Several amazing discoveries about compact metrizable spaces in ZF
aa r X i v : . [ m a t h . GN ] A ug Several amazing discoveries about compactmetrizable spaces in ZF Kyriakos Keremedis, Eleftherios Tachtsis and Eliza WajchDepartment of Mathematics, University of the AegeanKarlovassi, Samos 83200, [email protected] of Statistics and Actuarial-Financial Mathematics,University of the Aegean, Karlovassi 83200, Samos, [email protected] of MathematicsFaculty of Exact and Natural SciencesSiedlce University of Natural Sciences and Humanitiesul. 3 Maja 54, 08-110 Siedlce, [email protected] 5, 2020
Abstract
In the absence of the axiom of choice, the set-theoretic status ofmany natural statements about metrizable compact spaces is investi-gated. Some of the statements are provable in ZF , some are shown tobe independent of ZF . For independence results, distinct models of ZF and permutation models of ZFA with transfer theorems of Pincusare applied. New symmetric models are constructed in each of whichthe power set of R is well-orderable, the Continuum Hypothesis is sat-isfied but a denumerable family of non-empty finite sets can fail tohave a choice function, and a compact metrizable space need not beembeddable into the Tychonoff cube [0 , R . athematics Subject Classification (2010) : 03E25, 03E35, 54A35, 54E35,54D30 Keywords : Weak forms of the Axiom of Choice, metrizable space, to-tally bounded metric, compact space, permutation model, symmetricmodel.
In this paper, the intended context for reasoning and statements of theoremsis the Zermelo-Fraenkel set theory ZF without the axiom of choice AC . Thesystem ZF + AC is denoted by ZFC . We recommend [32] and [33] as agood introduction to ZF . To stress the fact that a result is proved in ZF or ZF + A (where A is a statement independent of ZF ), we shall write atthe beginning of the statements of the theorems and propositions ( ZF ) or( ZF + A ), respectively. Apart from models of ZF , we refer to some modelsof ZFA (or ZF in [15]), that is, we refer also to ZF with an infinite set ofatoms (see [20], [21] and [15]). Our theorems proved here in ZF are alsoprovable in ZFA ; however, we also mention some theorems of ZF that arenot theorems of ZFA .We denote by ω the set of all non-negative integers (i.e., finite ordinalnumbers of von Neumann). As usual, if n ∈ ω , then n + 1 = n ∪ { n } .Members of the set N = ω \ { } are called natural numbers. The power set ofa set X is denoted by P ( X ) . A set X is called countable if X is equipotent toa subset of ω . A set X is called uncountable if X is not countable. A set X is finite if X is equipotent to an element of ω . An infinite set is a set which isnot finite. An infinite countable set is called denumerable . A cardinal numberof von Neumann is an initial ordinal number of von Neumann. If X is a setand κ is a non-zero cardinal number of von Neumann, then [ X ] κ is the familyof all subsets of X equipotent to κ , [ X ] ≤ κ is the collection of all subsets of X equipotent to subsets of κ , and [ X ] <κ is the family of all subsets of X equipotent to a cardinal number of von Neumann which belongs to κ . For aset X , we denote by | X | the cardinal number of X in the sense of Definition11.2 of [20]. We recall that, in ZF , for every set X , the cardinal number | X | exists; however, X is equipotent to a cardinal number of von Neumann ifand only if X is well-orderable. For sets X and Y , the inequality | X | ≤ | Y | X is equipotent to a subset of Y .The set of all real numbers is denoted by R and, if it is not stated other-wise, R and every subspace of R are considered with the usual topology andwith the metric induced by the standard absolute value on R . In this subsection, we establish notation and recall several basic definitions.Let X = h X, d i be a metric space. The d - ball with centre x ∈ X andradius r ∈ (0 , + ∞ ) is the set B d ( x, r ) = { y ∈ X : d ( x, y ) < r } . The collection τ ( d ) = { V ⊆ X : ( ∀ x ∈ V )( ∃ n ∈ ω ) B d ( x, n ) ⊆ V } is the topology in X induced by d . For a set A ⊆ X , let δ d ( A ) = 0 if A = ∅ ,and let δ d ( A ) = sup { d ( x, y ) : x, y ∈ A } if A = ∅ . Then δ d ( A ) is the diameter of A in X . Definition 1.1.
