https://doi.org/10.1007/s00605-021-01582-0
Several results on compact metrizable spaces in ZF
Kyriakos Keremedis1 ·Eleftherios Tachtsis2 ·Eliza Wajch3
Received: 1 October 2020 / Accepted: 10 June 2021 / Published online: 29 June 2021
© The Author(s) 2021
Abstract
In the absence of the axiom of choice, the set-theoretic status of many natural state- ments about metrizable compact spaces is investigated. Some of the statements are provable inZF, some are shown to be independent ofZF. For independence results, distinct models ofZFand permutation models ofZFAwith transfer theorems of Pin- cus are applied. New symmetric models ofZFare constructed in each of which the power set ofRis well-orderable, the Continuum Hypothesis is satisfied but a denumer- able family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube[0,1]R.
Keywords Weak forms of the Axiom of Choice·Metrizable space·Totally bounded metric·Compact space·Permutation model·Symmetric model
Mathematics Subject Classification 03E25·03E35·54A35·54E35·54D30
Communicated by S.-D. Friedman.
B
Eliza Wajcheliza.wajch@gmail.com Kyriakos Keremedis kker@aegean.gr Eleftherios Tachtsis ltah@aegean.gr
1 Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece 2 Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean,
Karlovassi, Samos 83200, Greece
3 Institute of Mathematics, Faculty of Exact and Natural Sciences, Siedlce University of Natural Sciences and Humanities, ul. 3 Maja 54, 08-110 Siedlce, Poland
1 Preliminaries
1.1 The set-theoretic framework
In this paper, the intended context for reasoning and statements of theorems is the Zermelo–Fraenkel set theoryZFwithout the axiom of choice AC. The system ZF+ACis denoted byZFC. We recommend [32,33] as a good introduction toZF.
To stress the fact that a result is proved inZFor inZF+A(whereAis a statement independent ofZF), we shall write at the beginning of the statements of the theorems and propositions (ZF) or (ZF+A), respectively. Apart from models ofZF, we refer to some models ofZFA(or ZF0in [15]), that is, we refer also toZFwith an infinite set of atoms (see [15,20,21]). Our theorems proved here inZFare also provable in ZFA; however, we also mention some theorems ofZFthat are not theorems ofZFA.
Awell-ordered cardinal numberis an initial ordinal number, i.e., an ordinal which is not equipotent to any of its elements. Every well-orderable set is equipotent to a unique well-ordered cardinal number, called the cardinality of the well-orderable set.
By transfinite recursion over ordinalsα, we define:
ω0=ω(the set of all finite ordinal numbers);
ωα+1=H(ωα);
ωα =sup{ωβ :β < α}
=
{ωβ :β < α}
ifαis a non-zero limit ordinal, where, for a setA,H(A)is the Hartogs’ number ofA, i.e., the least ordinalαwhich is not equipotent to a subset of A. For each ordinal numberα,ωα is an infinite well- ordered cardinal number and, as it is customary, it is denoted byℵα. One usually uses ℵα when referring to the cardinality of an infinite well-orderable set, andωα when referring to the order-type of an infinite well-ordered set. Every well-ordered cardinal number is either a finite ordinal number or anℵαfor some ordinalα.
As usual, ifn ∈ ω, thenn+1 = n ∪ {n}. Members of the setN = ω\{0}are called natural numbers. The power set of a setXis denoted byP(X). A setXis called countableifXis equipotent to a subset ofω. A setXis calleduncountableifXis not countable. A setXisfiniteifXis equipotent to an element ofω. Aninfiniteset is a set which is not finite. An infinite countable set is calleddenumerable. IfXis a set andκ is a non-zero well-ordered cardinal number, then[X]κis the family of all subsets of Xequipotent toκ,[X]≤κis the collection of all subsets ofXequipotent to subsets of κ, and[X]<κ is the family of all subsets ofXequipotent to a (well-ordered) cardinal number inκ.
For setsXandY,
– |X| ≤ |Y|means thatX is equipotent to a subset ofY; – |X| = |Y|means thatXis equipotent toY; and – |X|<|Y|means that|X| ≤ |Y|and|X| = |Y|.
The set of all real numbers is denoted byRand, if it is not stated otherwise,R and every subspace ofRare considered with the usual topology and with the metric induced by the standard absolute value onR.
1.2 Notation and basic definitions
In this subsection, we establish notation and recall several basic definitions.
Let X = X,d be a metric space. The d-ball with centre x ∈ X and radius r∈(0,+∞)is the set
Bd(x,r)= {y∈ X :d(x,y) <r}.
The collection
τ(d)= {V ⊆X :(∀x∈V)(∃ε >0)Bd(x, ε)⊆V}
is thetopology on X induced by d. For a setA⊆X, letδd(A)=0 ifA= ∅, and let δd(A)=sup{d(x,y):x,y∈ A}if A= ∅. Thenδd(A)is thediameterofAinX.
Definition 1 LetX= X,dbe a metric space.
(i) Given a real numberε >0, a subsetDofXis calledε-denseor anε-netinXif X=
x∈DBd(x, ε).
(ii) Xis calledtotally boundedif, for every real numberε >0, there exists a finite ε-net inX.
(iii) Xis calledstrongly totally boundedif it admits a sequence(Dn)n∈Nsuch that, for everyn∈N,Dnis a finite1n-net inX.
(iv) (Cf. [24].)dis calledstrongly totally boundedifXis strongly totally bounded.
Remark 1 Every strongly totally bounded metric space is evidently totally bounded.
However, it was shown in [24, Proposition 8] that the sentence “Every totally bounded metric space is strongly totally bounded” is not a theorem ofZF.
Definition 2 LetX= X, τbe a topological space and letY ⊆X. Suppose thatBis a base ofX.
(i) The closure ofY inXis denoted by clτ(Y)or clX(Y).
(ii) τ|Y = {U∩Y :U ∈τ}.Y= Y, τ|Yis the subspace ofXwith the underlying setY.
(iii) BY = {U∩Y :U ∈B}.
Clearly, in Definition2(iii),BY is a base ofY. In Sect.5, it is shown thatBY need not be equipotent to a subset ofB.
