FORCING COUNTABLE NETWORKS FOR SPACES SATISFYING R(X^{ω}) = ω

I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

Abstract. We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that in V^{P} X is a countable union of 0dimensional subspaces of countable weight. We also show that this theorem is sharp in two different senses: (i) we can’t get rid of using generic extensions, (ii) we have to consider all finite powers of X.

1. Introduction

We use standard topological notation and terminology throughout, cf [4]. The following definitions are less well-known.

Definition 1.1. Given a topological space ⟨X, τ⟩ and a subspace Y ⊂ X a function f is called a neighbourhood assignment on Y iff f : Y ⟶ τ and y ∈ f(y) for each y ∈ Y.

Definition 1.2. A space Y is weakly separated if there is a neighbourhood assignment f on Y such that

∀y ≠ z ∈ Y (y ∉ f(z) ∨ z ∉ f(y)),

moreover

R(X) = sup{|Y | : Y ⊂ X is weakly separated}.

The notion of weakly separated spaces and the cardinal function R were introduced by Tkačenko in [7], where the following question was also raised: does R(X^{ω}) = ω (or even R(X) = ω) imply that X has a countable network (i.e. nw(X) = ω)? (Note that R(X^{ω}) = ω is equivalent to R(X^{n}) = ω for all n ∈ ω, moreover nw(X) = ω implies R(X^{ω}) = nw(X^{ω}) = ω.) Several consistent counterexamples to this were given, e.g. in [1], [2], [5] and [8, p. 43], but no ZFC counterexample

1991 Mathematics Subject Classification. 54A25,03E35.

Key words and phrases. net weight, weakly separated, Martin’s Axiom, forcing.

The preparation of this paper was supported by the Hungarian National Foundation for Scientific Research grant no. 16391 .

The second author was supported by DFG (grant Ko 490/7-1).

2 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

is known. (In [8] it is stated that under PFA the implication is valid, but no proof is given.) The counterexamples given in [5] and [8] from CH are also first countable.

Our main result here says that, at least for T_{2} spaces, a weaker version of Tkačenko’s conjecture is valid, namely R(X^{ω}) = ω implies that nw(X) = ω holds in a suitable c.c.c and hence cardinal preserving generic extension! In fact, in this extension X becomes σ-second countable, i.e. X is the union of countably many subspaces of countable weight.

In section 3 we show that the main result is sharp in different senses. Firstly, we force, for every natural number n, a 0-dimensional, first countable space X such that R(X^{n}) = ω, but nw(X) > ω in any cardinal preserving extension of the ground model. Secondly, we construct in ZFC a 0-dimensional T_{2} space X such that χ(X) = nw(X) = ω but X is not σ-second countable.

It easily follows from the proof of our main result that if MA_{κ} holds and X is a Hausdorff space with |X| + w(X) ≤ κ, then R(X^{ω}) = ω if and only if X is σ-second countable. We also prove that MA(Cohen) is not enough to yield this equivalence. To do this we use a result of Shelah (the proof of which presented here with his kind permission) saying that in any generic extension by Cohen reals the ideal of the first category subspaces of a space from the ground model is generated by the subspaces of first category from the ground model.

2. The main result

Theorem 2.1. Given a Hausdorff topological space X the following are equivalent:

(1) R(X^{ω}) = ω,

(2) there is a c.c.c poset P such that V^{P} ⊧ “nw(X) = ω”,

(3) there is a c.c.c poset P such that V^{P} ⊧ “X is a countable union of 0-dimensional subspaces of countable weight.”

Proof. Since the implications (3) ⇒ (2) ⇒ (1) are clear, it remains to prove only that (1) implies (3). So assume that X is Hausdorff with R(X^{ω}) = ω, and fix a base B of X and a well-ordering ≺ on X ∪ B.

The space X contains at most countably many isolated points by R(X) = ω, so it is enough to force an appropriate partition of X′, the set of non-isolated points of X. We say that a 4-tuple ⟨A, U , f , g⟩ is in P provided (i)–(v) below hold:

(i) A ∈ [X′]^{<ω} and U ∈ [B]^{<ω}, (ii) f and g are functions, (iii) f : A ⟶ ω, g : U ⟶ ω × ω, (iv) if g(U) = ⟨n, i⟩ and f(x) = n then x ∉ (\overline{U} \ U) whenever x ∈ A

and U ∈ U,

(v) if g(U) = g(V) = ⟨n, i⟩ and f(x) = n then x ∈ U iff x ∈ V whenever x ∈ A and U, V ∈ U.

Our idea is that f will guess the partition of X′ into countably many pieces, {F_{n} : n < ω}; if g(U) = ⟨n, i⟩ then U ∩ F_{n} will be clopen in the subspace F_{n}, and g(U) = g(V) = ⟨n, i⟩ implies U ∩ F_{n} = V ∩ F_{n}. Consequently, each F_{n} will have a countable clopen base.

For p ∈ P we write p = ⟨A^{p}, U^{p}, f^{p}, g^{p}⟩. If p, q ∈ P we set p ≤ q iff f^{p} ⊃ f^{q} and g^{p} ⊃ g^{q}.