Let X = h X, d i be a metric space.(i) Given a real number ε > , a subset D of X is called ε - dense or an ε - net in X if X = S x ∈ D B d ( x, ε ) .(ii) X is called totally bounded if, for every real number ε > , there existsa finite ε -net in X .(iii) X is called strongly totally bounded if it admits a sequence ( D n ) n ∈ N such that, for every n ∈ N , D n is a finite n -net in X .(iv) (Cf. [24].) d is called strongly totally bounded if X is strongly totallybounded. Remark . Every strongly totally bounded metric space is evidently totallybounded. However, it was shown in [24, Proposition 8] that the sentence“Every totally bounded metric space is strongly totally bounded” is not atheorem of ZF . 3 efinition 1.3. Let X = h X, τ i be a topological space and let Y ⊆ X .Suppose that B is a base of X .(i) The closure of Y in X is denoted by cl τ ( Y ) or cl X ( Y ) .(ii) τ | Y = { U ∩ Y : U ∈ τ } . Y = h Y, τ | Y i is the subspace of X with theunderlying set Y .(iii) B Y = { U ∩ Y : U ∈ B} .Clearly, in Definition 1.3 (iii), B Y is a base of Y . In Section 5, it is shownthat B Y need not be equipotent to a subset of B .In the sequel, boldface letters will denote metric or topological spaces(called spaces in abbreviation) and lightface letters will denote their under-lying sets. Definition 1.4.
A collection U of subsets of a space X is called:(i) locally finite if every point of X has a neighbourhood meeting onlyfinitely many members of U ;(ii) point-finite if every point of X belongs to at most finitely many mem-bers of U ;(iii) σ - locally finite (respectively, σ - point-finite ) if U is a countable union oflocally finite (respectively, point-finite) subfamilies. Definition 1.5.
A space X is called:(i) first-countable if every point of X has a countable base of neighbour-hoods;(ii) second-countable if X has a countable base.Given a collection { X j : j ∈ J } of sets, for every i ∈ J , we denote by π i the projection π i : Q j ∈ J X j → X i defined by π i ( x ) = x ( i ) for each x ∈ Q j ∈ J X j .If τ j is a topology in X j , then X = Q j ∈ J X j denotes the Tychonoff product ofthe topological spaces X j = h X j , τ j i with j ∈ J . If X j = X for every j ∈ J ,then X J = Q j ∈ J X j . As in [8], for an infinite set J and the unit interval [0 , of R , the cube [0 , J is called the Tychonoff cube . If J is denumerable, then4he Tychonoff cube [0 , J is called the Hilbert cube . In [12], all Tychonoffcubes are called Hilbert cubes. In [42], Tychonoff cubes are called cubes.We recall that if Q j ∈ J X j = ∅ , then it is said that the family { X j : j ∈ J } has a choice function, and every element of Q j ∈ J X j is called a choice function of the family { X j : j ∈ J } . A multiple choice function of { X j : j ∈ J } is every function f ∈ Q j ∈ J P ( X j ) such that, for every j ∈ J , f ( j ) is a non-empty finite subset of X j . A set f is called partial ( multiple ) choice function of { X j : j ∈ J } if there exists an infinite subset I of J such that f is a(multiple) choice function of { X j : j ∈ I } . Given a non-indexed family A ,we treat A as an indexed family A = { x : x ∈ A} to speak about a (partial)choice function and a (partial) multiple choice function of A .Let { X j : j ∈ J } be a disjoint family of sets, that is, X i ∩ X j = ∅ for eachpair i, j of distinct elements of J . If τ j is a topology in X j for every j ∈ J ,then L j ∈ J X j denotes the direct sum of the spaces X j = h X j , τ j i with j ∈ J . Definition 1.6. (Cf. [2], [34] and [26].)(i) A space X is said to be Loeb (respectively, weakly Loeb ) if the family ofall non-empty closed subsets of X has a choice function (respectively,a multiple choice function).(ii) If X is a (weakly) Loeb space, then every (multiple) choice function ofthe family of all non-empty closed subsets of X is called a ( weak ) Loebfunction of X .Other topological notions used in this article but not defined here arestandard. They can be found, for instance, in [8] and [42]. Definition 1.7.