In the sequel, boldface letters will denote metric or topological spaces (called spaces in abbreviation) and lightface letters will denote their underlying sets.
Definition 3 A collectionU of subsets of a spaceXis called:
(i) locally finiteif every point ofXhas a neighbourhood meeting only finitely many members ofU;
(ii) point-finiteif every point ofXbelongs to at most finitely many members ofU; (iii) σ-locally finite(respectively,σ-point-finite) ifU is a countable union of locally
finite (respectively, point-finite) subfamilies.
Definition 4 A spaceXis called:
(i) first-countableif every point ofX has a countable base of neighbourhoods;
(ii) second-countableifXhas a countable base.
Given a collection{Xj : j ∈ J}of sets, for everyi ∈ J, we denote byπi the projectionπi :
j∈J Xj → Xi defined byπi(x) = x(i)for eachx ∈
j∈J Xj. Ifτj is a topology onXj, thenX=
j∈JXj denotes the Tychonoff product of the topological spaces Xj = Xj, τj with j ∈ J. IfXj = Xfor every j ∈ J, then XJ =
j∈JXj. As in [8], for an infinite set J and the unit interval[0,1]ofR, the cube[0,1]J is called the Tychonoff cube. If J is denumerable, then the Tychonoff cube[0,1]J is called theHilbert cube. In [12], all Tychonoff cubes are called Hilbert cubes. In [42], Tychonoff cubes are called cubes.
We recall that if
j∈JXj = ∅, then it is said that the family{Xj : j ∈ J}has a choice function, and every element of
j∈JXj is called achoice function of the family {Xj : j ∈ J}. A multiple choice functionof {Xj : j ∈ J}is a function
f ∈
j∈JP(Xj)such that, for every j∈ J, f(j)is a non-empty finite subset ofXj. A set f is called apartial(multiple)choice functionof{Xj : j ∈ J}if there exists an infinite subsetI of J such that f is a (multiple) choice function of{Xj : j ∈ I}.
Given a non-indexed familyA, we treatAas an indexed familyA= {x :x∈A}to speak about a (partial) choice function and a (partial) multiple choice function ofA.
Let{Xj : j ∈ J}be a disjoint family of sets, that is,Xi∩Xj = ∅for each pairi,j of distinct elements of J. Ifτj is a topology onXj for every j ∈ J, then
j∈JXj
denotes the direct sum of the spacesXj = Xj, τjwith j ∈ J.
Definition 5 (Cf. [2,26,34].)
(i) A spaceXis said to beLoeb(respectively,weakly Loeb) if the family of all non- empty closed subsets ofXhas a choice function (respectively, a multiple choice function).
(ii) IfXis a (weakly) Loeb space, then every (multiple) choice function of the family of all non-empty closed subsets ofXis called a (weak)Loeb functionofX.
Other topological notions used in this article but not defined here are standard. They can be found, for instance, in [8,42].
Definition 6 A setXis called:
(i) acuf setifXis expressible as a countable union of finite sets (cf. [5,6,19] and [16, Form 419]);
(ii) Dedekind-finiteif X is not equipotent to a proper subset of itself (cf. [15, Note 94], [12, Definition 4.1] and [20, Definition 2.6]);Dedekind-infiniteif X is not Dedekind-finite (equivalently, if there exists an injection f : ω → X) (cf. [15, Note 94] and [12, Definition 2.13]);
(iii) amorphousifXis infinite and there does not exist a partition ofXinto two infinite sets (cf. [15, Note 57], [20, p. 52] and [12, E. 11 in Section 4.1]).
Definition 7 (Cf. [31].) A topological spaceX, τis called acuf spaceif Xis a cuf set.
1.3 The list of weaker forms of AC
In this subsection, for readers’ convenience, we define and denote most of the weaker forms ofACused directly in this paper. If a form is not defined in the forthcoming sections, its definition can be found in this subsection. For the known forms given in [15,16] or [12], we quote in their statements the form number under which they are recorded in [15] (or in [16] if they do not appear in [15]) and, if possible, we refer to their definitions in [12].
Definition 8 1. ACf i n([15, Form 62]): Every non-empty family of non-empty finite sets has a choice function.
2. ACW O([15, Form 60]): Every non-empty family of non-empty well-orderable sets has a choice function.
3. CAC([15, Form 8], [12, Definition 2.5]): Every denumerable family of non-empty sets has a choice function.
4. CAC(R)([15, Form 94], [12, Definition 2.9(1)]): Every denumerable family of non-empty subsets ofRhas a choice function.
5. CACω(R)(Cf. [29]): For every familyA= {An : n ∈ ω}such that, for every n∈ωand allx,y∈ An,∅ = An⊆P(ω)\{∅}andxy∈ [ω]<ω(denotes the operation of symmetric difference between sets), there exists a choice function of A.
6. IDI([15, Form 9], [12, Definition 2.13(ii)]): Every Dedekind-finite set is finite.
7. IDI(R)([15, Form 13], [12, Definition 2.13(2)]): Every Dedekind-finite subset of Ris finite.
8. WoAm([15, Form 133]): Every set is either well-orderable or has an amorphous subset.
9. Part(R): Every partition ofRis of size≤ |R|.
10. WO(R)([15, Form 79]):Ris well-orderable.
11. WO(P(R))([15, Form 130]):P(R)is well-orderable.
12. CACf i n ([15, Form 10], [12, Definition 2.9(3)]): Every denumerable family of non-empty finite sets has a choice function.
13. For a fixedn ∈ω\{0,1},CACn([15, Form 288(n)]): Every denumerable family ofn-element sets has a choice function.
14. CACW O: Every denumerable family of non-empty well-orderable sets has a choice function.
15. CMC([15, Form 126], [12, Definition 2.10]): Every denumerable family of non- empty sets has a multiple choice function.