Two conditions p and q from P are called twins provided |A^{p}| = |A^{q}|, |U^{p}| = |U^{q}| and denoting by η and by ρ the unique ≺-preserving bijections between A^{p} and A^{q}, and between U^{p} and U^{q}, respectively, we have

(1) η⌈A^{p} ∩ A^{q} = id, ρ⌈U^{p} ∩ U^{q} = id,

(2) ∀x ∈ A^{p} f^{p}(x) = f^{q}(η(x)).

(3) ∀U ∈ U^{p} g^{p}(U) = g^{q}(ρ(U)).

(4) ∀x ∈ A^{p} ∀U ∈ U^{p} ( x ∈ U iff η(x) ∈ ρ(U), and x ∈ \overline{U} iff η(x) ∈ \overline{ρ(U)} ).

Lemma 2.2. P = ⟨P, ≤⟩ satisfies c.c.c.

Proof of lemma 2.2. Let {p_{α} : α < ω_{1}} ⊂ P. Write p_{α} = ⟨A^{α}, U^{α}, f^{α}, g^{α}⟩. Using standard Δ-system and counting arguments we can assume that these conditions are pairwise twins. Let k = |A^{α}| and {a_{α,i} : i < k} be the ≺-increasing enumeration of A^{α}. For each α < ω_{1} and i < k put

U_{α,i}^{0} = {U ∈ U^{α} : ∃j g^{α}(U) = ⟨(f^{α}(a_{α,i}), j⟩ ∧ a_{α,i} ∈ U},
U_{α,i}^{1} = {U ∈ U^{α} : ∃j g^{α}(U) = ⟨(f^{α}(a_{α,i}), j⟩ ∧ a_{α,i} ∉ U},

and finally

V_{α,i} = ⋂\{U : U ∈ U_{α,i}^{0}\} ∩ ⋂\{X \ \overline{U} : U ∈ U_{α,i}^{1}\}.

By (iv) we have a_{α,i} ∉ \overline{U} for U ∈ U_{α,i}^{1}, so a_{α,i} ∈ V_{α,i}. Since R(X^{k}) = ω there are α < β < ω_{1} such that

(†) a_{α,i} ∈ V_{β,i} and a_{β,i} ∈ V_{α,i} for each i < k.

We claim that p_{α} and p_{β} are compatible in P. Let η and ρ be the functions witnessing that p_{α} and p_{β} are twins. Put A = A^{α} ∪ A^{β}, U = U^{α} ∪ U^{β}, f = f^{α} ∪ f^{β}, g = g^{α} ∪ g^{β} and p = ⟨A, U , f , g⟩. Since p_{α} and p_{β} are twins, p satisfies (i)–(iii) and p ≤ p_{α}, p_{β}. So all we have to do is to show that p satisfies (iv) and (v).

Claim . If U ∈ U_{α,i}^{0} ∪ U_{α,i}^{1}, then a_{α,i} ∈ U iff a_{α,i} ∈ ρ(U).

Proof of the claim. We know a_{α,i} ∈ U iff a_{β,i} ∈ ρ(U). Thus a_{α,i} ∈ U implies that ρ(U) ∈ U_{β,i}^{0} and so a_{α,i} ∈ V_{β,i} ⊂ ρ(U). On the other hand, if a_{α,i} ∉ U, then a_{β,i} ∉ ρ(U), hence ρ(U) ∈ U_{β,i}^{1}, and so a_{α,i} ∈ V_{β,i} ⊂ X \ \overline{ρ(U)}, i.e., a_{α,i} ∉ \overline{ρ(U)}.

Now we check (iv). Assume a_{α,i} ∈ A^{α}, V ∈ U^{β} with g^{β}(V) = ⟨f^{α}(a_{α,i}), l⟩ and a_{α,i} ∈ \overline{V}. We have to show that a_{α,i} ∈ V. Since a_{α,i} ∈ V_{β,i} by (†), we have V ∉ U_{β,i}^{1}. But f^{α}(a_{α,i}) = f^{β}(a_{β,i}) since p_{α} and p_{β} are twins, hence g^{β}(V) = ⟨f^{β}(a_{β,i}), l⟩ implies V ∈ U_{β,i}^{0} ∪ U_{β,i}^{1}, so V ∈ U_{β,i}^{0}. Hence a_{α,i} ∈ V_{β,i} ⊂ V by (†), which was to be proved.

Finally we check (v). Assume that a_{α,i} ∈ A^{α} and U, V ∈ U^{α} ∪ U^{β} are such that g(U) = g(V) = ⟨f(a_{α,i}), l⟩. Define the function ρ^{∗} : U ⟶ U^{α} by the formula ρ^{∗} = id⌈U^{α} ∪ (ρ)^{−1} Then a_{α,i} ∈ U iff a_{α,i} ∈ ρ^{∗}(U) and a_{α,i} ∈ V iff a_{α,i} ∈ ρ^{∗}(V) by the previous claim.

But g(ρ^{∗}(U)) = g(U) = g(V) = g(ρ^{∗}(V)) = ⟨f(a_{α,i}), l⟩, so a_{α,i} ∈ ρ^{∗}(U) iff a_{α,i} ∈ ρ^{∗}(V) for p_{α} satisfies (v). Thus a_{α,i} ∈ U iff a_{α,i} ∈ ρ^{∗}(U) iff a_{α,i} ∈ ρ^{∗}(V) iff a_{α,i} ∈ V, which proves (v).

