SEQUENTIALLY LINEARLY LINDELÖF SPACES

MENACHEM KOJMAN AND VICTORIA LUBITCH

Abstract. A topological Hausdorff space X is sequentially linearly Lindelöf if for every uncountable regular cardinal κ ≤ w(X) and every A ⊆ X of cardinality κ there exists B ⊆ A of cardinality κ which converges to a point. We prove that the existence of a good (μ, λ)-scale for a singular cardinal μ of countable cofinality and a regular λ > μ implies the existence of a sequentially linearly Lindelöf space of cardinality λ and weight μ which is not Lindelöf.

Corollaries of the main result are: (1) it is consistent to have linearly Lindelöf non Lindelöf spaces below the continuum; (2) it is consistent to have a realcompact linearly Lindelöf non Lindelöf space below 2^{ℵ_{ω}}; (3) it is consistent to have a Dowker topology on ℵ_{ω+1} in which every subset of cardinality ℵ_{n}, n > 0, has a converging subset of the same cardinality; (4) the nonexistence of sequentially linearly Lindelöf non-Lindelöf spaces implies the consistency of large cardinals.

1. Introduction

It is well-known that compactness of a topological space X is equivalent to:

(1)  CAP_{κ}(X) for all infinite regular κ

where CAP_{κ}(X) is the statement that every subset of X of cardinality κ has a point of complete accumulation. Omitting κ = ℵ_{0} from compactness one gets the following weaker property

(2)  CAP_{κ}(X) for all regular κ > ℵ_{0}

known as linear Lindelöfness, because it is equivalent to the property that every open cover of X which is linearly ordered by inclusion has a countable subcover. The property of being linearly Lindelöf but not Lindelöf will be abbreviated by LLnL and LL will abbreviate linear Lindelöfness. Three LLnL spaces were shown to exist in ZFC and a fourth, realcompact space was constructed from an additional assumption [14, 2, 1, 12].

CAP_{κ}(X) holds trivially for all regular κ > w(X), thus (2) is equivalent to

(2′)  CAP_{κ}(X) for all regular ℵ_{0} < κ ≤ w(X)

If one strengthens compactness by replacing CAP_{κ}(X) with SCAP_{κ}(X), which means: “for every A ⊆ X of cardinality κ there exists B ⊆ A of cardinality κ that

1991 Mathematics Subject Classification. Primary: 54A20, 03E04, 54A35; Secondary: 03E55, 54G20, 03E45.

Key words and phrases. Linearly Lindelöf spaces, PCF-theory, realcompact spaces, complete accumulation, singular cardinals, square principle, inner models, large cardinals.

This research was supported by the Israel Science Foundation, grant No. 177/01-01.

converges to a point x ∈ X”, one gets the following property of “chain compactness”:

(3)  SCAP_{κ}(X) for all infinite regular κ

A Hausdorff space X is chain-compact if and only if it is scattered compact by [15]. A set A ⊆ X converges to x ∈ X if |A \ u| < |A| for all open u ∋ x. For a regular κ, the property SCAP_{κ}(X) is equivalent to “every sequence f : κ → X has a converging cofinal subsequence”.

Spaces in which SCAP_{κ}(X) holds for all k ≤ λ are called initially λ-chain compact and were considered in [23, 24].

In the present paper we consider the property which is obtained from compactness by applying both modifications above, or, equivalently, which is obtained from initial w(X)-chain compactness by omitting κ = ℵ_{0}:

(4)  SCAP_{κ}(X) for all regular ℵ_{0} < κ ≤ w(X)

Let us call a space X sequentially^{1} Linearly Lindelöf iff it satisfies (4).

Proposition 1. (1) for all regular κ,

SCAP_{κ}(X) ∧ CAP_{κ}(Y) ⇒ CAP_{κ}(X × Y)

(2) Suppose X is SLL and Y is LL with w(Y) ≤ w(X). Then X × Y is LL.