16. CMCω ([15, Form 350]): Every denumerable family of denumerable sets has a multiple choice function.
17. CUC([15, Form 31], [12, Definition 3.2(1)]): Every countable union of countable sets is countable.
18. CUCf i n(Form [10 A] of [15], [12, Definition 3.2(3)]): Every countable union of finite sets is countable.
19. UT(ℵ0,cu f,cu f)([16, Form 419]): Every countable union of cuf sets is a cuf set.
(Cf. also [6].)
20. UT(ℵ0,ℵ0,cu f)([16, Form 420]): Every countable union of countable sets is a cuf set. (Cf. also [6].)
21. BPI([15, Form 14], [12, Definition 2.15(1)]): Every Boolean algebra has a prime ideal.
22. DC([15, Form 43], [12, Definition 2.11(1)]): For every non-empty setXand every binary relationρonX if, for eachx ∈ X there existsy∈ X such thatxρy, then there exists a sequence(xn)n∈Nof points ofX such thatxnρxn+1for eachn ∈N.
Remark 2 The following are well-known facts inZF:
(i) CACf i n andCUCf i n are both equivalent to the sentence: Every infinite well- ordered family of non-empty finite sets has a partial choice function (see Form [10 O] of [15] and [12, Diagram 3.4, p. 23]). Moreover,CACf i nis equivalent to Form [10 E] of [15], that is, to the sentence: Every denumerable family of non-empty finite sets has a partial choice function. It is known thatIDIimpliesCACf i n and this implication is not reversible inZF(cf. [12, pp. 324–324]).
(ii) CACis equivalent to the sentence: Every denumerable family of non-empty sets has a partial choice function (see Form [8 A] of [15]).
(iii) BPIis equivalent to the statement that all products of compact Hausdorff spaces are compact (see Form [14 J] of [15] and [12, Theorem 4.37]).
(iv) CMCω is equivalent to the following sentence: Every denumerable family of denumerable sets has a multiple choice function.
Remark 3 (a) It was proved in [19] that the following implications are true inZFand none of the implications are reversible inZF:
CMC→UT(ℵ0,cu f,cu f)→CMCω →vDCP(ℵ0),
wherevDCP(ℵ0)is van Douwen’s choice principle: “Every denumerable family {An,≤n :n ∈ω}of linearly ordered sets, each of which is order-isomorphic to the setZ,≤of integers with the standard linear order≤, has a choice function”
(cf. [7], [12, p, 79], [15, Form 119]).
(b) Clearly, UT(ℵ0,cu f,cu f) implies UT(ℵ0,ℵ0,cu f). In [6, proof to Theo- rem 3.3] a model of ZFA was shown in which UT(ℵ0,ℵ0,cu f) is true and UT(ℵ0,cu f,cu f)is false.
(c) It was proved in [31] that the following equivalences hold inZF:
(i) UT(ℵ0,cu f,cu f)is equivalent to the sentence: Every countable product of one-point Hausdorff compactifications of infinite discrete cuf spaces is metriz- able (equivalently, first-countable).
(ii) UT(ℵ0,ℵ0,cu f)is equivalent to the sentence: Every countable product of one-point Hausdorff compactifications of denumerable discrete spaces is metrizable (equivalently, first-countable).
Let us pass to definitions of forms concerning metric and metrizable spaces.
Definition 9 1. CAC(R,C): For every disjoint familyA = {An : n ∈ N}of non- empty subsets ofR, if there exists a family{dn :n ∈N}of metrics such that, for everyn∈N,An,dnis a compact metric space, thenAhas a choice function.
2. CAC(C,M): If{Xn,dn : n ∈ ω}is a family of non-empty compact metric spaces, then the family{Xn:n∈ω}has a choice function.
3. M(T B,W O): For every totally bounded metric spaceX,d, the set X is well- orderable.
4. M(T B,S): Every totally bounded metric space is separable.
5. M(T B,ST B): Every totally bounded metric space is strongly totally bounded.
6. M(I C,D I): Every infinite compact metrizable space is Dedekind-infinite.
7. MP([15, Form 383]): Every metrizable space is paracompact.
8. M(σ−p.f.)([15, Form 233]): Every metrizable space has aσ-point-finite base.
9. M(σ−l.f.)(Form [232 B] of [15]): Every metrizable space has aσ-locally finite base.
Definition 10 The following forms will be calledforms of typeM(C,).
1. M(C,S): Every compact metrizable space is separable.
2. M(C,2): Every compact metrizable space is second-countable.
3. M(C,ST B): Every compact metric space is strongly totally bounded.
4. M(C,L): Every compact metrizable space is Loeb.
5. M(C,W O): Every compact metrizable space is well-orderable.
6. M(C, → [0,1]N): Every compact metrizable space is embeddable in the Hilbert cube[0,1]N.
7. M(C, → [0,1]R): Every compact metrizable space is embeddable in the Tychonoff cube[0,1]R.
8. M(C,≤ |R|): Every compact metrizable space is of size≤ |R|.
9. M(C,W(R)): For every infinite compact metrizable space X, τ,τ andRare equipotent.
10. M(C,B(R)): Every compact metrizable space has a base of size≤ |R|.
11. M(C,|BY| ≤ |B|): For every compact metrizable spaceX, every baseBofXand every compact subspaceYofX,|BY| ≤ |B|.
12. M([0,1],|BY| ≤ |B|): For every baseBof the interval[0,1]with the usual topol- ogy and every compact subspaceYof[0,1],|BY| ≤ |B|.
13. M(C, σ−l.f): Every compact metrizable space has aσ-locally finite base.
14. M(C, σ−p.f): Every compact metrizable space has aσ-point-finite base.
The notation of typeM(C,)was introduced in [22] and was also used in [23], but not all forms from the definition above were defined in [22,23]. The formsM(C,L) andM(C,W O)were denoted byC M LandC M W Oin [27]. Most forms from Def- inition10are new here. That the new formsM(C, → [0,1]N),M(C, → [0,1]R), M(C,W(R)),M(C,B(R)), M(C,|BY| ≤ |B|)andM([0,1],|BY| ≤ |B|)are all important is shown in Sect.4.
Apart from the forms defined above, we also refer to the following forms that are not weaker thanACinZF:
Definition 11 1. LW([15, Form 90]): For every linearly ordered setX,≤, the set Xis well-orderable.