A set X is called:(i) a cuf set if X is expressible as a countable union of finite sets (cf. [5],[6], [19] and [16, Form 419]);(ii) Dedekind-finite if X is not equipotent to a proper subset of itself (cf.[15, Note 94], [12, Definition 4.1] and [20, Definition 2.6]); Dedekind-infinite if X is not Dedekind-finite (equivalently, if there exists an in-jection f : ω → X ) (cf. [15, Note 94] and [12, Definition 2.13]);5iii) amorphous if X is infinite and there does not exist a partition of X into two infinite sets (cf. [15, Note 57],[20, p. 52] and [12, E. 11 inSection 4.1]). Definition 1.8. (Cf. [31].) A topological space h X, τ i is called a cuf space if X is a cuf set. AC In this subsection, for readers’ convenience, we define and denote most of theweaker forms of AC used directly in this paper. If a form is not defined inthe forthcoming sections, its definition can be found in this subsection. Forthe known forms given in [15], [16] or [12], we quote in their statements theform number under which they are recorded in [15] (or in [16] if they do notappear in [15]) and, if possible, we refer to their definitions in [12]. Definition 1.9. AC fin ([15, Form 62]): Every non-empty family ofnon-empty finite sets has a choice function.2. AC W 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 ofnon-empty sets has a choice function.4.
CAC ( R ) ([15, Form 94], [12, Definition 2.9(1)]): Every denumerablefamily of non-empty subsets of R has a choice function.5. CAC △ ω ( R ) (Cf. [29]): For every family A = { A n : n ∈ ω } such that,for every n ∈ ω and all x, y ∈ A n , ∅ 6 = A n ⊆ P ( ω ) \{∅} and x △ y ∈ [ ω ] <ω ( △ denotes the operation of symmetric difference between sets), thereexists a choice function of A .6. IDI ([15, Form 9], [12, Definition 2.13(ii)]): Every Dedekind-finite setis finite.7.
IDI ( R ) ([15, Form 13], [12, Definition 2.13(2)]): Every Dedekind-finitesubset of R is finite.8. WoAm ([15, Form 133]): Every set is either well-orderable or has anamorphous subset. 6.
Part ( R ) : Every partition of R is of size ≤ | R | .10. WO ( R ) ([15, Form 79]): R is well-orderable.11. WO ( P ( R )) ([15, Form 130]): P ( R ) is well-orderable.12. CAC fin ([15, Form 10], [12, Definition 2.9(3)]): Every denumerablefamily of non-empty finite sets has a choice function.13. For a fixed n ∈ ω \ { , } , CAC n ([15, Form 288(n)]): Every denumer-able family of n -element sets has a choice function.14. CAC
W O : Every denumerable family of non-empty well-orderable setshas a choice function.15.
CMC ([15, Form 126], [12, Definition 2.10]): Every denumerable familyof non-empty sets has a multiple choice function.16.
CMC ω ([15, Form 350]): Every denumerable family of denumerablesets has a multiple choice function.17. CUC ([15, Form 31], [12, Definition 3.2(1)]): Every countable unionof countable sets is countable.18.
