ON POSSIBLE NON-HOMEOMORPHIC SUBSTRUCTURES OF THE REAL LINE

P.D.WELCH

Abstract. We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. This requires large cardinals, and we obtain an exact consistency strength: Theorem 1 The following are equiconsistent:

(i) ZF C + ∃κ a Jónsson cardinal ;

(ii) ZF C+∃M a sufficiently elementary submodel of the universe of sets with ℝ_{M} not homeomorphic to ℝ. The reverse direction is a corollary to: Theorem 2 𝔠 is Jónsson ⟺ ∃M ≺ H(𝔠^{+})∃X_{M} hereditarily separable, hereditarily Lindelöf, T_{3} with X ≠ X_{M}.

We further consider the large cardinal consequences of the existence of a topological space with a proper substructure homeomorphic to Baire space.

1. Introduction

Tall in [8], and Junqueira & Tall [3] consider the question of the behaviour of a topological space X = ⟨X, T ⟩ when X ∈ M, where M is a submodel, not necessarily transitive, of the universe of all sets. The topology they consider for X_{M} is that generated by the open sets of T that happen to be in M. Thus X_{M} = ⟨X ∩M, T_{M}⟩ where T_{M} is that generated by {U ∩ M | U ∈ T ∩ M}. In general this will not coincide with the usual relative topology. We refer to this ⟨X_{M}, T_{M}⟩, somewhat incorrectly, as a “substructure” of ⟨X, T ⟩. (In fact, as they observe, it suffices to take M a submodel of the class of sets of hereditary cardinality θ, H(θ), where ⟨X, T ⟩ ∈ H(θ).) We refer the reader to these papers for further discussion and motivation of these ideas. The purpose of this note is to be brief, and state a theorem that characterises Jónsson cardinals:

Definition 1.1. (i) A Jónsson algebra A = ⟨A, ⟨f_{n}⟩_{n∈ω}⟩, is an algebra on A of finitary functions f_{n} : [A]^{n} ⟶ A that has no proper subalgebra of the same cardinality as that of A. (ii) A cardinal κ is a Jónsson cardinal if there is no Jónsson algebra A with κ ⊆ A.

In the first part of this note we prove the theorems of the abstract which involve this concept. The reader is assumed familiar with, or to have access to a copy of [6] and [8]. Our definitions and terminology are standard. In the final part of the paper we look at a question of Tall’s from [8] concerning the possibility of proper topological substructures being homeomorphic to the irrationals. The “large cardinal” theorem there, has a proof that presupposes some basic familiarity with core model theory.

Date: March 29, 2001 Draft.

1991 Mathematics Subject Classification. 03C62,03E55,06A05,03C55,54F05; Secondary 12L99.

Key words and phrases. real continuum, subspaces, Jónsson cardinals.

It is known that the existence of a Jónsson cardinal has mild large cardinal strength (Ramsey cardinals are Jónsson, and so their strength is less than that of a measurable cardinal). One may prove outright in ZF C that no ℵ_{n} (n < ω) is Jónsson. It is not known whether ℵ_{ω} may be Jónsson, although it is known it would require very large cardinals to render this consistent. The first candidate for the continuum 𝔠 to be Jónsson is ℵ_{ω_{1}}. (Simply because cf(𝔠) > ω and no regular cardinal below ℵ_{ω_{1}} is Jónsson by results of Shelah [7], Woodin, and Tryba [9].)

2. Jónsson cardinals and the continuum

The following (Theorem 2 of the abstract) should be compared to Theorem 4 of [8].

Theorem 2.1. 𝔠 is Jónsson ⟺ ∃M ≺ H(𝔠^{+})∃X_{M} hereditarily separable, hereditarily Lindelöf, T_{3} with X ≠ X_{M}