Now let G be a P -generic filter and let F = ⋃{f^{p} : p ∈ G} and G = ⋃{g^{p} : p ∈ G}. For n < ω let F_{n} = F^{−1}{n}.

Lemma 2.3. dom(F) = X′ and dom(G) = B.

Proof. Let p = ⟨A, U , f , g⟩ ∈ P, x ∈ X′ \ A and U ∈ B \ U. Let A^{∗} = A ∪ {x}, U^{∗} = U ∪ {U} and n = max ran(f) + 1. Let dom(f^{∗}) = A^{∗}, f^{∗} ⊃ f and f^{∗}(x) = n, dom(g^{∗}) = U^{∗}, g^{∗} ⊃ g and g^{∗}(U) = ⟨n + 1, 0⟩. Then it is easy to check p^{∗} = ⟨A^{∗}, U^{∗}, f^{∗}, g^{∗}⟩ ∈ P and obviously p^{∗} ≤ p. So the lemma holds because a generic filter intersects every dense set.

For m ∈ ω let

B_{m} = {U ∩ F_{m} : U ∈ B and G(U) = ⟨m, i⟩ for some i ∈ ω}.

Lemma 2.4. B_{m} is a countable, clopen base of the subspace F_{m} of X′.

Proof. If U ∈ B, G(U) = ⟨m, i⟩ then U ∩ F_{m} is clopen in F_{m} by (iv). If U, V ∈ B, G(U) = G(V) = ⟨m, i⟩ then U ∩ F_{m} = V ∩ F_{m} by (v). So B_{m} is countable.

Finally we show that it is a base of F_{m}. So fix x ∈ F_{m} and V ∈ B with x ∈ V. Let p = ⟨A, U , f , g⟩ ∈ P such that f(x) = m. Since X is Hausdorff and x is non-isolated in X, we can choose U ∈ B \ U such that x ∈ U ⊂ V and \overline{U} ∩ A = \{x\}.

Let U^{∗} = U ∪ {U}. Define the function g^{∗} : U^{∗} ⟶ ω × ω such that g^{∗} ⊃ g and g^{∗}(U) = ⟨m, k⟩ where k = min{l : ran g ⊂ l × l}. Then p^{∗} = ⟨A, U^{∗}, f, g^{∗}⟩ is an extension of p in P and p^{∗}^{∥}—x ∈ U ∩ F_{m} ⊂ V ∩ F_{m} ∧ U ∩ F_{m} ∈ B_{m}. Consequently, if p^{∥}— “x ∈ F_{m} ∩ V ” then we also have p^{∥}— “∃U ∈ B_{m}(x ∈ U ⊂ F_{m} ∩ V)”, which completes the proof.

Thus theorem 2.1 is proved.

It is easy to check that the above proof needs the genericity of G over |X| + |B| many dense sets only, and this immediately yields the following result.

Corollary 2.5. If MA_{κ} holds then for a Hausdorff space X with |X|+ w(X) ≤ κ the following are equivalent:

(1) R(X^{ω}) = ω,

(2) nw(X) = ω,

(3) X is the union of countably many 0-dimensional subspaces of countable weigh,

(4) X is the union of countably many separable metrizable subspaces.

3. Sharpness of the main result

Our aim in this section is to examine how sharp the above main result is. The co-finite topology on any uncountable set X clearly satisfies R(X^{ω}) = ω, while, in any extension, nw(X) = |X|. This show that in the proof of (1) ⟶ (2) of 2.1 the Hausdorffness of X cannot be replaced by T_{1}.

The next result in this section implies that, at least in ZFC, the exponent ω in proving (1) ⟶ (2) in theorem 2.1 can not be lowered.

Theorem 3.1. For each uncountable cardinal κ and natural number m there is a c.c.c poset P of cardinality κ such that in V^{P} there is a 0-dimensional first countable topological space X = ⟨κ, τ⟩ such that R(X^{m}) = ω but R(X^{m+1}) = κ, hence nw(X) = κ in any cardinal preserving extension.

In [5, theorem 3.5] we constructed a c.c.c poset ⟨P^{κ}, ≤⟩ which adds to the ground model a 0-dimensional, first countable topology τ on κ such that R(X^{ω}) = ω and w(X) = κ for X = ⟨κ, τ⟩. The conditions in P^{κ} are finite approximations of the space X and the property R(X^{ω}) = ω is guaranteed by some Δ-system and amalgamation arguments. Here we will use a subset P of P^{κ} with the inherited order. To ensure

6 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

R(X^{m+1}) = κ we thin out P^{κ} in the following way. We fix a family D = {d_{α} : α < κ} of pairwise disjoint elements of κ^{m+1} with the intention to make D discrete in X^{m+1}. A condition p ∈ P^{κ} is put into P if and only if every neighbourhood given by p witnesses that D is discrete. The main step of the proof is to show that P is large enough to allow the Δ-system and amalgamation arguments to work in showing R(X^{m}) = ω.

