ξ}} (as defined in the proof of Lemma 2.6) is dense in ω^{<ω}. Then there is a branch r of ω^{<ω} such that r ∩ D_{C_{ξ}} ≠ ∅ for each ξ < κ. Thus ⋃_{s∈r} f_{τ} (s) ⊃ ⋃_{ξ<κ} C_{ξ} = X. Since {f_{X}(s) : s ∈ r} is closed discrete in ⋃_{s∈r} f_{τ} (s), we have that {f_{X}(s) : s ∈ r} is closed discrete in X.

3 Forcing with trees of height ω

The results of the previous section suggest that some basic facts about D-spaces may be independent of ZFC. While we do not have such a result, we present in this section two facts about D-spaces and forcing which may be of some use. These facts concern forcing with trees of height ω, and apply the approach of the previous section.

Theorem 3.1. If X is a D-space and T is a tree of height ω, then X remains a D-space after forcing with T.

Proof. Let Ṅ be a T -name for an ONA on X, and let B be the set of open subsets of X which are forced by some condition to be in the range of the realization of Ṅ. Let ⟨p_{α} : α < κ⟩ be a wellordering of the elements of T such that shorter elements are listed before longer ones. We define recursively on α closed discrete sets D_{α} (α < κ) and functions f_{α}: D_{α} → T and h_{α}: D_{α} → B such that, letting Y_{α} be the set of x in any D_{β} (β < α) such that f_{β}(x) ≥ p_{α}:

• for all x in any D_{α}, f_{α}(x) ≤ p_{α} and f_{α}(x)⊩Ṅ(x̌) = h_{α}(x)ˇ ;

• for all α < κ and for all y ∈ X, either there exist β < α and x ∈ D_{β} ∩ Y_{α} such that y ∈ h_{β}(x) or there exists an x ∈ D_{α} such that y ∈ h_{α}(x);

• if β < α and x ∈ D_{β} ∩ Y_{α}, then h_{β}(x) ∩ D_{α} = ∅.

(Note that since f_{β}(x) ≤ p_{β} for each β < κ and each x ∈ X, p_{β} ≥ p_{α} whenever D_{β} ∩ Y_{α} is nonempty; in particular there are only finitely many such β, so Y_{α} is closed discrete.)

Supposing that we have constructed D_{α}, f_{α} and h_{α} for all β < α, let

E_{α} = ⋃{h_{β}(x) | β < α ∧ x ∈ D_{β} ∩ Y_{α}}.

We define a new ONA N_{α} as follows. For each x ∈ E_{α}, let N_{α}(x) = E_{α}. For each x ∈ X \ E_{α}, pick a condition p(x) ≤ p_{α} and an element B(x) ∈ B such that p(x)⊩Ṅ(x̌) = B(x)ˇ , and let N_{α}(x) = B(x). Then there is a closed discrete set D_{α}^{∗} such that N_{α}[D_{α}^{∗}] = X. Let D_{α} = D_{α}^{∗} \ E_{α}. For each x ∈ D_{α}, let f_{α}(x) = p(x) and let h_{α}(x) = B(x). This completes the construction.

Let g be a V -generic path through T. For each α ∈ κ, let C_{α} be the set of x ∈ D_{α} such that f_{α}(x) ∈ g. Let

C = ⋃{C_{α} | α ∈ κ}.

By genericity Ṅ_{g}[C] = X. We will we done once we show that C is closed discrete.

Pick a point y in X. There is a x ∈ C_{β} for some β ∈ κ such that y ∈ Ṅ_{g}(x). Since f_{β}(x) ∈ g, Ṅ_{g}(x) = h_{β}(x). Fix γ < κ such that p_{γ} ∈ g, γ > β and p_{γ} ≤ f_{β}(x). Since D_{α} ∩ h_{β}(x) = ∅ for all α > β with p_{α} ≤ f_{β}(x), y is not in the closure of

⋃{D_{α} | γ ≤ α < κ, p_{α} ∈ g},

which contains ⋃{C_{α} : γ ≤ α < κ}. On the other hand,

C ⊆ ⋃{D_{α} : p_{α} > p_{γ}} ∪ ⋃{C_{α} | γ ≤ α < κ}.

Since ⋃{D_{α} : p_{α} > p_{γ}} is a finite union of closed discrete sets, y is not in the closure of ⋃{D_{α} : p_{α} > p_{γ}} \ {y}, either, which shows that C is closed discrete.