CUC fin ([15, Form [10 A]], [12, Definition 3.2(3)]): Every countableunion of finite sets is countable.19. UT ( ℵ , cuf, cuf ) ([16, Form 419]): Every countable union of cuf setsis a cuf set. (Cf. also [6].)20. UT ( ℵ , ℵ , cuf ) ([16, Form 420]): Every countable union of countablesets is a cuf set. (Cf. also [6].)21. vDCP ( ℵ ) ( [15, Form 119], [12, p, 79], [7]): Every denumerable family {h A n , ≤ n i : n ∈ ω } of linearly ordered sets, each of which is order-isomorphic to the set h Z , ≤i of integers with the standard linear order ≤ , has a choice function.22. BPI ([15, Form 14], [12, Definition 2.15(1)]): Every Boolean algebrahas a prime ideal. 73. DC ([15, Form 43], [12, Definition 2.11(1)]): For every non-empty set X and every binary relation ρ on X if, for each x ∈ X there exists y ∈ X such that xρy , then there exists a sequence ( x n ) n ∈ N of points of X such that x n ρx n +1 for each n ∈ N . Remark . The following are well-known facts in ZF :(i) CAC fin and
CUC fin are both equivalent to the sentence: Every in-finite well-ordered family of non-empty finite sets has a partial choicefunction (see [15, Form [10 O]] and [12, Diagram 3.4, p. 23]). Moreover,
CAC fin is equivalent to Form [10 E] of [15], that is, to the sentence:Every denumerable family of non-empty finite sets has a partial choicefunction. It is known that
IDI implies
CAC fin and this implication isnot reversible in ZF (cf. [12, pp. 324–324]).(ii) CAC is equivalent to the sentence: Every denumerable family of non-empty sets has a partial choice function (see [15, Form [8 A]]).(iii)
BPI is equivalent to the statement that all products of compact Haus-dorff spaces are compact (see [15, Form [14 J]] and [12, Theorem 4.37]).(iv)
CMC ω is equivalent to the following sentence: Every denumerablefamily of denumerable sets has a multiple choice function. Remark . (a) It was proved in [19] that the following implications aretrue in ZF and none of the implications is reversible in ZF : CMC → UT ( ℵ , cuf, cuf ) → CMC ω → vDCP ( ℵ ) . (b) Clearly, UT ( ℵ , cuf, cuf ) implies UT ( ℵ , ℵ , cuf ) . In [6, proof to The-orem 3.3] a model of ZFA was shown in which UT ( ℵ , ℵ , cuf ) is trueand UT ( ℵ , cuf, cuf ) is false.(c) It was proved in [31] that the following equivalences hold in ZF :(i) UT ( ℵ , cuf, cuf ) is equivalent to the sentence: Every countableproduct of one-point Hausdorff compactifications of infinite dis-crete cuf spaces is metrizable (equivalently, first-countable).(ii) UT ( ℵ , ℵ , cuf ) is equivalent to the sentence: Every countableproduct of one-point Hausdorff compactifications of denumerablediscrete spaces is metrizable (equivalently, first-countable).8et us pass to definitions of forms concerning metric and metrizablespaces. Definition 1.12. CAC ( R , C ) : For every disjoint family A = { A n : n ∈ N } of non-empty subsets of R , if there exists a family { d n : n ∈ N } of metrics such that, for every n ∈ N , h A n , d n i is a compact metricspace, then A has a choice function.2. CAC ( C, M ) : If {h X n , d n i : n ∈ ω } is a family of non-empty compactmetric spaces, then the family { X n : n ∈ ω } has a choice function.3. M ( T B, W O ) : For every totally bounded metric space h X, d i , the set X is well-orderable.4. M ( T B, S ) : Every totally bounded metric space is separable.5. ICMDI : Every infinite compact metrizable space is Dedekind-infinite.6. MP ([15, Form 383]): Every metrizable space is paracompact.7. M ( σ − p.f. ) ([15, Form 233]): Every metrizable space has a σ -point-finite base.8. M ( σ − l.f. ) ([15, Form [232 B]]): Every metrizable space has a σ -locallyfinite base. Definition 1.13.
The following forms will be called forms of type M ( C, (cid:3) ) .1. M ( C, S ) : Every compact metrizable space is separable.2. M ( C, : 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 , N ) : Every compact metrizable space is embeddable in theHilbert cube [0 , N .7. M ( C, ֒ → [0 , R ) : Every compact metrizable space is embeddable in theTychonoff cube [0 , R . 9. M ( C, ≤ | R | ) : Every compact metrizable space is of size ≤ | R | .9. M ( C, W ( R )) : For every infinite compact metrizable space h X, τ i , τ and R are equipotent.10. M ( C, B ( R )) : Every compact metrizable space has a base of size ≤ | R | .11. M ( C, |B Y | ≤ |B| ) : For every compact metrizable space X , every base B of X and every compact subspace Y of X , |B Y | ≤ |B| .12. M ([0 , , |B Y | ≤ |B| ) : For every base B of the interval [0 , with theusual topology and every compact subspace Y of [0 , , |B Y | ≤ |B| .13. M ( C, σ − l.f ) : Every compact metrizable space has a σ -locally finitebase.14. M ( C, σ − p.f ) : Every compact metrizable space has a σ -point-finitebase.The notation of type M ( C, (cid:3) ) was started in [22] and continued in [23]but not all forms from the definition above were defined in [22] and [23]. Theforms M ( C, L ) and M ( C, W O ) were denoted by CM L and