Proof of theorem 3.1. First we recall some definitions and lemmas from the proof of [5, theorem 3.5]. A quadruple ⟨A, n, f, g⟩ is said to be in P_{0}^{κ} provided (a)–(b) below hold:

(a) A ∈ [κ]^{<ω}, n ∈ ω, f and g are functions,

(b) f : A × A × n ⟶ 2, g : A × n × A × n ⟶ 3,

For p ∈ P_{0}^{κ} we write p = ⟨A^{p}, n^{p}, f^{p}, g^{p}⟩. If p, q ∈ P_{0}^{κ} we set p ≤ q iff f^{p} ⊇ f^{q} and g^{p} ⊇ g^{q}. If p ∈ P_{0}^{κ}, α ∈ A^{p}, i < n^{p} we defined U(α, i) = U^{p}(α, i) = {β ∈ A^{p} : f^{p}(β, α, i) = 1}.

A quadruple ⟨A, n, f, g⟩ ∈ P_{0}^{κ} is put in P^{κ} iff (i)–(ii) below are also satisfied:

(i) ∀α ∈ A ∀i < j < n α ∈ U(α, j) ⊂ U(α, i), (ii) ∀α ≠ β ∈ A ∀i, j < n

g(α, i, β, j) = 0 if and only if U(α, i) ⊂ U(β, j),
g(α, i, β, j) = 1 if and only if U(α, i) ∩ U(β, j) = ∅,
g(α, i, β, j) = 2 if α ∈ U(β, j) and β ∈ U(α, i).

Definition 3.2. ([5, definition 3.6]) Assume that p_{i} = ⟨A^{i}, n^{i}, f^{i}, g^{i}⟩ ∈ P_{0}^{κ} for i ∈ 2. We say that p_{0} and p_{1} are twins iff n_{0} = n_{1}, |A_{0}| = |A_{1}| and taking n = n_{0} and denoting by σ the unique <_{On}-preserving bijection between A_{0} and A_{1} we have

(1) σ⌈A_{0} ∩ A_{1} = id_{A_{0}∩A_{1}}.

(2) σ is an isomorphism between p_{0} and p_{1}, i.e. ∀α, β ∈ A_{0}, ∀i, j < n f_{0}(α, β, i) = f_{1}(σ(α), σ(β), i), g_{0}(α, i, β, j) = g_{1}(σ(α), i, σ(β), j),

We say that σ is the twin function of p_{0} and p_{1}. Define the smashing function \overline{σ} of p_{0} and p_{1} as follows: \overline{σ} = σ∪id_{A_{1}}. The function σ^{∗} defined by the formula σ^{∗} = σ ∪ σ^{−1}⌈A_{1} is called the exchange function of p_{0} and p_{1}.

Definition 3.3. ([5, definition 3.7]) Assume that p_{0} and p_{1} are twins and ε : A^{p_{1}} \ A^{p_{0}} ⟶ 2. A common extension q ∈ P^{κ} of p_{0} and p_{1} is called an ε-amalgamation of the twins p_{0} and p_{1} provided

∀α ∈ A^{p_{0}}△A^{p_{1}} f^{q}(α, σ^{∗}(α), i) = ε(\overline{σ}(α)).

Lemma 3.4. ([5, lemma 3.8]) If p_{0}, p_{1} ∈ P^{κ} are twins and ε : A^{p_{1}} \ A^{p_{0}} ⟶ 2, then p_{0} and p_{1} have an ε-amalgamation in P^{κ}.

In [5] we used the poset P^{κ} = ⟨P^{κ}, ≤⟩. Here we will apply a subset P of P^{κ}. To define it let {d_{α} : α < κ} be a family of pairwise disjoint elements of [κ]^{m+1} such that κ \ ⋃{d_{α} : α < κ} is still infinite. Write d_{α} = {d_{α,i} : i ≤ m}.

Definition 3.5. A condition p = ⟨A, n, f, g⟩ ∈ P^{κ} is in P iff it satisfies (1) and (2) below:

(1) d_{α} ⊂ A or d_{α} ∩ A = ∅ for each α < κ.

(2) if α < β < κ, d_{α} ∪ d_{β} ⊂ A, then there is an i = i_{α,β} ≤ m such that

d_{α,i} ∉ U^{p}(d_{β,i}, 0) and d_{β,i} ∉ U^{p}(d_{α,i}, 0).

Let P = ⟨P, ≤⟩.

We define X as expected. Let G be a P-generic filter and let F = ⋃{f^{p} : p ∈ G}. For each α < κ and n ∈ ω let V (α, i) = {β < κ : F (β, α, i) = 1}. Put B_{α} = {V (α, i) : i < κ} and B = ⋃{B_{α} : α < κ}. We choose B as the base of X = ⟨κ, τ⟩. By standard density arguments we can see that X is first countable and 0-dimensional.

It is easy to see that R(X^{m+1}) = κ, in fact s(X^{m+1}) = κ. Indeed, by 3.5(2), {d_{α} : α < κ} is discrete in X^{m+1}, as witnessed by the open neighborhoods V (d_{α,0}, 0) × V (d_{α,1}, 0) ⋯ × V (d_{α,m}, 0).

Finally we need to show that P satisfies c.c.c and V^{P} ⊧ R(X^{m}) = ω. Clearly, both of these statement follow from the next lemma.