Proof. Let κ be a regular cardinal and assume SCAP_{κ}(X), CAP_{κ}(Y). Let A ⊆ X × Y be given with |A| = κ. Since κ is regular, there exists B ⊆ A with |B| = κ so that B is a constant function from X to Y, or B^{−1} := {(y, x) : (x, y) ∈ B} is a constant function from Y to X or B is a 1-1 function from X to Y. In either of the first two cases B has trivially a point of complete accumulation in X × Y. In the third case assume, by thinning B out and using SCAP_{κ}(X), that Pr_{X}[B] := {x : (∃y)[(x, y) ∈ B]} converges to x_{0} ∈ X, namely, | Pr_{X}[B] \ u| < κ for all open u ∋ x_{0}. By CAP_{κ}(Y), fix y_{0} ∈ Y, a point of complete accumulation of P r_{Y} [B]. Now (x_{0}, y_{0}) is clearly a point of complete accumulation of B, hence of A. This proves (1).

(2) follows immediately from (1). □

The Sorgenfrey line K is Lindelöf, hence LL, and the diagonal of K^{2} is a closed discrete uncountable subset of K^{2}. Hence CAP_{ℵ_{1}}(K^{2}) fails and K^{2} is not LL. From (2) above, the Sorgenfrey line is not SLL.

We shall need the following very simple

Fact 2. Suppose κ_{0}, κ_{1}, …, κ_{m} are cardinals, κ is a regular cardinal and \overline{t} = ⟨t_{α} : α < κ⟩ is a sequence in ∏_{n≤m}(κ_{n} + 1). Then there exists t ∈ ∏_{n≤m}(κ_{n} + 1) and a (cofinal) subsequence of \overline{t} that converges to t.

Proof. Successively thin out \overline{t} so that for each n ≤ m the sequence ⟨t_{α}(n) : α < κ⟩ is either constant or strictly increasing of order-type κ. □

1It would have been more appropriate to use the name chain linearly Lindelöf, but since linearly Lindelöf spaces are sometimes called “chain Lindelöf” in the literature, this would be confusing.

1.1. The results. The property of being SLL but not Lidelöf is abbreviated by SLLnL. We shall construct SLLnL spaces from PCF-theory principles called “good scales”. A slight variation in the construction provides a Dowker space X of cardinality ℵ_{ω+1} which satisfies SCAP_{κ}(X) for all ℵ_{0} < κ < ℵ_{ω}.

The first corollary of the construction is the consistent existence of SLLnL spaces below the continuum, which in particular settles the question of whether LLnL spaces can exist below the continuum. If one assumes the consistency of large cardinals, then the consistency of infinitely many different SLLnL spaces below the continuum follows. Finally, SLLnL spaces serve in proving the consistency of a realcompact LLnL space below 2^{ℵ_{ω}}. A realcompact LLnL topology on 2^{ℵ_{ω}} itself is known to follow from the assumption 2^{ℵ_{0}} = 2^{ℵ_{ω}} [1].

At the moment we do not know if SLLnL spaces exists just in ZFC. But we have meta-mathematical consequences of the nonexistence of SLLnL spaces. If there is no SLLnL space of cardinality ℵ_{ω+1} then there exists a strong cardinal in an inner model; if there is no SLLnL space at all, then for every n > 0 there is an inner model with n Woodin cardinals. These results indicate that the consistency of not having SLLnL spaces would necessarily require stronger assumptions than the consistency of ZFC. At the moment it is not known whether it is consistent (from any assumptions) that no SLLnL topology exists on ℵ_{ω+1}.

2. SLLnL spaces X from good pcf-scales

Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of uncountable regular cardinals. Their limit, μ, is a singular cardinal of countable cofinality. Let ∏_{n} κ_{n} = {f | f : ω → μ and (∀n)(f(n) < κ_{n})}. For f, g ∈ ∏_{n} κ_{n} and m ∈ ω write f <_{m} g iff f(n) < g(n) for all n > m and write f =_{m} g if f(n) = g(n) for all n > m. Let f <^{∗} g iff f <_{m} g for some m; let f =^{∗} g iff f =_{m} g for some m. The structure (∏_{n} κ_{n}, <^{∗}) is a quasi-ordered set. A sequence \overline{f} = ⟨f_{α} : α < λ⟩ ⊆ ∏_{n} κ_{n} is called <^{∗}-increasing if for all α < β < λ it holds that f_{α} <^{∗} f_{β} and a sequence \overline{f} = ⟨f_{α} : α < λ⟩ ⊆ ∏_{n} κ_{n} is called <^{∗}-cofinal in ∏_{n} κ_{n} if for all f ∈ ∏_{n} κ_{n} there exists α < λ so that f <^{∗} f_{α}.