CM W O in [27].Most forms from Definition 1.13 are new here. That the new forms M ( C, ֒ → [0 , N ) , M ( C, ֒ → [0 , R ) , M ( C, W ( R )) , M ( C, B ( R )) , M ( C, |B Y | ≤ |B| ) and M ([0 , , |B Y | ≤ |B| ) are all important is shown in Section 4.Apart from the forms defined above, we also refer to the following formsthat are not weaker than AC in ZF : Definition 1.14. LW ([15, Form 90]): For every linearly ordered set h X, ≤i , the set X is well-orderable.2. CH (the Continuum Hypothesis): ℵ = ℵ . Remark . It is well known that AC and LW are equivalent in ZF ; how-ever, LW does not imply AC in ZFA (see [15] and [20, Theorems 9.1 and9.2]). 10
Introduction
Although mathematicians are aware that a lot of theorems of
ZFC that areincluded in standard textbooks on general topology (e.g., in [8] and [42])may fail in ZF and many amazing disasters in topology in ZF have been dis-covered, new non-trivial results showing significant differences between truthvalues in ZFC and in ZF of some given propositions can be still surprising.In this article, we show new results concerning forms of type M ( C, (cid:3) ) in ZF .The main aim of our work is to establish in ZF the set-theoretic strengthof the forms of type M ( C, (cid:3) ) , as well as relationships between these formsand relevant ones. Taking care of the readability of the article, in the forth-coming Subsections 2.2–2.4, we include some known facts and few definitionsfor future references. In particular, in Subsection 2.4, we give definitions ofpermutation models (called also Fraenkel-Mostowski models) and formulateonly this version of a transfer theorem due to Pincus (called here the PincusTransfer Theorem ) (cf. Theorem 2.19) which is applied in this article. Themain new results of the article are included in Sections 3–5. Section 6 con-tains a list of open problems that suggest a direction for future research inthis field.In Section 3, we construct new symmetric ZF -models in each of whichthe conjunction CH ∧ WP ( P ( R )) ∧ ¬ CAC fin is true.In Section 4, we investigate relationships between the forms M ( T B, W O ) , M ( T B, S ) , ICMDI and M ( C, S ) . Among other results of Section 4, byshowing appropriate permutation models and using the Pincus Transfer The-orem, we prove that the conjunctions BPI ∧ ICMDI ∧ ¬
IDI , ( ¬ BPI ) ∧ ICMDI ∧ ¬
IDI and UT ( ℵ , ℵ , cuf ) ∧ ¬ ICMDI have ZF -models (see The-orems 4.13, 4.14 and 4.23, respectively). We deduce that the conjunction UT ( ℵ , ℵ , cuf ) ∧ ¬ M ( C, S ) has a ZF -model (see Corollary 4.24). We dis-cuss a relationship between M ( C, S ) and CUC . Taking the opportunity,we fill in a gap in [15] and [16] by proving that
WoAm implies
CUC (seeProposition 4.18).Among a plethora of results of Section 5, we show that
CAC fin im-plies neither M ( C, S ) nor M ( C, ≤ | R | ) (see Proposition 5.1), and M ( C, S ) is equivalent to each one of the conjunctions: CAC fin ∧ M ( C, σ − l.f. ) , CAC fin ∧ M ( C, ST B ) and CAC ( R , C ) ∧ M ( C, ≤ | R | ) (see Theorems 5.2and 5.4, respectively). We deduce that M ( C, σ − l.f ) is unprovable in ZF M ( C, S ) and M ( C, ≤ | R | ) are equiv-alent in every permutation model (see Corollary 5.5). Furthermore, we provethat M ( C, S ) and M ( C, ֒ → [0 , N ) are equivalent (see Theorem 5.8). Weshow that, surprisingly, M ( C, |B Y | ≤ |B| ) and M ( C, B ( R )) are independentof ZF (see Theorem 5.11). In Theorem 5.12, we show that M ( C, B ( R )) isequivalent to the conjunction M ( C, ֒ → [0 , R ) ∧ Part ( R ) , CAC ( R ) impliesthat M ( C, S ) and M ( C, B ( R )) are equivalent; moreover, the statement that M ( C, S ) and M ( C, W ( R )) are equivalent is equivalent to M ( C, B ( R )) . Themodels of ZF + WO ( P ( R )) + ¬ CAC fin constructed in Section 3 are appliedto a proof that
Part ( R ) does not imply M ( C, ֒ → [0 , R ) in ZF (see Theorem5.13). We list below some known theorems for future references.