Lemma 3.6. If {p_{γ} : γ < ω_{1}} ⊂ P, {c_{γ} : γ < ω_{1}} ⊂ κ^{m} and j_{0}, …, j_{m−1} are natural numbers, then there are ordinals α < β < ω_{1} and a condition p ∈ P such that (+) p ≤ p_{α}, p_{β} and p^{∥}—c_{α} ∈ ∏_{i<m} V (c_{β}(i), j_{i}) ∧ c_{β} ∈ ∏_{i<m} V (c_{α}(i), j_{i}).

Proof. We can assume that c_{γ} ⊂ A^{p_{γ}} holds for each γ < ω_{1}.

Pick α and β such that p_{α} and p_{β} are twins, and denoting by ρ their twin function we have ρ′′c_{α} = c_{β} and {ρ′′d_{ξ} : d_{ξ} ⊂ A^{p_{α}}} = {d_{ζ} : d_{ζ} ⊂ A^{p_{β}}}.

Let x = {ξ < κ : d_{ξ} ⊂ A^{p_{α}}} and y = {ξ < κ : d_{ξ} ⊂ A^{p_{β}}}. Define the function ε : A^{p_{α}} \ A^{p_{β}} ⟶ 2 by the stipulations ε(ν) = 1 iff ν ∈ c_{α}. By lemma 3.4 p_{α} and p_{β} has an ε-amalgamation p in P^{κ}. Since C = κ \ ⋃{d_{ξ} : ξ < κ} is infinite, we can assume that A^{p} \ (A^{p_{α}} ∪ A^{p_{β}}) ⊂ C.

First we show that p ∈ P. Observe that for any ξ < κ we have d_{ξ} ∩A^{p} ≠ ∅ if and only if d_{ξ} ∩(A^{p_{α}} ∪A^{p_{β}}) ≠ ∅ if and only if (d_{ξ} ⊂ A^{p_{α}} ∨ d_{ξ} ⊂ A^{p_{β}}) by A^{p} \ (A^{p_{α}} ∪ A^{p_{β}}) ⊂ C. Thus 3.5(1) holds. To check 3.5(2) assume that ξ ≠ ζ < κ and d_{ξ} ∪ d_{ζ} ⊂ A^{p}. Then d_{ξ} ∪ d_{ζ} ⊂ A^{p_{α}} ∪ A^{p_{β}} and

8 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

since p_{α} and p_{β} are in P we can assume that d_{ξ} ⊂ A^{p_{α}} and d_{ζ} ⊂ A^{p_{β}}. If d_{ξ} ∩ A^{p_{β}} ≠ ∅ then d_{ξ} ⊂ A^{p_{β}} as well. Therefore d_{ξ} ∪ d_{ζ} ⊂ A^{p_{β}}, and so 3.5(2) holds for ξ and ζ because p_{β} ∈ P. Thus we can assume that d_{ξ} ⊂ A^{p_{α}} \ A^{p_{β}}, and similarly that d_{ζ} ⊂ A^{p_{β}} \ A^{p_{α}}. If ρ′′d_{ξ} ≠ d_{ζ}, then ρ′′d_{ξ} = d_{μ} for some μ ∈ y \ {ζ}. Since p_{β} ∈ P, there is i ≤ m such that d_{μ,i} ∉ U^{p_{β}}(d_{ζ,i}, 0) and d_{ζ,i} ∉ U^{p_{β}}(d_{μ,i}, 0). So, by the definition of ε-amalgamation, d_{ξ,i} ∉ U^{p}(d_{ζ,i}, 0) and d_{ζ,i} ∉ U^{p}(d_{ξ,i}, 0). On the other hand, if ρ′′d_{ξ} = d_{ζ}, then there is i ≤ m such that ε(d_{ξ,i}) = 0, for |d_{ξ}| = m + 1 > m = |ε^{−1}{1}|. So, by the definition of ε-amalgamation, d_{ξ,i} ∉ U^{p}(d_{ζ,i}, 0) and d_{ζ,i} ∉ U^{p}(d_{ξ,i}, 0). Thus p ∈ P.

Finally we show that p^{∥}— “c_{α} ∈ ∏_{i<m} V (c_{β}(i), j_{i})∧c_{β} ∈ ∏_{i<m} V (c_{α}(i), j_{i})”. Indeed, c_{β}(i) ∈ U^{p}(c_{α}(i), j_{i}) and c_{α}(i) ∈ U^{p}(c_{β}(i), j_{i}) for each i < m because ρ(c_{α,i}) = c_{β,i} and either c_{α,i} = c_{β,i} or ε(c_{α,i}) = 1.

Theorem 3.1 is proved.

Next we show that the use of forcing in the implications (1) ⟶ (3) and (2) ⟶ (3) from theorem 2.1 is essential because in 3.8 we shall produce a ZFC example of a 0-dimensional, first countable space X that satisfies nw(X) = ω (hence R(X^{ω}) = ω) but still X is not σ-second countable. To achieve this we need the following lemma. If X = ⟨X, τ⟩ is a topological space, D(X) denotes the discrete topology on X. If A and B are sets, let Fin(A, B) be the family of functions mapping a finite subset of A into B.