Let μ be singular of countable cofinality and let λ > μ be regular. A (μ, λ)-scale is a pair (\overline{κ}, \overline{f}) where \overline{κ} = ⟨κ_{n} : n < ω⟩ is a strictly increasing sequence of uncountable regular cardinals with sup\{κ_{n} : n < ω\} = μ and \overline{f} = ⟨f_{α} : α < λ⟩ ⊆ ∏_{n} κ_{n} is <^{∗}-increasing and <^{∗}-cofinal in ∏_{n} κ_{n}.

A function g : ω → On is an exact upper bound (eub) of a <^{∗}-increasing sequence ⟨f_{α} : α < θ⟩ ⊆ On^{ω}, where θ is a limit ordinal, if f_{α} <^{∗} g for all α < θ and for all g′ <^{∗} g there is some α < θ so that g′ <^{∗} f_{α}. For example, if (\overline{κ}, \overline{f}) is a (κ, λ)-scale then the function g with g(n) = κ_{n} is an exact upper bound of \overline{f}. If g_{1} and g_{2} are exact upper bounds of a <^{∗}-increasing \overline{f} then g_{1} =^{∗} g_{2}.

We write f < g for f, g : ω → On if f(n) < g(n) for all n. A sequence ⟨h_{α} : α < δ⟩ is <-increasing if α < β < δ ⇒ h_{α} < h_{β}.

Lemma 3. Suppose ⟨f_{α} : α < δ⟩ ⊆ On^{ω} is a <^{∗}-increasing sequence, δ is a limit ordinal and cfδ = θ > ℵ_{0}. Then the following conditions are equivalent:

(1) there exists an eub g of \overline{f} so that cfg(n) = θ for all n;

(2) there exists a <-increasing sequence \overline{h} = ⟨h_{i} : i < θ⟩ ⊆ On^{ω} so that for all i < θ there is α < δ with h_{i} <^{∗} f_{α} and for all α < δ there is i < θ with

f_{α} <^{∗} h_{i};

(3) For every unbounded set C ⊆ α there is some m_{0} ∈ ω and an unbounded set A ⊆ C with otpA = θ so that f_{α}(n) <_{m_{0}} f_{β}(n) for all α < β in A.

Proof. (1)⇒(2): Suppose \overline{f} and θ are as above and g is an eub of \overline{f} with cfg(n) = θ for all n. For each n fix a sequence ⟨β_{i}^{n} : i < θ⟩, strictly increasing with supremum g(n). For i < θ define h_{i} : ω → On by h_{i}(n) = β_{i}^{n}. Thus i < j < θ implies h_{i} < h_{j}, and sup{h_{i} : i < θ} = g.

For every f that satisfies f <^{∗} g the set {n ∈ ω : f(n) < g(n)} is co-finite in ω, and for every n in this set there is some i < θ so that f(n) < β_{i}. Since θ is regular uncountable and \overline{h} increases in <, there is some fixed i < θ for which f_{α} <^{∗} h_{i}. Conversely, since g is an eub of \overline{f} and h_{i} < g for i < θ, there is some α < δ for which h_{i} <^{∗} f_{α}. This proves (2).

(2)⇒(3): Suppose \overline{h} is given as in (2) and that C ⊆ α is unbounded. For each i < θ let α(i) ∈ C be chosen such that h_{i} <^{∗} f_{α(i)} and so that i < j ⇒ α(i) < α(j). Let A = {α(i) : i < θ}. So otpA = θ and also A is unbounded in α. By thinning out \overline{h} we may assume that f_{α(i)} <^{∗} h_{i+1}. Let m(i) be so that h_{i} <_{m(i)} f_{α(i)} <_{m(i)} h_{i+1}. Since θ > ℵ_{0} is regular, we may assume, by shrinking A and re-enumerating it increasingly, that there is some fixed m_{0} ∈ ω for which m(i) = m_{0} for all i < θ. Suppose that α(i) < α(j) are in A. Then

f_{α(i)} <_{m_{0}} h_{i+1} ≤ h_{j} <_{m_{0}} f_{α(j)}

as required.