Theorem 2.1. (Cf. [36].)
CAC implies M ( T B, S ) in ZF . Theorem 2.2. (Cf. [23].) ( ZF ) (i) Let X = h X, d i be an uncountable compact separable metric space.Then | X | = | R | .(ii) CAC fin follows from each of the statements: M ( C, S ) , M ( C, ≤ | R | ) and “For every compact metric space h X, d i , either | X | ≤ | R | or | R | ≤| X | ”. Theorem 2.3. ( ZF ) (a) ([28, Theorem 8].) The statements M ( C, S ) , CAC ( C, M ) are equiva-lent.(b) ([28, Corollary 1(a)].) CAC ( C, M ) implies CAC fin . Theorem 2.4. ([9, Corollary 4.8], Urysohn’s Metrization Theorem.) ( ZF ) If X is a second-countable T -space, then X is metrizable. Theorem 2.5. (Cf. [35], [39], [1], [3]. )(i) ( ZFC ) Every metrizable space has a σ -locally finite base. ii) ( ZF ) If a T -space X is regular and has a σ -locally finite base, then X is metrizable.Remark . That it holds in
ZFC that a T -space is metrizable if and onlyif it is regular and has a σ -locally finite base was originally proved by Nagatain [35], Smirnov in [39] and Bing in [1]. It was shown in [3] that it is provablein ZF that every regular T -space which admits a σ -locally finite base ismetrizable. It was established in [14] that M ( σ − l.f. ) is an equivalent to M ( σ − p.f. ) and implies MP . Using similar arguments, one can prove that M ( C, σ − l.f. ) and M ( C, σ − p.f ) are also equivalent in ZF . In [10], a modelof ZF + DC was shown in which MP fails. In [4], a model of ZF + BPI was shown in which MP fails. This implies that, in each of the above-mentioned ZF - models constructed in [10] and [4], there exists a metrizablespace which fails to have a σ -point-finite base. This means that M ( σ − l.f ) is unprovable in ZF . In Section 4, it is clearly explained that M ( C, σ − f.l ) is also unprovable in ZF . Theorem 2.7. ( ZF ) (i) (Cf. [27].) A compact metrizable space is Loeb iff it is second-countableiff it is separable. In consequence, the statement M ( C, L ) , M ( C, S ) and M ( C, are all equivalent.(ii) (Cf. [30].) If X is a compact second-countable and metrizable space,then X ω is compact and separable. In particular, the Hilbert cube [0 , N is a compact, separable metrizable space.(iii) (Cf. [23].) BPI implies M ( C, S ) and M ( C, ≤ | R | ) . Similarly to [31], we make use of the following idea several times in the sequel.Suppose that A = { A n : n ∈ N } is a disjoint family of non-empty sets, A = S n ∈ N A n and ∞ / ∈ A . Let X = A ∪ {∞} . Suppose that ( ρ n ) n ∈ N is asequence such that, for each n ∈ N , ρ n is a metric on A n . Let d n ( x, y ) =min { ρ n ( x, y ) , n } for all x, y ∈ A n . We define a function d : X × X → R asfollows: 13 ∗ ) d ( x, y ) = if x = y ; max { n , m } if x ∈ A n , y ∈ A m and n = m ; d n ( x, y ) if x, y ∈ A n ; n if x ∈ A and y = ∞ or x = ∞ and y ∈ A . Proposition 2.8.