Lemma 3.7. If Z ⊂ X^{ω} and Z is somewhere dense in the space D(X)^{ω}, then w(Z) = w(X).

Proof. Fix a natural number n ∈ ω and a function f : n ⟶ X such that Z is dense in the basic open set U_{f} = {g ∈ X^{ω} : f ⊂ g} of D(X)^{ω}. This means that

∀f′ ∈ Fin(ω \ n, X)∃g ∈ Z f ∪ f′ ⊂ g. (†)

From now on we forget about the D(X)^{ω} topology, we will use only (†). Without loss of generality we can assume that Z ⊂ U_{f}. Let Z be a base of Z in the subspace topology of X^{ω}.

Let π_{m} : X^{ω} ⟶ X be the projection to the m^{th} factor, i.e. π_{m}(g) = g(m). Set X = {π_{n}(U) : U ∈ Z}. Since w(Z) ≤ w(X), it is enough to show that X is a base of X.

Claim . If U is open in Z then π_{n}(U) is open in X.

Proof of the claim . Let x ∈ π_{n}(U). We need to show that π_{n}(U) contains a neighbourhood of x. Pick g ∈ U with x = g(n). Then, by the definition of the product topology on X^{ω}, there is a function σ which maps a finite subset of ω \ n into the family of non-empty open subsets of X such that

g ∈ Z ∩ ⋂_{m∈dom(σ)} π_{m}^{−1} σ(m) ⊂ U. (⋆)

We can assume that n ∈ dom(σ). Let g′ = g⌈(dom σ \ {n}). By (†), for each x′ ∈ σ(n) there is h_{x′} ∈ Z such that

f ∪ {⟨n, x′⟩} ∪ f′ ⊂ h_{x′}. (⋆⋆)

Now (⋆) implies h_{x′} ∈ U and so x′ ∈ π_{n}(U). Thus x ∈ σ(n) ⊂ π_{n}(U), which was to be proved.

To show that X is a base let x ∈ V ⊂ X, V open. By (†) we can find a point g ∈ Z with f ⊂ g and g(n) = x. The family Z is a base of Z in the subspace topology of X^{ω}, so there is U ∈ Z such that g ∈ U ⊂ Z ∩ π_{n}^{−1} V. Thus x ∈ π_{n}(U) ⊂ V and π_{n}(U) is open by the previous claim.

Thus X is a base of X, and so w(X) ≤ |X | ≤ |Z|. Since w(Z) ≤ ω w(X) = w(X), we are done.

After this preparation we can give the ZFC example promised above.

Theorem 3.8. There is 0-dimensional Hausdorff space Y such that χ(Y) nw(Y) = ω, but Y is not σ-second countable.

Proof. By [5, theorem 3.1] there is a 0-dimensional Hausdorff space X of size 2^{ω} such that χ(X) nw(X) = ω, but w(X) = 2^{ω}. We show that Y = X^{ω} is as required. Clearly χ(Y) = χ(X) = ω and nw(Y) = nw(X) = ω.

Assume that Y = ⋃_{k<ω} Z_{k}. The Baire category theorem implies that some Z_{k} is somewhere dense in D(X)^{ω}. Then w(Z_{k}) = w(X) = 2^{ω} by lemma 3.7.

By corollary 2.5, if Martin’s Axiom holds, then every Hausdorff space X of size and weight < 2^{ω} is σ-second countable if and only if nw(X) = ω. The next theorem shows that MA(Cohen) is not enough to get this equivalence. Note that for a first countable space X we have w(X) ≤ |X|.

Theorem 3.9. If ZFC is consistent, then so is ZFC + MA(Cohen) +“there is a first countable 0-dimensional Hausdorff space Y such that nw(Y) = ω and |Y | < 2^{ω} but Y is not σ-second countable ”.

The proof is based on theorem 3.7 above and theorem 3.11 below.

10 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

Definition 3.10. Given a topological space Z, let I(Z) be the σ-ideal generated by the nowhere dense subsets of Z. The elements of I(Z) are called first category in Z.

The next result is due to Saharon Shelah [6] and it is included here with his kind permission.

Theorem 3.11. If Z is a topological space and the forcing notion P = Fn(κ, 2, ω) adds κ Cohen reals to the ground model, then the ideal I^{V^{P}} (Z) is generated by I^{V} (Z), that is, for each T ∈ I^{V^{P}} (Z) there is T′ ∈ I^{V} (Z) with T ⊂ T′.

Proof. Since I^{V^{P}} (Z) is σ-generated by the nowhere dense subsets of Z in V^{P}, we can assume that T is nowhere dense. Let Ṫ be a P -name of T such that 1_{P}^{∥}—“Ṫ is nowhere dense”.

For each m ∈ ω define, in V, the subset B_{m} of Z as follows:

B_{m} = {x ∈ Z : ∃p ∈ P |p| = m ∧ p^{∥}—x ∈ Ṫ }. (∗)

Clearly 1_{P}^{∥}—Ṫ ⊂ ⋃_{m∈ω} B_{m}, so it is enough to show that every B_{m} is

nowhere dense. Assume on the contrary that there are an open set U ⊂ Z and m ∈ ω such that B_{m} is dense in U.