Proof: (⇒) Let 𝔠 be Jónsson. Then ∃M ≺ ⟨H_{𝔠^{+}}, ∈⟩ with 𝔠 ∈ M, |M ∩ 𝔠| = 𝔠, M ∩ 𝔠 ≠ 𝔠. That such an M with these properties exists is a standard fact about Jónsson cardinals, see e.g. [4] §8. As H_{𝔠^{+}} ⊧ ∃f : ℝ ⟷ 𝔠 ∈ On, it follows that ℝ ∩ M ≠ ℝ. However, as [3] show, for the case of first countable spaces X, we have that in fact X ∩ M = X_{M}. Hence ℝ_{M} ≠ ℝ. (⇐) Suppose 𝔠 is not Jónsson. It suffices by the proof of [8] Theorem 4, to show that [0, 1] ⊆ M. For this it is enough to show 𝔠 ⊆ M. H_{c^{+}} ⊧ “∃⟨𝔠, f_{n}⟩_{n<ω} a Jónsson algebra on 𝔠”. Hence M is a model of the above sentence. Let ⟨𝔠, g_{n}⟩_{n<ω} ∈ M be a witness to this. (It is unproblematic to assume that the field of the algebra is 𝔠 itself.) Then each g_{n} ∈ M, and, if g_{n} : [𝔠]^{n} ⟶ 𝔠, ⟨M ∩ 𝔠, g_{n} ↾ [M ∩ 𝔠]^{n}⟩_{n<ω} is a subalgebra of ⟨𝔠, g_{n}⟩_{n<ω}. By definition of ⟨g_{n}⟩ we have M ∩ 𝔠 = 𝔠.

Q.E.D.(Theorem 2.1)

Corollary 2.2. If 𝔠 is not Jónsson, and if X_{M} is an uncountable complete separable metric space, then X_{M} = X.

Corollary 2.3. If 𝔠 is not Jónsson, and X_{M} is homeomorphic to an uncountable Borel subspace of ℝ, then X_{M} = X.

In [8] already some anti-large cardinal assumption was used. In the former corollary the weaker hypothesis “𝔠 is not Jónsson” replaces “CH + ¬O^{#}” of [8] Theorem 21 a), and just “¬O^{#}” of Cor. 25., op.cit. The proofs are the same.

In [6] it is asked:

Question: What is a lower bound on the consistency strength of:

(∗) ∃2^{𝔠}-many non-homeomorphic ℝ_{M} of size 𝔠?

They show ([6] Theorems 16 and 8 respectively):

(i) Con(ZF C + ∃ a Ramsey cardinal ) ⇒ Con(ZF C + (∗)). (ii) ZF C + ∃ℝ_{M}(|ℝ_{M}| = 𝔠 ∧ ℝ_{M} is not homeomorphic to ℝ) ⊢ O^{#} exists.

We raise this latter lower bound to that of a Jónsson cardinal.

Theorem 2.4. ZF C + ∃ℝ_{M}(|ℝ_{M}| = 𝔠 ∧ ℝ_{M} is not homeomorphic to ℝ) ⊢ 𝔠 is a Jónsson cardinal

Proof: This is just the argument of (⇐) of Theorem 2.1. Q.E.D.

In their model of (i) in which (∗) holds, 𝔠 has been made equal to the Ramsey cardinal κ of the ground model by c.c.c. forcing. In such a model κ is Jónsson (see [1]). As Jónsson cardinals are equiconsistent with Ramseys, this shows that [6] were using exactly the right hypothesis. In their model 𝔠 is weakly inaccessible.

Their proof that there were at least 2^{𝔠} non-homeomorphic ℝ_{M} amongst their 2^{𝔠} many different submodels M, used Lavrentieff’s theorem on extending putative homeomorphisms to G_{δ}’s and a counting argument. We next observe that in their construction all the ℝ_{M} are non-homeomorphic - again using Lavrentieff, but using a further property of indiscernibles arising from Ramsey cardinals:

Definition 2.1. Let A = ⟨L_{κ}[A], A, R⃗, f⃗, … ⟩ be a first order structure with κ ⊆ |A|. I ⊆ κ is a good set of indiscernibles for A if for all γ ∈ I

(i) ⟨L_{γ}[A ∩ γ], ∈, A ∩ γ, R⃗ ∩ γ, f⃗ ↾ γ, … ⟩ ≺ A; (ii) I\γ is a set of indiscernibles for the structure ⟨L_{δ}[A], ∈, A, R⃗, f⃗, … ⟨ξ⟩_{ξ≤γ}⟩. Fact 2.1. Suppose κ is a Ramsey cardinal. Then for evey first order structure A as above there is a good sequence of indiscernibles of order type κ. (See, e.g. [2] Ch.16).

Using this and the notation of [6]’s Theorem 16, if I is decomposed into ⟨I_{α} | α < κ⟩ a sequence of mutually disjoint subsets each of size κ, then each M_{α} =_{df} {τ_{G} | τ is a ℙ name τ ∈ H(I_{α})} has a version of ℝ : ℝ_{M}. (We are taking here H(Y) to be defined as the Skolem Hull of Y in H(λ^{+}).) We have then:

Lemma 2.5. The collection of ℝ_{α} =_{df} ℝ_{M_{α}} are all pairwise non-homeomorphic.