A similar argument shows the following result, where we start with a Lindelöf space in the ground model. If T is a tree and S is a subset of T, we say that S can be refined to an antichain if there is a function a: S → T such that a(s) ≤ s for all s ∈ S, and such that the range of S is an antichain.

Theorem 3.2. If X is a Lindelöf space and T is a tree of height ω such that every countable subset of T can be refined to an antichain, then X is a D-space after forcing with T.

Proof. Let Ṅ be a T -name for an ONA on X, and let B be the set of open subsets of X which are forced by some condition to be in the range of the realization of Ṅ. Let ⟨p_{α} : α < κ⟩ be a wellordering of the elements of T such that shorter elements are listed before longer ones. We define recursively on α countable sets D_{α} ⊆ X (α < κ) and functions f_{α}: D_{α} → T and h_{α}: D_{α} → B such that, letting Y_{α} be the set of x in any D_{β} (β < α) such that f_{β}(x) ≥ p_{α}:

• for all x in any D_{α}, f_{α}(x) ≤ p_{α} and f_{α}(x)⊩Ṅ(x̌) = h_{α}(x)ˇ ;

• for all α < κ and for all y ∈ X, either there exist β < α and x ∈ D_{β} ∩ Y_{α} such that y ∈ h_{β}(x) or there exists an x ∈ D_{α} such that y ∈ h_{α}(x);

• if β < α and x ∈ D_{β} ∩ Y_{α}, then h_{β}(x) ∩ D_{α} = ∅;

• the range of each f_{α} is an antichain.

(Note that since f_{β}(x) ≤ p_{β} for each β < κ and each x ∈ X, p_{β} ≥ p_{α} whenever D_{β} ∩ Y_{α} is nonempty; in particular there are only finitely many such β, so Y_{α} is finite.)

Supposing that we have constructed D_{α}, f_{α} and h_{α} for all β < α, let

E_{α} = ⋃{h_{β}(x) | β < α ∧ x ∈ D_{β} ∩ Y_{α}}.

We define a new ONA N_{α} as follows. For each x ∈ E_{α}, let N_{α}(x) = E_{α}. For each x ∈ X \ E_{α}, pick a condition p(x) ≤ p_{α} and an element B(x) ∈ B such that p(x)⊩Ṅ(x̌) = B(x)ˇ , and let N_{α}(x) = B(x). Then there is a countable set D_{α}^{∗} such that N_{α}[D_{α}^{∗}] = X. Let D_{α} = D_{α}^{∗} \ E_{α}. For each x ∈ D_{α}, let f_{α}(x) be a condition below p(x) in such a way that the range of f_{α} is an antichain, and let h_{α}(x) = B(x). This completes the construction.

Let g be a V -generic path through T. For each α ∈ κ, let C_{α} be the set of x ∈ D_{α} such that f_{α}(x) ∈ g. Let

C = ⋃{C_{α} | α ∈ κ}.

By genericity Ṅ_{g}[C] = X. We will we done once we show that C is closed discrete.

Pick a point y in X. There is a x ∈ C_{β} for some β ∈ κ such that y ∈ Ṅ_{g}(x). Since f_{β}(x) ∈ g, Ṅ_{g}(x) = h_{β}(x). Fix γ < κ such that p_{γ} ∈ g, γ > β and p_{γ} ≤ f_{β}(x). Since D_{α} ∩ h_{β}(x) = ∅ for all α > β with p_{α} ≤ f_{β}(x), y is not in the closure of

⋃{D_{α} | γ ≤ α < κ, p_{α} ∈ g},

which contains ⋃{C_{α} : γ ≤ α < κ}. On the other hand, letting

Z = ⋃{{z ∈ D_{α} | f_{α}(z) ∈ g} : α < γ},
C ⊆ Z ∪ ⋃{C_{α} | γ ≤ α < κ}.

Since Z is a finite set, y is not in the closure of Z \ {y}, either, which shows that C is closed discrete.

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Instituto de Matemática e Estatística da Universidade de São Paulo (IME-USP) Rua do Matão, 1010 - Cidade Universitria CEP 05508-090 São Paulo - SP - Brasil lucia@ime.usp.br laurichi@ime.usp.br

Department of Mathematics and Statistics Miami University, Oxford, Ohio 45056, United States larsonpb@muohio.edu