Now, by finite induction, we can define open sets U ⊃ U_{0} ⊃ U_{1} ⋯ ⊃ U_{m} and conditions q_{0}, …, q_{m} with pairwise disjoint domains such that for each j ≤ m and for each f ∈ P if dom(f) = ⋃_{i<j} dom(q_{j}) then

f ∪ q_{j}^{∥}—Ṫ ∩ U_{j} = ∅. (†)

Since B_{m} is dense in U and U_{m} ⊂ U there is x ∈ B_{m} ∩ U_{m}. Then, by the definition of B_{m}, we have a condition p ∈ P with |p| = m such that p^{∥}—x ∈ Ṫ. But the domains of the q_{j} are pairwise disjoint, so there is j ≤ m with dom(p) ∩ dom(q_{j}) = ∅. Thus p ∪ q_{j} ∈ P. Let

f = p⌈ ⋃_{i<j} dom(q_{i}). By (†) f ∪q_{j}^{∥}—Ṫ ∩U_{j} = ∅, so p∪q_{j}^{∥}—Ṫ ∩U_{m} = ∅ as

well. But this contradicts x ∈ U_{m} and p^{∥}—x ∈ Ṫ, and thus the theorem is proved.

Proof of theorem 3.9. Using again [5, theorem 3.1] we have a 0-dimensional Hausdorff space X such that χ(X) nw(X) = ω, but w(X) = 2^{ω}. Let Y = X^{ω} and Z = D(X)^{ω}. Then add κ > 2^{ω} Cohen reals to the ground model. In the generic extension clearly χ(Y) = nw(Y) = ω will remain valid.

Since Z ∉ I^{V} (Z) by the Baire Category theorem, we have Z ∉ I^{V^{P}} (Z) as well by theorem 3.11.

Therefore, if Y = ⋃{Y_{k} : k < ω} holds in V^{P}, then Y_{k} ∉ I^{V^{P}} (Z) for some k ∈ ω, i.e.. Y_{k} is somewhere dense in Z. Thus there is a natural number n ∈ ω and a function f : n ⟶ X such that

∀f′ ∈ Fin(ω \ n, X) ∃g ∈ Y_{k} f ∪ f′ ⊂ g. (†)

But (†) implies that Y_{k} is also dense in the basic open set (V_{f})^{V^{P}} = {g ∈ X^{ω} ∩ V^{P} : f ⊂ g} of (D(X)^{ω})^{V^{P}}. Thus, applying lemma 3.7 in V^{P} we have w(Y_{k}) = w(X) > ω. Thus, in V^{P}, MA(Cohen) holds, and still the first countable, 0-dimensional T_{2} space Y is not σ-second countable, though w(Y) = (2^{ω})^{V} < κ = (2^{ω})^{V^{P}}.

4. Examples for higher cardinals

In this section we generalize the constructions of [5, theorem 3.1] and 3.8 for cardinals greater than ω.

Theorem 4.1. Let κ and λ be cardinals, cf(κ) = κ. Then there is a 0dimensional Hausdorff space X_{λ}^{κ} such that χ(X_{λ}^{κ}) = κ, nw(X_{λ}^{κ}) = λ^{<κ} and w(X_{λ}^{κ}) = λ^{κ}.

Proof. For each f ∈ Fn(κ, λ, κ) put U_{f} = {g ∈ ^{κ}λ : f ⊂ g}. Write U(g, α) = U_{g⌈α} for g ∈ ^{κ}λ and α < κ. For g ≠ h ∈ ^{κ}λ define Δ(g, h) = min{α : g(α) ≠ h(α)}. Consider the topological space C_{λ}^{κ} = ⟨^{κ}λ, τ⟩ that has as a base {U_{f} : f ∈ ^{<κ}λ}. Let

Y = {g ∈ ^{κ}λ : ∃α < κ ( g(β) = 0 iff β ≥ α )}

and

Z = {g ∈ ^{κ}λ : 0 ∉ ran(g)}.

Clearly Y and Z are disjoint, |Y | = λ^{<κ}, |Z| = λ^{κ}, Z is closed and nowhere dense in C_{λ}^{κ}. Let X = Y ∪ Z. Our required space will be X_{λ}^{κ} = ⟨X, ρ⟩, where ρ refines the topology τ_{X}. To define ρ put

X_{g} = ⋃{U_{f} : ∃α < κ f = g⌈α ∪ {⟨α, 0⟩}}

for g ∈ Z and X_{g} = ∅ for g ∈ Y. For g ∈ X and α < κ let V (g, α) = (U(g, α) \ X_{g}) ∩ X. Let the neighbourhood base of g ∈ X in ρ be

B_{g} = {V (g, α) : α < κ}.

First we note that B = ⋃{B_{x} : x ∈ X} is a base of a topology because

(†) ∀α < κ ∀h ∈ V (g, α) \ {g} U(h, Δ(g, h) + 1) ∩ X ⊂ V (g, α).

12 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

Since V (g, α) ∩ Y = U(g, α) ∩ Y and V (h, α) ∩ Z = U(h, α) ∩ Z for each g ∈ Y, h ∈ Z and α < κ we have that ⟨Y, τ⟩ = ⟨Y, ρ⟩ and ⟨Z, τ⟩ = ⟨Z, ρ⟩. Thus

nw(⟨X, ρ⟩) = nw(⟨Y, ρ⟩)+nw(⟨Z, ρ⟩) ≤ w(⟨Y, τ⟩)+w(⟨Z, τ⟩) ≤ w(⟨C_{λ}^{κ}⟩) = λ^{<κ}.