Proof: As Kunen and Tall argue, by Lavrentieff’s theorem, if ℝ_{α} were homeomorphic with ℝ_{β} via a homoeomorphism g say, g could be extended to a homeomorphism between two G_{δ} subsets of ℝ; g̃ : G_{α} ⟶ G_{β}. But such a homeomorphism, and G_{α}, G_{β}, are essentially coded by a real r say. But any real of V [G] (where G is the generic for the c.c.c. forcing ℙ adding κ many reals) is added at some initial stage of the forcing. We may thus factor ℙ as ℙ_{0} ∗ ℙ_{1} with r being added by G_{0} =_{df} G ∩ ℙ_{0}. Let ṙ be a ℙ-name (and a ℙ_{0}-name) for r. But for all sufficiently large γ, the sets I_{γ} are good indiscernibles for ⟨L_{κ}[A], ∈, ℙ, ⊩_{ℙ}, ℙ_{0}, ⊩_{ℙ_{0}}, ṙ, ⟨ξ⟩_{ξ<γ}⟩ where ℙ_{0}, ṙ ∈ V_{γ} and A ⊆ κ is assumed to code up V_{κ} so that for any strong inaccessible η < κ we have V_{η} = L_{η}[A ∩ η]. They are thus easily seen to be good indiscernibles in V [G_{0}]. Now the argument finishes as in [6]: let f : κ ↔ ℝ be a bijection in V [G] named by a term in H(Ø); if i > γ is any indiscernible in I_{β}\I_{α}, we have f(i) ∈ ℝ_{β}\ℝ_{α}. But then g̃^{−1}(f(i)) ∈ ℝ_{α} and thus i ∈ M_{α} although i cannot be named by any term defined from the indiscernibles of I_{α}. Q.E.D.

To make 𝔠 singular and Jónsson requires the consistency of larger cardinals.

Theorem 2.6. Con(ZF C + 𝔠 > cf(𝔠) = λ + ∃ℝ_{M}(|ℝ_{M}| = 𝔠 ∧ ℝ_{M} ≠ ℝ) ⟺ Con(ZF C + ∃⟨κ_{i} | i < λ⟩ an increasing sequence of measurable cardinals, with κ_{0} > λ).

Proof: (Sketch) (⇒) The assumption implies again that 𝔠 is Jónsson, by the argument of Theorem 2.1, and that together with the cofinality hypothesis gives an inner model with the sequence of λ measurable cardinals by Koepke [5]. (⇐) Assuming we have measures U_{i} on κ_{i} find a sequence ⟨X_{i} | i < λ⟩ of measure one sets which are mutually coherently indiscernible, and use the argument of [6]: make 𝔠 = sup_{i}{κ_{i}} by c.c.c. forcing adding reals, and get 2^{𝔠} many models with different versions of ℝ_{M}. Q.E.D.

3. On the possibility of a topological space with a proper substructure homeomorphic to Baire space

In [8] an interesting question is raised:

Question: Can there be ⟨X, T ⟩ ∈ M ≺ H(λ^{+}) with X_{M} homeomorphic to the irrationals, but X_{M} ≠ X?

It is easy to see by the reasoning above, that if the question had a positive answer with λ = 𝔠 then 𝔠 would have to be Jónsson (and hence at least ℵ_{ω_{1}}). However the possibility exists that X may be of size λ > 𝔠, with |M| = 𝔠 and X_{M} is homeomorphic to Baire space. This would, prima facie only imply that ⟨λ, 𝔠⟩ ⟶ ⟨𝔠, < 𝔠⟩ - a Chang conjecture like property weaker than that of a Jónsson cardinal. We cannot answer the above question, but can show that large cardinals are still involved.

Theorem 3.1. Suppose the answer to the question is positive, and let X, T, λ, M witness this. Then there is an inner model with λ a 𝔠-Erdős cardinal.