(3)⇒(1): Suppose A ⊆ δ is unbounded of order type θ and m_{0} ∈ ω satisfies α < β in A implies that f_{α} <_{m_{0}} f_{β}. Let g(n) = sup{f_{α}(n) : α ∈ A}. Now g is clearly an eub of \overline{f} and cfg(n) = θ for all n > m_{0}. To obtain this for all n, the values g(0), …, g(m_{0}) can be re-defined as θ. □

A <^{∗}-increasing \overline{f} = ⟨f_{α} : α < δ⟩ with cfδ = θ > ℵ_{0} that satisfies one (equivalently, all) of the conditions in Lemma 3 is called flat. Suppose (\overline{k}, \overline{f}) is a (μ, λ)-scale for a singular μ of countable cofinality and a regular λ > μ. An ordinal α < λ is called a “flat point in \overline{f}” if cfα > ℵ_{0} and \overline{f}↾α is flat. A (μ, λ)-scale is called good if for all α < λ with μ > cfα > ℵ_{0}, α is a flat point in \overline{f} and f_{α} is an eub of \overline{f}↾α.

Theorem 4. Suppose μ is singular of cofinality ω and λ > μ is regular. If there exists a good (μ, λ)-scale there exists a Tychonov, sequentially linearly Lindelöf non-Lindelöf space of cardinality λ and weight μ.

Proof. Suppose \overline{f} = ⟨f_{α} : α < λ⟩ ⊆ ∏_{n} κ_{n} is a good (μ, λ)-scale. Define

(5)  X = {f ∈ ∏_{n}(κ_{n} + 1) : (∃α < λ)(f =^{∗} f_{α})}

The topology on X is the topology inherited from the usual product topology on ∏_{n}(κ_{n} + 1).

The cardinality of X is clearly λ and the weight of X is μ. The space X is also clearly Tychonov.

Lemma 5. X is not Lindelöf.

Proof. Let U = {u_{n,α} : α < κ_{n}} where u_{n,α} := {f ∈ X : f(n) < α}. Clearly, |U| = μ. For every f ∈ X there exists f_{α} ∈ \overline{f} so that f =^{∗} f_{α}, thus for some n it holds that f(n) = f_{α}(n) < κ_{n}. Thus f ∈ u_{n,f(n)+1}. This shows that U is a cover of X.

To see that U has no smaller subcover, fix V ⊆ U with |V| = θ for some θ < μ. Let a function g ∈ ∏_{n} κ_{n} be defined as follows:

g(n) = \begin{cases}0 & if κ_{n} ≤ θ \\ sup\{α : u_{n,α} ∈ V\} & if κ_{n} > θ \end{cases}

Since sup_{n} κ_{n} = μ and θ < μ, there is some m so that for all n > m it holds that θ < κ_{n}. For n > m, since κ_{n} is regular and |V| = θ < κ_{n}, it holds that g(n) < κ_{n}. Since \overline{f} is a scale, there exists some α < λ and m_{0} so that g(n) < f_{α}(n) for all n > m_{0}. Let f ∈ X be defined by

f(n) = \begin{cases}κ_{n} & if n ≤ max\{k, m_{0}\} \\ f_{α}(n) & if n > k \end{cases}

The function f is indeed an element of X since f ∈ ∏_{n}(κ_{n} + 1) and f =^{∗} f_{α}. Now one can check that f ∉ ⋃ V: Suppose u_{n,α} ∈ V. If n ≤ max{k, m_{0}} then f(n) = κ_{n} > α; and if n > max{k, m_{0}} then f(n) = f_{α}(n) > g(n) = sup{β : u_{n,β} ∈ V} ≥ α. So f ∉ u_{α,n}. This shows that V is not a cover, and therefore that X is not Lindelöf. □

Lemma 6. X is sequentially linearly lindelöf. In fact, for every regular ℵ_{0} < κ < μ and A ⊆ X with |A| = κ there exists B ⊆ A with |B| = κ which converges in the box topology on X.

Proof. Let A ∈ [X]^{κ} be given for some regular uncountable κ < μ. For every g ∈ A fix α(g) < λ and m(g) < ω so that g =_{m(g)} f_{α(f)}. By shrinking A we may assume that m(g) is some fixed m_{0} for all g ∈ A, and that g ↦ α(g) is either constant or 1-1.