2. CH(theContinuum Hypothesis): 2ℵ0 = ℵ1.
Remark 4 It is known thatACandLWare equivalent inZF; however,LWdoes not implyACinZFA(see [20, Theorems 9.1 and 9.2]).
2 Introduction
2.1 The content of the article in brief
Although mathematicians are aware that a lot of theorems ofZFCthat are included in standard textbooks on general topology (e.g., in [8,42]) may fail inZFand many amazing disasters in topology inZFhave been discovered, new non-trivial results showing significant differences between truth values inZFCand inZFof some given propositions can be still surprising. In this paper, we show new results concerning forms of typeM(C,)inZF. The main aim of our work is to establish inZFthe set-theoretic strength of the forms of type M(C,), as well as to clarify possible relationships between those forms and relevant ones. Taking care of the readability of the article, in the forthcoming Sects.2.2–2.4, we include some known facts and few definitions for future references. In particular, in Sect.2.4, we give definitions of permutation models (also called Fraenkel–Mostowski models) and formulate a version of a transfer theorem due to Pincus [38], called here thePincus Transfer Theorem(cf.
Theorem7), which will be useful for the transfer of certainZFA-independence results toZF. The main new results of the paper are included in Sects.3–5. Section6contains a list of open problems that suggest a direction for future research in this field.
In Sect.3, we construct a (infinite) class of new symmetricZF-models in each of which the conjunctionCH∧WO(P(R))∧ ¬CACf i n is true (see Theorem8).
In Sect.4, we investigate relationships between the formsM(T B,W O),M(T B,S), M(I C,D I)andM(C,S). Among other results of Sect.4, by using appropriate per- mutation models and the Pincus Transfer Theorem, we prove that the conjunctions BPI∧M(I C,D I)∧ ¬IDI,(¬BPI)∧M(I C,D I)∧ ¬IDIandUT(ℵ0,ℵ0,cu f)∧
¬M(I C,D I)haveZF-models (see Theorems11,12and13, respectively). We deduce thatUT(ℵ0,ℵ0,cu f)∧¬M(C,S)has aZF-model (see Corollary5). The latter result provides apartial answerto the open problem of whether or notCUCimpliesM(C,S) inZF. The status of the reverse implication is also unknown (see the discussion in Remark14(a)). Taking the opportunity, we also fill a gap in [15,16] by proving that WoAmimpliesCUC(see Proposition8).
In Sect. 5, among a plethora of results, we show that CACf i n implies neither M(C,S)norM(C,≤ |R|)inZF(see Proposition10), andM(C,S)is equivalent to each one of the conjunctions:CACf i n∧M(C, σ−l.f.),CACf i n∧M(C,ST B)and CAC(R,C)∧M(C,≤ |R|)(see Theorems14and15, respectively). We deduce that M(C, σ−l.f)is unprovable inZF(see Remark16). We also deduce thatM(C,S)and M(C,≤ |R|)are equivalent in every permutation model (see Corollary6). Further- more, we prove thatM(C,S)andM(C, → [0,1]N)are equivalent (see Theorem16).
We show that, surprisingly,M(C,|BY| ≤ |B|)andM(C,B(R))are independent of ZF(see Theorem17). In Theorem18, we show thatM(C,B(R))is equivalent to the conjunctionM(C, → [0,1]R)∧Part(R)and that, under the assumption ofCAC(R), M(C,S),M(C,W(R)), andM(C,B(R)), are pairwise equivalent. The symmetric models ofZF+WO(P(R))+ ¬CACf i n constructed in the proof of Theorem8of Sect.3are applied to a proof thatPart(R)does not implyM(C, → [0,1]R)inZF (see Theorem19).
2.2 A list of several known theorems
We list below some known theorems for future references.
Theorem 1 (Cf. [36].)CACimpliesM(T B,S)inZF.
Theorem 2 (Cf. [23].)(ZF)
(i) LetX= X,dbe an uncountable compact separable metric space. Then|X| =
|R|.
(ii) CACf i n follows from each of the statements:M(C,S),M(C,≤ |R|)and “For every compact metric spaceX,d, either|X| ≤ |R|or|R| ≤ |X|”.
Theorem 3 (ZF)
(a) ([28, Theorem 8].)The statementsM(C,S),CAC(C,M)are equivalent.
(b) ([28, Corollary 1(a)].)CAC(C,M)impliesCACf i n.
Theorem 4 ([9, Corollary 4.8], Urysohn’s Metrization Theorem.) (ZF) If X is a second-countable T3-space, thenXis metrizable.
Theorem 5 (Cf. [1,3,35,39].)
(i) (ZFC)Every metrizable space has aσ-locally finite base.
(ii) (ZF)If a T1-spaceXis regular and has aσ-locally finite base, thenXis metrizable.
Remark 5 The fact that, inZFC, aT1-space is metrizable if and only if it is regular and has aσ-locally finite base was originally proved by Nagata in [35], Smirnov in [39] and Bing in [1]. It was shown in [3] that it is provable inZFthat every regularT1- space which admits aσ-locally finite base is metrizable. It was established in [14] that M(σ−l.f.)is equivalent toM(σ−p.f.)and impliesMP. Using similar arguments, one can prove thatM(C, σ−l.f.)andM(C, σ −p.f)are also equivalent inZF. In [10], a model ofZF+DCwas shown in whichMPfails. In [4], a model ofZF+BPI was shown in whichMPfails. This implies that, in each of the above-mentionedZF- models constructed in [4,10], there exists a metrizable space which fails to have a σ-point-finite base. This means thatM(σ−l.f)is unprovable inZF. In Sect.4, it is clearly explained thatM(C, σ − f.l)is also unprovable inZF.
Theorem 6 (ZF)
(i) (Cf.[27].) A compact metrizable space is Loeb iff it is second-countable iff it is separable. In consequence, the statementsM(C,L),M(C,S)andM(C,2)are all equivalent.
(ii) (Cf.[30].) IfXis a compact second-countable and metrizable space, thenXω is compact and separable. In particular, the Hilbert cube[0,1]N is a compact, separable metrizable space.