Proof: We suppose there is no inner model with a measurable cardinal κ ≤ λ, for otherwise we have nothing to do: in the core model K below a measurable cardinal every V -cardinal \overline{λ} ≥ κ is a Ramsey cardinal, and hence certainly a 𝔠-Erdős cardinal! We show then that in this core model K λ satisfies the partition relation λ ⟶ (𝔠)_{2}^{<ω}. This suffices. Recall that λ ⟶ (𝔠)_{2}^{<ω} is defined to mean that for every function F : [λ]^{<ω} ⟶ 2 there is a set H ⊆ λ with |H| = 𝔠 and for each n < ω we have |F “[H]^{n}| = 1, that is H is homogeneous for F. We assume only a slight familiarity with the construction of core models. Such a core model is of the form K = L[E] where E is a sequence of extenders. The point is that, unlike for Gödel’s constructible universe L, we must have a failure of condensation. Suppose F ∈ K is a function as above, which is a counterexample to the partition property in K. We may assume that F has been chosen least in the natural ordering of H(λ^{+})^{K} and hence is a definable point in the substructure M, as is also the predicate E ↾ λ′ coding the whole model’s construction up to λ′, the latter a sufficiently large ordinal so that F ∈ L_{λ′}[E ↾ λ′]. As X_{M} is assumed homeomorphic to Baire space, M has size 𝔠. |M ∩ 𝔠| < 𝔠, as otherwise M would contain all of ℝ. Let σ : N ⟶ M be the inverse of the transitive collapse of M, with N transitive. Set σ(\overline{λ}, \overline{E}, \overline{F}) = λ, E, F. We then note On ∩ N ≥ 𝔠 and N ⊧ “\overline{λ} is the largest cardinal”. Hence \overline{λ} ≥ 𝔠. The point now is to note that L_{𝔠}[\overline{E}] ≠ K_{𝔠} =_{df} L_{𝔠}[E] (the core model constructed up to 𝔠). For if it were, as the least ordinal moved by σ, crit(σ) ≤ δ ≤ 𝔠 for some regular δ. We should then be able to define an iterable K-ultrafilter on an ordinal β < 𝔠, and hence an elementary j : K ⟶ K. This implies there is an inner model with a measurable cardinal contrary to our global hypothesis. Hence if \overline{K} =_{df} K^{N} then \overline{K}_{𝔠} ≠ K_{𝔠}. If we now perform the comparison iteration of K with \overline{K} we shall see that there is a “mouse” structure M of cardinality < 𝔠 that iterates past \overline{K}. If

C = {κ_{i}|κ_{i} is the critical point used at stage i in the comparison }

then standard arguments show that C is cub in 𝔠 and moreover forms an unbounded set of good indiscernibles for the structure ⟨\overline{K}\overline{λ}, ∈ \overline{F} ⟩. Let D = σ“C. Then D is a good set of indiscernibles of order type 𝔠 for ⟨K_{λ}, ∈ F ⟩. We now appeal to the Jensen Indiscernibles Lemma (see [2] 16.10), and use the uncountable cofinality of 𝔠, to claim that there is E ⊇ D with E ∈ K and elements of E good indiscernibles for the same structure. But this entails E being a homogeneous set for the function F. This contradicts our choice of F. Q.E.D.

References

[1] K.J.Devlin, Some weak versions of large cardinal axioms in Annals of Mathematical Logic, 5, 1973, 291-325.

[2] A.J.Dodd, The Core Model, London Mathematical Society Lecture Notes in Mathematics Series, 61, CUP, 1982.

[3] L.R.Junqueira & F.D.Tall, The topology of elementary submodels, in Topology & its Applications, 82, (1998), 239-266.

[4] A. Kanamori, The Higher Infinite, Perspectives in Mathematical Logic, 1994, Springer Verlag, Berlin.

[5] P. Koepke, Some applications of short core models, in Annals of Pure & Applied Logic 37, (1988), 179-204.

[6] K.Kunen & F.D.Tall, The real line in elementary submodels of set theory, in Journal of Symbolic Logic, 65, No.2, June 2000, 683-691.

[7] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, (1994), Oxford University Press, Oxford.

[8] F.D.Tall, If it looks and smells like the reals …, in Fundamenta Mathematicae, 163 No.1, (2000), 1-11.

[9] J. Tryba, On Jónsson cardinals of uncountable cofinality, in Israel Journal of Mathematics, 94, (1984), 315-324.

Dept. of Mathematics, University of Bristol, Bristol BS8 1TW, England

Institut für Formale Logik, Währingerstr 25, A-1090 Wien, Austria

E-mail address: welch@logic.univie.ac.at