In the first case there is some fixed α_{0} < λ so that g =_{m_{0}} f_{α_{0}} for all g ∈ A. Thus, |{g↾{0, 1, …, m_{0}} : g ∈ A}| = κ and by Fact 2 there exists some t ∈ ∏_{i≤m_{0}}(κ_{n} + 1) and B ∈ [A]^{κ} so that {g↾{0, 1, …, m_{0}} : g ∈ B} converges to t. Now define

f(n) = \begin{cases}t(n) & if 1 < n ≤ m_{0} \\ f_{α_{0}}(n) & if n > m_{0} \end{cases}

The function f satisfies f =_{m_{0}} f_{α_{0}}, hence f ∈ X. Also, B converges to f in the box topology on X.

In the second case g ↦ α(g) is 1-1, and, by shrinking A further, it may be assumed that the set {α(g) : g ∈ A} has order-type κ. Let ⟨α_{i} : i < κ⟩ be the increasing enumeration of {α(g) : g ∈ A} and let us denote by g_{i} the unique g ∈ A for which α(g) = α_{i}.

Letting δ = sup\{α_{i} : i < κ\} we have that cfδ = κ and δ < μ. Since (\overline{κ}, \overline{f}) is good, δ is a flat point in \overline{f}. Therefore, f_{δ} is an exact upper bound of \overline{f}↾δ and for all sufficiently large n it holds that cff_{δ}(n) = κ. Also, since {α_{i} : i < κ} is unbounded in δ, clause (3) in Lemma 3 implies that by shrinking A further, we can find some m_{1} so that for all n > m_{1} the sequence f_{α_{i}}(n) is strictly increasing with limit f_{δ}(n). Let m = max{m_{0}, m_{1}}.

Consider the sequence ⟨g_{i}↾{0, 1, …, m} : i < κ}. Further shrinking gives, by Fact 2, that this sequence converges to some t ∈ ∏_{n≤m}(κ_{n} + 1). Let f ∈ ∏_{n} κ_{n} be defined by:

f(n) = \begin{cases}t(n) & if n ≤ m \\ f_{δ}(n) & if n > m \end{cases}

Now{g_{i} : i < κ} converges to f in the box topology on X, and f =_{m} f_{δ}. Thus f ∈ X and the proof is complete. □

□

2.1. A Dowker space X with SCAP_{κ}(X) for all ℵ_{0} < κ < w(X). In [7] it was shows that the M. E. Rudin Dowker space X^{R} [16] contains a closed and cofinal Dowker subspace X^{D} of cardinality ℵ_{ω+1} which is defined inside X^{R} by means of an (ℵ_{ω}, ℵ_{ω+1})-scale (\overline{κ}, \overline{f}) = ⟨f_{α} : α < ℵ_{ω+1}⟩. Recall that

(6)  X^{R} = {g ∈ ∏_{n}(κ_{n} + 1) : (∃m)(∀n)[ℵ_{0} < cfg(n) < ℵ_{m}]}

and is Dowker under the box topology. The space X^{D} is defined by:

(7)  X^{D} = {g ∈ X^{R} : (∃α < ℵ_{ω+1})(g =^{∗} f_{α})}

If one uses a good (ℵ_{ω}, ℵ_{ω+1})-scale in this construction, than the resulting space satisfies additional properties:

Theorem 7. Suppose that the (\overline{κ}, \overline{f}) is a good (ℵ_{ω}, ℵ_{ω+1})-scale. Then X^{D} in (7) is Dowker and satisfies SCAP_{κ}(X) for all ℵ_{0} < κ < ℵ_{ω}.

Proof. The space X^{D} is Dowker by [7]. Let X be the space defined in (5) from (\overline{κ}, \overline{f}) and observe that X^{D} = X^{R} ∩ X. Therefore, By Lemma 6, for every A ⊆ X^{D} with |A| = κ, ℵ_{0} < κ < ℵ_{ω}, there exists B ⊆ A of cardinality κ which converges to some g ∈ X in the box topology. The function g ∈ X is found in the proof of Lemma 6 so that g = t ∪ g↾(m_{0}, ω), where t ∈ ∏_{n≤m_{0}}(κ_{n} + 1) is chosen by applying Fact 2. By the proof of Fact 2, since cfg(n) > ℵ_{0} for all g ∈ A ⊆ X^{D}, cft(n) > ℵ_{0} for all n ≤ m_{0}. The function g↾(m_{0}, ω) is either equal to f↾(m_{0}, ω) for some f ∈ A ⊆ X^{D} or else has values of cofinality κ only — according to whether g is constructed in the first or the second case in the proof. Therefore there exists some m so that ℵ_{0} < cfg(n) < ℵ_{m} for all n < ω, hence g ∈ X^{R}. Since g ∈ X, it follows that g ∈ X^{D}. □