(iii) (Cf.[23].)BPIimpliesM(C,S)andM(C,≤ |R|).
2.3 Frequently used metrics
Similarly to [31], we make use of the following idea several times in the sequel.
Suppose thatA= {An :n ∈N}is a disjoint family of non-empty sets,A= A and∞ ∈/ A. Let X = A∪ {∞}. Suppose that(ρn)n∈Nis a sequence such that, for eachn∈N,ρnis a metric onAn. Letdn(x,y)=min{ρn(x,y),1n}for allx,y∈ An. We define a functiond: X×X →Ras follows:
(∗)d(x,y)=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0 ifx=y;
max{1n,m1} ifx∈ An,y∈ Am andn=m;
dn(x,y) ifx,y∈ An;
1
n ifx∈ Anandy= ∞orx= ∞andy∈ An.
Proposition 1 The function d, defined by (∗), has the following properties:
(i) d is a metric on X (cf.[31]);
(ii) if, for every n∈N, the spaceAn, τ(ρn)is compact, then so is the spaceX, τ(d) (cf.[31]);
(iii) the spaceX, τ(d)has aσ-locally finite base;
(iv) ifAdoes not have a choice function, the spaceX, τ(d)is not separable.
Metrics defined by(∗)were used, for instance, in [26,27,31], as well as in several other papers not cited here.
2.4 Permutation models and the Pincus Transfer Theorem
Let us clarify definitions of the permutation models we deal with. We refer to [20, Chapter 4] and [21, Chapter 15, p. 251] for the basic terminology and facts concerning permutation models.
Suppose we are given a modelMofZFA+ACwith an infinite setAof all atoms ofM, and a groupGof permutations of A. For a setx ∈ M, we denote by TC(x) the transitive closure ofx inM. Every permutationφof Aextends uniquely to an
∈-automorphism (usually denoted also byφ) ofM. Forx∈M, we put:
fixG(x)= {φ∈G:(∀t ∈x)φ(t)=t}and symG(x)= {φ∈G:φ(x)=x}.
We refer the readers to [20, Chapter 4, pp. 46–47] for the definitions of the concepts of anormal filterand anormal ideal.
Definition 12 (i) Thepermutation modelNdetermined byM,Gand a normal filter Fof subgroups ofGis defined by the equality:
N = {x∈M:(∀t ∈TC({x}))(symG(t)∈F)}.
(ii) Thepermutation modelN determined byM,Gand a normal idealIof subsets of the set of all atoms ofMis defined by the equality:
N = {x∈M:(∀t ∈TC({x}))(∃E ∈I)(fixG(E)⊆symG(t))}.
(iii) (Cf. [20, p. 46] and [21, p. 251].) A permutation model (or, equivalently, a Fraenkel–Mostowski model) is a classN which can be defined by (i).
Remark 6(a) LetFbe a normal filter of subgroups ofGand letx∈M. If symG(x)∈ F, thenxis calledsymmetric. If every element of TC({x})is symmetric, thenx is calledhereditarily symmetric(cf. [20, p. 46] and [21, p. 251]).
(b) Given a normal idealI of subsets of the set Aof atoms ofM, the filterFI of subgroups ofGgenerated by{fixG(E): E ∈ I}is a normal filter such that the permutation model determined byM,GandFI coincides with the permutation model determined byM,GandI(see [20, p. 47]). Forx ∈M, a setE∈Isuch that fixG(E)⊆symG(x)is called asupportofx.
In the forthcoming sections, we describe and apply several permutation models.
For example, we apply the permutation model which appeared in [26, the proof to Theorem 2.5] and was also used in [27], theBasic Fraenkel Model(labeled asN1 in [15]) and theMostowski Linearly Ordered Model(labeled asN3 in [15]). Let us give definitions of these models and recall some of their properties for future references.
Definition 13 (Cf.[26].)LetMbe a model ofZFA+AC. LetAbe the set of all atoms ofMand letI= [A]<ω. Assume that:
(i) Ais expressed as
n∈NAnwhere{An:n ∈N}is a disjoint family such that, for everyn∈N,
An=
an,x :x ∈S
0,1 n
andS(0,1n)is the circle of the Euclidean planeR2, ρeof radius1n, centered at 0;
(ii) Gis the group of all permutations of Athat rotate theAn’s by an angleθn ∈R. Then the permutation modelNcr determined byM,Gand the normal idealIwill be called theconcentric circles permutation model.
Remark 7 We need to recall some properties ofNcr for applications in this paper. Let us use the notation from Definition13. In [26, proof of Theorem 2.6], it was proved that{An :n ∈ N}does not have a multiple choice function inNcr. In [27, proof of Theorem 3.5], it was proved thatIDIholds inNcr, soCACf i nalso holds inNcr (see Remark2(i)).
Definition 14 (Cf.[15, p. 176]and[20, Section 4.3].)LetMbe a model ofZFA+AC.
LetAbe the set of all atoms ofMand letI= [A]<ω. Assume that:
(i) Ais a denumerable set;
(ii) Gis the group of all permutations of A.
Then theBasic Fraenkel ModelN1 is the permutation model determined byM,G andI.
Remark 8 It is known that, inN1, the set Aof all atoms is amorphous, soIDIfails (see [20, p. 52] and [15, pp, 176–177]). It is also known thatBPIis false inN1 but CACf i n is true inN1 (see [15, p. 177]).
Definition 15 (Cf.[15, p. 182]and[20, Section 4.6].)LetMbe a model ofZFA+AC.
LetAbe the set of all atoms ofMand letI= [A]<ω. Assume that:
(i) the set Ais denumerable and there is a fixed ordering≤in Asuch thatA,≤ is order isomorphic to the set of all rational numbers equipped with the standard linear order;
(ii) Gis the group of all order-automorphisms ofA,≤.
Then theMostowski Linearly Ordered ModelN3 is the permutation model determined byM,GandI.
Remark 9 It is known that the power set of the set of all atoms is Dedekind-finite in N3, soIDIfails inN3 (see [15, pp. 182–183]). However,BPIandCACf i n are true inN3 (see [15, p, 183]).