The function d , defined by ( ∗ ), has the following proper-ties:(i) d is a metric on X (cf. [31]);(ii) if, for every n ∈ N , the space h A n , τ ( ρ n ) i is compact, then so is thespace h X, τ ( d ) i (cf. [31]);(iii) the space h X, τ ( d ) i has a σ -locally finite base;(iv) if A does not have a choice function, the space h X, τ ( d ) i is not separa-ble. Metrics defined by ( ∗ ) were used, for instance, in [26], [27], [31], as wellas in several other papers not cited here. Let us clarify definitions of the permutation models we deal with. We referto [20, Chapter 4] and [21, Chapter 15, p. 251] for the basic terminology andfacts concerning permutation models.Suppose we are given a model M of ZFA + AC with an infinite set A ofall atoms of M , and a group G of permutations of A . For a set x ∈ M , wedenote by TC( x ) the transitive closure of x in M . Then every permutation φ of A extends uniquely to an ∈ -automorphism (usually denoted also by φ )of M . For x ∈ M , we put: fix G ( x ) = { φ ∈ G : ( ∀ t ∈ x ) φ ( t ) = t } and sym G ( x ) = { φ ∈ G : φ ( x ) = x } . We refer the readers to [20, Chapter 4, pp. 46–47] for the definitions of theconcepts of a normal filter and a normal ideal .14 efinition 2.9. (i) The permutation model N determined by M , G anda normal filter F of subgroups of G is defined by the equality: N = { x ∈ M : ( ∀ t ∈ TC( { x } ))(sym G ( t ) ∈ F ) } . (ii) The permutation model N determined by M , G and a normal ideal I of subsets of the set of all atoms of M is defined by the equality: N = { x ∈ M : ( ∀ t ∈ TC( { x } ))( ∃ E ∈ I )(fix G ( E ) ⊆ sym G ( t )) } . (iii) (Cf. [20, p. 46] and [21, p. 251].) A permutation model (or, equiv-alently, a Fraenkel-Mostowski model ) is every class N which can bedefined by (i). Remark . ( a ) Let F be a normal filter of subgroups of G and let x ∈ M .If sym G ( x ) ∈ F , then x is called symmetric . If every element of TC( { x } ) issymmetric, then x is called hereditarily symmetric (cf. [20, p. 46] and [21,p. 251]).( b ) Given a normal ideal I of subsets of the set A of atoms of M , thefilter F I of subgroups of G generated by { fix G ( E ) : E ∈ I} is a normal filtersuch that the permutation model determined by M , G and F I coincides withthe permutation model determined by M , G and I (see [20, p. 47]). For x ∈ M , a set E ∈ I such that fix G ( E ) ⊆ sym G ( x ) is called a support of x .In the forthcoming sections, we describe and apply several permutationmodels. For example, we apply the permutation model which appeared in[26, the proof to Theorem 2.5] and was also used in [27], the Basic FraenkelModel (labeled as N in [15]) and the Mostowski Linearly Ordered Model (labeled as N in [15]). Let us give definitions of these models and recallsome of their properties for future references. Definition 2.11. (Cf. [26].) Let M be a model of ZFA + AC . Let A bethe set of all atoms of M and let I = [ A ] <ω . Assume that:(i) A is expressed as S n ∈ N A n where { A n : n ∈ N } is a disjoint family suchthat, for every n ∈ N , A n = { a n,x : x ∈ S (0 , n ) } and S (0 , n ) is the circle of the Euclidean plane h R , ρ e i of radius n ,centered at ; 15ii) G is the group of all permutations of A that rotate the A n ’s by an angle θ n ∈ R .Then the permutation model N cr determined by M , G and the normal ideal I will be called the concentric circles permutation model . Remark . We need to recall some properties of N cr for applications inthis paper. Let us use the notation from Definition 2.11. In [26, the proofto Theorem 2.5], it was proved that { A n : n ∈ N } does not have a multiplechoice function in N cr . In [27, the proof to Theorem 3.5], it was proved that IDI holds in N cr , so CAC fin also holds in N cr (see Remark 1.10(i)). Definition 2.13. (Cf. [15, p. 176] and [20, Section 4.3].) Let M be a modelof ZFA + AC . Let A be the set of all atoms of M and let I = [ A ] <ω . Assumethat:(i) A is a denumerable set;(ii) G is the group of all permutations of A .Then the Basic Fraenkel Model N is the permutation model determined by M , G and I . Remark . It is known that, in N , the set A of all atoms is amorphous,so IDI fails (see [20, p. 52] and [15, pp, 176–177]). It is also known that