3. The consistency of SLLnL spaces

Suppose that μ is a singular cardinal of countable cofinality. A famous theorem of PCF-theory is the existence of a (μ, μ^{+})-scale. The existence of a good (μ, μ^{+})- scale, however, is not a theorem of ZFC — it is consistent, from a supercompact cardinal, that there is no good (ℵ_{ω}, ℵ_{ω+1})-scale [22, 5] (for a proof from a larger large cardinal the model in [13] suffices, since Chang’s conjecture for (ℵ_{ω+1}, ℵ_{ω}) easily contradicts a good (ℵ_{ω}, ℵ_{ω+1})-scale).

Definition 8. Suppose that μ is a singular cardinal of countable cofinality. Let pp_{J_{bd}}(μ) denote the set of all regular λ > μ for which there exists a (μ, λ)-scale^{2}.

The following summarizes the relevant facts about pp_{J_{bd}} μ:

Proposition 9. For every singular μ of countable cofinality:

(1) pp_{J_{bd}} μ is an interval of regular cardinals which contains μ^{+} [19, 3, 9].

(2) For every element λ ∈ pp_{J_{bd}} μ except, maybe, the largest element, there exists a good (μ, λ)-scale [19].

(3) For every α < ω_{1} it is consistent, relative to large cardinal axioms, that pp_{J_{bd}} ℵ_{ω} contains all regular cardinals λ ∈ [ℵ_{ω+1}, ℵ_{ω+α}) [21].

(4) it is consistent (if a supercompact cardinal is consistent) that there is no good (ℵ_{ω}, ℵ_{ω+1})-scale [22, 4].

(5) □_{μ}, and even the weaker principle μ^{+} ∈ I[μ^{+}], implies the existence of a good (μ, μ^{+}) scale [19, 4].

By (1),(2) and (5) above, a necessary condition for the nonexistence of a good (μ, μ^{+})-scale is that there is no (μ, μ^{++})-scale and that □_{μ} fails.

Theorem 10. (1) (Jensen) If □_{μ} fails for some singular μ with countable co-

finality, then there is an inner model with a strong cardinal [10, 11].

(2) (Schimmerling, Steel, Zeeman) If □_{μ} fails for a singular μ so that (∀κ < μ)(κ^{ℵ_{0}} < μ), then for each n > 0 there exists an inner model with n Woodin cardinals (Corollary 5 in [18]; see also the references therein).

It follows that the statement “there is no good (ℵ_{ω}, ℵ_{ω+1})-scale” has consistency strength of at least a strong cardinal, and that the statement “there are no good (μ, μ^{+})-scales for any singular μ of cofinality ℵ_{0}” has consistency strength of at least n Woodin cardinals for each n (consider the first μ of cofinality ω which satisfies (∀κ < μ)(κ^{ℵ_{0}} < μ)).

By Theorem 4 we have:

Theorem 11. If there is no SLLnL space then for all n there is an inner model with n Woodin cardinals. If there is no SLLnL space of cardinality ℵ_{ω+1} then there is an inner model with a strong cardinal.

Theorem 12. For every ordinal α < ω_{1} it is consistent, relative to large cardinal axioms, that:

(1) For every β < α there is SLLnL topology on ℵ_{ω+β+1}.

(2) For every β < α there is SLLnL topology on ℵ_{ω+β+1} and ℵ_{ω+α} < 2^{ℵ_{0}}.

Proof. Start with a model in which max pp_{J_{bd}} ℵ_{ω} > ℵ_{ω+α} [21]. By Proposition 9 there is a good ℵ_{ω+β+1}-scale for each β < α. By Theorem 4 there exists a SLLnL space of cardinality ℵ_{ω+β+1}. This proves (1)

To prove (2) add ℵ_{ω+α+1} Cohen reals to obtain ℵ_{ω+α} < 2^{ℵ_{0}}. Since CCC forcing does not change pp_{J_{bd}} ℵ_{ω}, the spaces constructed above exist also in the forcing extension. □

2The notation pp_{J_{bd}} μ is not self-explanatory in this context, but it is a standard PCF-theory notation that we do not wish to change.