It is well known that, in any permutation model, the power set of any pure set (that is, a set with no atoms in its transitive closure) is well-orderable (see, e.g., [15, p.
176]). This can be deduced from the following helpful proposition:
Proposition 2 (Cf. [20, Item (4.2), p. 47].)LetNbe the permutation model determined by M, G and a normal filter F. For every x ∈ N, x is well-orderable in N iff fixG(x)∈F.
Remark 10 If a statementAis satisfied in a permutation model, then to show that there exists aZF-model in whichAis satisfied, we use transfer theorems due to Pincus (cf.
[37,38]). Pincus transfer theorems, together with definitions of aboundable formula and aninjectively boundable formulathat are involved in the theorems, are included in [15, Note 103].
To our transfer results, we apply mainly the following fragment of the third theorem from [15, p. 286]:
Theorem 7 (The Pincus Transfer Theorem.) (Cf. [37,38] and [15, p. 286].)Letbe a conjunction of statements that are either injectively boundable orBPI. Ifhas a permutation model, thenhas aZF-model.
In the definition of an injectively boundable formula, the notion of injective cardi- nality is involved (see [37], [15, Item (3), p. 284]). Let us recall the definition of the latter notion.
Definition 16 For a setx, theinjective cardinalityofx, denoted by|x|−, is the (well- ordered) cardinal number defined as follows:
|x|−=sup{κ :κ is a well-ordered cardinal equipotent to a subset ofx}.
Now, we are in a position to pass to the main body of the paper.
3 New symmetric models of ZF
Suppose that Φ is a form that is satisfied in a ZFA-model. Even if fulfills the assumptions of the Pincus Transfer Theorem, it might be complicated to check it and to see well aZF-model in whichis satisfied. This is why it is good to give a direct relatively simple description of a ZF-model satisfying. By the proof of Theorem8below, we shall obtain an infinite class of symmetric models, each satisfying CH∧WO(P(R))∧¬CACf i n. In Sect.5, models of this class are applied to a proof that the conjunctionPart(R)∧ ¬M(C, → [0,1]R)has aZF-model (see the forthcoming Theorem19).
For the convenience of readers, before embarking on the proof of Theorem8, let us recall in brief the construction of symmetric extension models. Assume thatM is a countable transitive model ofZFCand thatP,≤ ∈M is a poset with a maximum element denoted by1P; such a posetP,≤inMis said to be a notion of forcing. Let MPbe the (proper) class of allP-names, which are defined by transfinite recursion withinM(cf. [32, Definitions 2.5, 2.6, pp. 188–189]). We will denote aP-name byx˙ and, following the notation of [32, Definition 2.10, p. 190], forx∈ M, we will denote byxˇthe canonical name{ ˇy,1P :y∈x}forx.
Ifφis an order-automorphism ofP,≤, thenφcan be extended to an automorphism φ˜ofMPdefined by recursion,
φ(˜ x)˙ = { ˜φ(y), φ(˙ p) : ˙y,p ∈ ˙x}.
For everyx ∈ M,φ(˜ x)ˇ = ˇx. We shall henceforth useφto denote both the automor- phism ofP,≤and the automorphismφ˜of theP-names.
LetGbe a group of order-automorphisms ofP,≤, and also letΓ be a normal filter onG, that is,Γ is a filter of subgroups ofGclosed under conjugation (i.e., for allφ∈G andH ∈Γ,φHφ−1∈Γ). AP-namex˙is calledΓ-symmetricif symG(x)˙ ∈Γ, where symG(x)˙ is the stabilizer of the namex, i.e. the subgroup˙ {φ∈ G:φ(x)˙ = ˙x}ofG.
˙
xis calledhereditarilyΓ-symmetricifx˙isΓ-symmetric and, for every ˙y,p ∈ ˙x,y˙ is hereditarilyΓ-symmetric. The class of hereditarilyΓ-symmetricP-names (which, in view of the above, is defined by transfinite recursion over the rank ofx) is denoted˙ by HSΓ.
LetGbe aP-generic filter overM, and also let N = { ˙xG: ˙x∈HSΓ},
wherex˙Gdenotes the value of the namex˙byG, i.e.x˙G = { ˙yG : ∃p ∈G( ˙y,p ∈ ˙x)}.
ThenNis a transitive model ofZFandM ⊆N⊆M[G](cf. [20, Section 5.2, p. 64]), where M[G] = { ˙xG : ˙x ∈ MP}is the generic extension model of M.N is called a symmetric extensionofM generated byΓ.
In the proof of Theorem8below, we will write “1” instead of “1P”, “(hereditarily) symmetric” instead of “(hereditarily)Γ-symmetric”, and “HS” instead of “HSΓ”.
Theorem 8 Let n, ∈ω\{0,1}. There is a symmetric model Nn,ofZFsuch that Nn,| ∀m∈n(2ℵm = ℵm+1)∧ ¬CAC.
Hence, it is also the case that
Nn,|CH∧WO(P(R))∧ ¬CACf i n.
Proof By [32, Theorem 6.18, p. 216], we may fix a countable transitive model M of ZFC+ ∀m∈n(2ℵm = ℵm+1). Our plan is to construct a symmetric extension model Nn,ofM with the required properties.
We use as our notion of forcing the setP=Fn(ω××ωn×ωn,2, ωn)of all partial functions pwith|p|<ℵn, dom(p)⊆ω××ωn×ωnand ran(p)⊆2 = {0,1}, partially ordered by reverse inclusion, i.e., for p,q ∈P,p≤qif and only if p ⊇q.
The posetP,≤has the empty function as its maximum element, which we denote by1. Furthermore, sinceωnis a regular cardinal, it follows from [32, Lemma 6.13, p. 214] thatP,≤is anωn-closed poset. Therefore, by [32, Theorem 6.14, p. 214], forcing withP,≤adds no new subsets ofωm form∈ n, and hence it adds no new reals or sets of reals, but it does add new subsets ofωn. Moreover, by [32, Corollary 6.15, p. 215], we have thatP,≤preserves cofinalities≤ ωn, and hence cardinals
≤ωn.