Obviously χ(⟨X, ρ⟩) = κ.

Finally we show that w(⟨X, ρ⟩) = λ^{κ}. This will follow if we show that B is an irreducible base for X, by lemma [5, 2.6]. We claim that {B_{x} : x ∈ X} is an irreducible decomposition of the base B (see definition [5, 2.3]). Since Y is discrete in ρ it is enough to show that if g ≠ h are from Z with g ∈ V (h, α) for some α < κ then V (h, α) ⊄ V (g, 0). Let δ = Δ(g, h). Then U(g, δ + 1) ⊂ V (h, α) by (†). Consider the element y of Y defined by the formulas y⌈δ + 1 = g⌈δ + 1 and y(δ + 1) = 0. Then y ∈ X_{g} and so y ∉ V (g, 0). On the other hand y ∈ U(g, δ + 1), so V (h, α) ⊄ V (g, 0).

Lemma 4.2. If X is any topological space and Z ⊂ X^{κ}, α < κ, f : α ⟶ X are such that

∀f′ ∈ Fin(κ \ α, X) ∃g ∈ Z f ∪ f′ ⊂ g, (‡)

then w(Z) ≥ w(X).

The proof is similar to that of lemma 3.7, so we omit it.

Let us recall that given a cardinal μ the Singular Cardinal Hypothesis (SCH) is said to hold below μ provided ν^{cf(ν)} = 2^{cf(ν)}ν^{+} for each singular cardinal ν ≤ μ. By [3, lemma 1.8.1], if μ is regular and SCH holds below μ, then log(μ^{+}) = min{ν : 2^{ν} ≥ μ^{+}} is also regular. It is well-known that the failure of SCH requires the consistency of large cardinals, therefore the assumption of our next lemma is quite reasonable. Also note that SCH trivially holds below ℵ_{ω}.

Theorem 4.3. If μ and log(μ^{+}) are both regular cardinals then there is a 0-dimensional T_{2} space Y such that χ(Y) nw(Y) ≤ μ, but Y is not the union of μ subspaces of weight μ.

Proof. Let ρ = log(μ^{+}). Applying theorem 4.1 for κ = ρ and λ = 2 we get a space X with χ(X) nw(X) = 2^{<ρ} ≤ μ < w(X). Let Y = X^{ρ} and consider a partition Y = ⋃_{α<ρ} Y_{α}. Since κ = ρ is regular, applying the technique of the standard proof of the Baire Category theorem we can see that the space C_{|X|}^{ρ} is not the union of ρ nowhere dense subspaces. Therefore there are ordinals α, ξ < ρ and a function f : α ⟶ X such that

∀f′ ∈ Fn(ρ \ α, X, ρ) ∃g ∈ Y_{ξ} f ∪ f′ ⊂ g. (‡’)

But (‡’) clearly implies (‡). So, by lemma 4.2, w(Y_{ξ}) ≥ w(X) > μ. Thus Y satisfies the requirements. The theorem is proved.

Problem 1. (ZFC) If μ is a cardinal, is there a space X such that χ(X) nw(X) = μ (or just nw(X) = μ), but X is not the union of < 2^{μ} many subspaces of weight < 2^{μ} (or just w(X) = 2^{μ})?

Remark . The simplest case of Problem 1 left open by theorem 4.1 is that 2^{ω} = ω_{2}, 2_{1}^{ω} = ω_{3} and μ = ω_{1}. Indeed, if X_{λ}^{κ} is the space constructed in theorem 4.1 and nw(X_{λ}^{κ}) = λ^{<κ} ≤ ω_{1}, then λ ≤ ω_{1} and κ ≤ ω. So w(X_{λ}^{κ}) = λ^{κ} ≤ ω_{1}^{ω} = ω_{2} < 2^{ω_{1}}.

References

[1] K. Ciesielski, On the netweigth of subspaces, Fund. Math., 117 (1983), no 1, 37–46.

[2] A. Hajnal, I. Juhász, Weakly separated subspaces and networks, Logic ’78, Studies in Logic, 97, 235–245.

[3] T. Jech, Set Theory, Academic Press, New York, 1978.

[4] I. Juhász, Cardinal Functions - Ten Years Later, Math. Center Tracts 123, Amsterdam, 1980.

[5] I. Juhász, L. Soukup, Z. Szentmiklóssy, What makes a space have large weight?, Topology and its Applications, 57 (1994) 271-285.

[6] S. Shelah, private communication

[7] M. G. Tkačenko, Chains and cardinals, Dokl. Akad. Nauk. SSSR 239 (l978) no 3, 546-549

[8] S. Todorčevič, Partition Problems in Topology, Contemporary Mathematics, vol 84, Providence, 1989

Mathematical Institute of the Hungarian Academy of Sciences

  E-mail address: juhasz@math-inst.hu

Mathematical Institute of the Hungarian Academy of Sciences

  E-mail address: soukup@math-inst.hu

Eötvös Loránd University, Department of Analysis