4. Realcompact LLnL spaces below 2^{ℵ_{ω}}

Every Lindelöf space is realcompact ([8], Theorem 3.11.12), so one may ask if the requirement of realcompactness makes a LL space Lindelöf. In [1] a realcompact LLnL space was constructed on 2^{ℵ_{ω}} from the assumption 2^{ℵ_{ω}} = 2^{ℵ_{0}}. In this section we use SLLnL spaces to construct a realcompact LLnL space below 2^{ℵ_{ω}}.

Recall that a Bernstein set is a subset of ℝ which meets every closed uncountable subset of ℝ. It is well known or else it is an easy exercise in diagonalization of length 2^{ℵ_{0}} that:

Fact 13. ℝ is the union of 2^{ℵ_{0}} pairwise disjoint Bernstein sets.

Theorem 14. If there exists a SLLnL topology on 2^{ℵ_{0}} there exists a realcompact LLnL topology which extends the usual topology on ℝ.

Proof. Suppose that X is SLLnL of cardinality 2^{ℵ_{0}}. Fix, by Fact 13, a partition ℝ = ⋃_{x∈X} S(x) of ℝ in which each S(x) is a Bernstein set. Let F : ℝ → X be defined by F (r) = x ⟺ r ∈ S(x). Now let H ⊆ ℝ × X be the graph of F with the induced topology from the product topology on ℝ × X:

H = {⟨r, F (r)⟩ : r ∈ ℝ}

Clearly, Pr_{R} ↾H is a continuous 1-1 function from H onto ℝ and Pr_{X} ↾X is a continuous surjection. Since there exists a continuous 1-1 function from H onto ℝ, H is hereditarily realcompact (by, e.g., [8] Theorem 3.11.14). Since X is a continuous non-Lindelöf image of H, H is not Lindelöf. The topology on H can be regarded as an extension of the usual topology on ℝ via Pr_{R} ↾H.

Let us see that H is linearly Lindelöf. Let κ > ℵ_{0} be regular and assume that C ⊆ H and |C| = κ. C = F ↾A for some A ⊆ ℝ with |A| = κ. Let a point x_{0} ∈ X be chosen as follows: By shrinking A we may assume that F ↾A is either 1-1 or constant. If F ↾A is 1-1 we may assume by shrinking A further that {F (r) : r ∈ A} converges to some x ∈ X and we let x_{0} = x. In the other case let x_{0} be the constant value of F ↾A. Let D ⊆ R be the set of all complete accumulation points of A. Since |A| = κ and κ > ℵ_{0} is regular, D is an uncountable closed subset of ℝ. Therefore, since S(x_{0}) is a Bernstein set, there exists some r_{0} ∈ D ∩ S(x_{0}). Now (r_{0}, x_{0}) ∈ H and is a point of complete accumulation of C. □

Corollary 15. It is consistent that 2^{ℵ_{ω}} > ℵ_{ω+1} and that there exists a realcompact LLnL space of cardinality ℵ_{ω+1}.

Proof. Start with a model of V = L and add λ Cohen subsets to ω_{1} for some λ > ℵ_{ω+1}. Since this forcing is ω_{1}-complete, no new countable subsets are added, and the good (ℵ_{ω}, ℵ_{ω+1})-scale from L is preserved. Then add ℵ_{ω+1} Cohen subsets to ω_{0}. In the resulting model 2^{ℵ_{0}} = ℵ_{ω+1} < 2^{ℵ_{1}} = λ ≤ 2^{ℵ_{ω}}, and a good (ℵ_{ω}, ℵ_{ω+1})- scale exists. By Theorem 4 there exists a SLLnL topology on ℵ_{ω+1} and by Theorem 14 there exists a realcompact LLnL topology on ℵ_{ω+1} in this model. □

Similarly one can get the consistency of a realcompact LLnL topology on ℵ_{ω+α+1} = 2^{ℵ_{0}} < 2^{ℵ_{ω}} for an arbitrary 0 < α < ω_{1} — from a large cardinal assumption.

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Ben-Gurion University of the Negev, Beer-Sheva, Israel