LetG be aP-generic filter over M, and let M[G]be the corresponding generic extension model ofM. By [32, Theorem 4.2, p. 201],ACis trueM[G]. In view of the observations of the previous paragraph, for every modelN withM ⊆ N ⊆ M[G], we have the following:
N | ∀m∈n(2ℵm = ℵm+1).
InM[G], fork∈ω,t∈, andi ∈ωn, we define the following sets, together with their canonical names:
1. ak,t,i = {j ∈ωn: ∃p∈G(p(k,t,i,j)=1)},
˙
ak,t,i = { ˇj,p : j ∈ωn∧ p∈P∧p(k,t,i,j)=1}. 2. Ak,t = {ak,t,i :i ∈ωn},
A˙k,t = {˙ak,t,i,1 :i ∈ωn}. 3. Ak = {Ak,s :s∈},
A˙k = { ˙Ak,s,1 :s∈}.
4. A= {Ak:k∈ω}, A˙= { ˙Ak,1 :k∈ω}.
Now, every permutationφofω××ωninduces an order-automorphism ofP,≤ as follows: for every p∈P,
domφ(p)= {φ(k,t,i),j : k,t,i,j ∈dom(p)},
φ(p)(φ(k,t,i),j)=p(k,t,i,j). (1)
LetG be the group of all order-automorphisms ofP,≤induced (as in (1)) by all those permutationsφofω××ωnwhich are defined as follows:
For everyk∈ ω, letσk be a permutation ofand also letηk be a permutation of ωn. We define
φ(k,t,i)= k, σk(t), ηk(i) (2) for allk,t,i ∈ω××ωn. By (2), it follows that, for allφ∈G,k∈ω, andt ∈,
φ(A˙k,t)= ˙Ak,σk(t), (3) and thus, for allφ∈Gandk∈ω,
φ(A˙k)= ˙Ak. (4)
Hence, for everyφ∈G,
φ(A)˙ = ˙A. (5)
For everyE ∈ [ω××ωn]<ω, we let fixG(E)= {φ∈G: ∀e∈E(φ(e)=e)}and we also letΓ be the filter of subgroups ofGgenerated by the filter base{fixG(E):E ∈ [ω××ωn]<ω}. It is not hard to verify thatΓ is a normal filter onG, so we leave this to interested readers. Ifx˙is aP-name,E ∈ [ω××ωn]<ω, and fixG(E)⊆symG(x),˙ then we callEasupportofx. Let˙
Nn,= { ˙xG : ˙x∈HS}
be the symmetric extension model of M. By the definitions ofΓ and HS, it is clear that everyx˙∈HS has a support in the above sense.
In view of the observations at the beginning of the proof, we have Nn,| ∀m∈n(2ℵm = ℵm+1),
and thus
Nn,|CH∧WO(P(R)).
Claim Fork ∈ω,t ∈, andi ∈ωn, the setsak,t,i,Ak,t,Ak, andA, are all elements ofNn,. Moreover,Ais denumerable inNn,.
Proof Fixk ∈ ω,t ∈ , andi ∈ ωn. By the definition ofG, it easily follows that E = {k,t,i}is a support ofa˙k,t,iandA˙k,t, and since (by (4) and (5))φ(A˙k)= ˙Ak
andφ(A)˙ = ˙Afor allφ ∈G, we conclude thatak,t,i, Ak,t,Ak, andAall belong to the modelNn,.
Furthermore, f˙= {op(m,ˇ A˙m),1 :m ∈ω}, where op(σ, τ)is the name for the ordered pairσG, τGgiven in [32, Definition 2.16, p. 191], is an HS-name for the mapping f = {m,Am :m∈ω}since, for everyφ∈G,φfixes f˙(pointwise), and all names in f˙are hereditarily symmetric. Thus,Ais denumerable inNn,.
Claim The denumerable familyA= {Ak :k ∈ω}has no partial choice function in Nn,. Hence,
Nn,| ¬CAC.
Proof By way of contradiction, we assume thatAhas an infinite subfamily inNn,, B= {Am :m∈W}for some infinite setW ⊆ω, which has a choice function inNn,, f say. Clearly,Bhas a canonical HS-name, namelyB˙ = { ˙Am,1 :m∈W}. We let f˙be an HS-name for f. There existsp∈Gsuch that
p“f˙is a choice function forB˙”. (6) LetE ∈ [ω××ωn]<ωbe a support of f˙. SinceW is infinite andEis finite, there existsm0∈W such thatE∩({m0} ××ωn)= ∅. Lett0be the unique element of such that f(Am0)=Am0,t0.
Letq∈Gbe such thatq ≤ pand
q f˙(A˙m0)= ˙Am0,t0. (7) Since|q|<ℵn, there existsk ∈ ωn such that, for alli ∈ ωnwithi ≥kand for all t ∈and j ∈ωn,m0,t,i,j∈/dom(q). We letσm0 be the following-cycle:
σm0 :0→1→ · · · →−1→0,
and we also let η : [0,k) → [k,2k)be an order-isomorphism. Then ηinduces a permutationηm0ofωndefined by
ηm0(i)=
⎧⎪
⎨
⎪⎩
η(i), ifi ∈ [0,k);
η−1(i), ifi ∈ [k,2k);
i, if 2k≤i.
We define aψ∈Gby stipulating, for allm,t,i ∈ω××ωn,
ψ(m,t,i)=
m0, σm0(t), ηm0(i), ifm=m0; m,t,i, ifm=m0.
(So, form=m0,σmandηmare the identity permutations ofandωn, respectively—
recall the definition ofG.) Then the following hold:
(a) ψ∈fixG(E), and henceψ(f˙)= ˙f (sinceEis a support of f˙), (b) ψ(A˙m0,t0)= ˙Am0,σm0(t0),
(c) qandψ(q)are compatible conditions. Thus,q∪ψ(q)is a well-defined extension ofq,ψ(q), and p.
By (4), (a), (b), and (7), we obtain that
ψ(q) f˙(A˙m0)= ˙Am0,σm0(t0), (8) and from (c), together with Eqs. (6), (7) and (8), we conclude that
q∪ψ(q)“f˙is a choice function forB˙∧ ˙f(A˙m0)= ˙Am0,t0
∧ ˙f(A˙m0)= ˙Am0,σm0(t0). (9) But then, (9) yields a contradiction. Indeed, using DCin M[G](recall that M[G]
satisfies the fullAC) and the proof of [32, Lemma 2.3, pp. 186–187], we may construct aP-generic filterHoverMsuch thatq∪ψ(q)∈ H. Then, by (9) and [32, Theorem 3.6(2), p. 200], we deduce that, inM[H], the following hold:
(i) f˙H is a choice function forB˙H, (ii) f˙H((A˙m0)H)=(A˙m0,t0)H, and (iii) f˙H((A˙m0)H)=(A˙m0,σm0(t0))H,
where B˙H = {(A˙m)H : m ∈ W},(A˙m0)H = {(A˙m0,t)H : t ∈ }, (A˙m0,t)H = {(˙am0,t,i)H :i ∈ωn}(t∈), and(a˙m0,t,i)H = {j ∈ωn: ∃r∈ H(r(m0,t,i,j)=1)}
(t ∈,i∈ωn).
However, since t0 = σm0(t0)(recall that σm0 is the cycle (0,1, . . . ,l −1), so σm0 moves every element of ), we have that, for anyP-generic filter Q over M, (A˙m0,t0)Q∩(A˙m0,σm0(t0))Q = ∅. If not, then there exist aP-generic filterQoverM and anx∈(A˙m0,t0)Q∩(A˙m0,σm0(t0))Q. Thenx=(a˙m0,t0,i)Qandx =(a˙m0,σm0(t0),i)Q
for some ordinalsi,i∈ωn. Let
D= {r∈P: ∃j ∈ωn(r(m0,t0,i,j)=r(m0, σm0(t0),i,j))}.
ThenD∈ Mand it is fairly easy to verify thatDis dense inP. Hence,Q∩D= ∅.
Letting r ∈ Q∩ D, we obtain a j ∈ (˙am0,t0,i)Q(˙am0,σm
0(t0),i)Q, contradicting (˙am0,t0,i)Q=(˙am0,σm0(t0),i)Q. Thus,(A˙m0,t0)Q∩(A˙m0,σm0(t0))Q = ∅, as required.1
In particular, for theP-generic filterHoverM, we have (A˙m0,t0)H∩(A˙m0,σm0(t0))H = ∅,
so(A˙m0,t0)H =(A˙m0,σm0(t0))H. This, together with (ii) and (iii), yields that f˙His not a function, contradicting property (i) of f˙H.
Hence,Ahas no partial choice function in the model Nn,, finishing the proof of the claim.
The above arguments complete the proof of the theorem.
1 Note that the above argument yields that, for everys∈P(so also fors=q∪ψ(q)),s A˙m0,t0∩
˙ Am0,σm
0(t0)= ˇ∅(cf. [32, Definition 3.1, p. 194]).
4 M(IC,DI)and M(TB,WO)
Since every compact metric space is totally bounded and every infinite separable Haus- dorff space is Dedekind-infinite, let us begin our investigations of the forms of type M(C,)with a deeper look at the formsM(T B,W O),M(T B,S)andM(I C,D I).
We include a simple proof of the following proposition for completeness.
Proposition 3 (ZF)LetX= X,dis a totally bounded metric space such that X is well-orderable. ThenXis separable.
Proof Since X is well-orderable, so is the setY =
n∈N(Xn × {n}). Let≤ be a fixed well-ordering inY. For everym ∈ N, let ym = xm,km ∈ Xkm × {km}be the first element of Y,≤ such that X =
{Bd(xm(i),m1) : i ∈ km}. The set D=
m∈N{xm(i):i ∈km}is countable and dense inX.
Theorem 9 (ZF)
(i) M(T B,W O)→M(T B,S)andM(C,W O)→M(C,S). None of these impli- cations are reversible.
(ii) M(T B,W O)→M(C,W O)→M(C,S)→M(I C,D I).
(iii) (Cf. [23, Theorem 7 (i)].)CAC→M(T B,S)→M(C,S).
(iv) NeitherM(T B,W O)norM(T B,S)implyCAC.
Proof It follows from Proposition3that the implications from (i) are both true. It is known that, in Feferman’s modelM2 in [15],CACis true butRis not well-orderable (see [15, p. 140]). Then[0,1]is a compact, metrizable but not well-orderable space in M2. HenceM(T B,S)∧ ¬M(T B,W O)andM(C,S)∧ ¬M(C,W O)are both true inM2. This completes the proof to (i). In view of (i), it is obvious that (ii) holds. It is known from [23] that (iii) also holds. It follows from the first implication of (i) that to prove (iv), it suffices to show thatM(T B,W O)does not implyCAC.
It was shown in [23, proof of Theorem 15] that there exists a modelMofZF+
¬CACin which it is true that if a metric spaceX= X,dis sequentially bounded (i.e., every sequence of points ofX has a Cauchy’s subsequence), thenXis well- orderable and separable. By [23, Theorem 7 (vii)], every totally bounded metric space is sequentially bounded. This shows that there exists a modelM of ZFin which
M(T B,W O)∧ ¬CACis true. Hence (iv) holds.
ThatM(I C,D I)does not implyM(C,S)is shown in Proposition10(iv). It is unknown whetherM(C,W O)is equivalent to or weaker thanM(T B,W O)inZF.
To compareM(C,S)withM(T B,S), we recall that it was proved in [24] that the implicationM(T B,S)→ CAC(R)holds inZF; however, the implicationCAC→ M(T B,S)of Theorem1is not reversible inZF. On the other hand, it is known that CAC(R)and M(C,S) are independent of each other in ZF (see, e.g., [23]). The following proposition, together with the fact thatM(T B,S)impliesM(C,S), shows thatM(T B,S)is essentially stronger thanM(C,S)inZF.
Proposition 4 (ZF)
(i) (Cf.[24, Proposition 8].)M(T B,ST B)impliesCAC(R).