BPI is false in N but CAC fin is true in N (see [15, p. 177]). Definition 2.15. (Cf. [15, p. 182] and [20, Section 4.6].) Let M be a modelof ZFA + AC . Let A be the set of all atoms of M and let I = [ A ] <ω . Assumethat:(i) the set A is denumerable and there is a fixed ordering ≤ in A such that h A, ≤i is order isomorphic to the set of all rational numbers equippedwith the standard linear order;(ii) G is the group of all order-automorphisms of h A, ≤i .Then the Mostowski Linearly Ordered Model N is the permutation modeldetermined by M , G and I . Remark . It is known that the power set of the set of all atoms isDedekind-finite in N , so IDI fails in N (see [15, pp. 182–183]). How-ever, BPI and
CAC fin are true in N (see [15, p, 183]).16t is well known that, in any permutation model, the power set of any pureset (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 helpfulproposition: Proposition 2.17. (Cf. [20, Item (4.2), p. 47].) Let N be the permutationmodel determined by M , G and a normal filter F . For every x ∈ N , x iswell-orderable in N iff fix G ( x ) ∈ F .Remark . If a statement A is satisfied in a permutation model, to showthat there exists a ZF -model in which A is satisfied, we use transfer theoremsdue to Pincus (cf. [37] and [38]). Pincus transfer theorems, together withdefinitions of a boundable formula and an injectively boundable formula thatare involved in the theorems, are included in [15, Note 103].To our transfer results, we apply mainly the following fragment of thethird theorem from [15, p. 286]: Theorem 2.19. (The Pincus Transfer Theorem.) (Cf. [37], [38] and [15,p. 286].) Let Φ be a conjunction of statements that are either injectivelyboundable or BPI . If Φ has a permutation model, then Φ has a ZF -model. In the definition of an injectively boundable formula, a notion of an in-jective cardinality is involved. This notion is given, for instance, in [15, Item(3), p. 284]. Let us formulate its equivalent definition below.
Definition 2.20.
For a set x , the injective cardinality of x is the von Neu-mann cardinal number | x | − defined as follows: | x | − = sup { κ : κ is a von Neumann cardinal equipotent to a subset of x } . Now, we are in the position to pass to the main body of the article.
Suppose that Φ is a form that is satisfied in a ZFA -model. Even if Φ fulfillsthe assumptions of the Pincus Transfer Theorem, it might be complicated tocheck it and to see well a ZF -model in which Φ is satisfied. This is why it isgood to give a direct relatively simple description of a ZF -model satisfying Φ . In the proof to Theorem 3.1 below, we show a class of symmetric models17atisfying CH ∧ WO ( P ( R )) ∧ ¬ CAC fin . In Section 5, models of this classare applied to a proof that the conjunction
Part ( R ) ∧ ¬ M ( C, ֒ → [0 , R ) hasa ZF -model (see Theorem 5.13). Theorem 3.1.
Let n, ℓ ∈ ω \ { , } . There is a symmetric model N n,ℓ of ZF such that N n,ℓ | = ∀ m ∈ n (2 ℵ m = ℵ m +1 ) ∧ ¬ CAC ℓ . Hence, it is also the case that N n,ℓ | = CH ∧ WO ( P ( R )) ∧ ¬ CAC fin . Proof.
Let us use the terminology and results from [32, Chapter VII] and [20,Chapter 5]. By [32, Theorem 6.18, p. 216], we can fix a countable transitivemodel M of ZFC + ∀ m ∈ n (2 ℵ m = ℵ m +1 ) . Our plan is to construct asymmetric extension model N n,ℓ of M with the required properties.Let P = Fn( ω × ℓ × ω n × ω n , , ω n ) be the set of all partial functions p with | p | < ℵ n , dom( p ) ⊆ ω × ℓ × ω n × ω n and ran( p ) ⊆ { , } , partiallyordered by reverse inclusion, i.e., for p, q ∈ P , p ≤ q if and only if p ⊇ q .The poset h P , ≤i has the empty function as its maximum element, whichwe denote by . Furthermore, since ω n is a regular cardinal, it follows from[32, Lemma 6.13, p. 214] that h P , ≤i is an ω n -closed poset. Hence, by [32,Theorem 6.14, p. 214], forcing with P adds no new subsets of ω m for m ∈ n ,and hence it adds no new reals or sets of reals, but it does add new subsetsof ω n . Furthermore, by [32, Corollary 6.15, p. 215], we have that P preservescofinalities ≤ ω n , and hence cardinals ≤ ω n .Let G be a P -generic filter over M , and let M [ G ] be the correspondinggeneric extension model of M . In view of the above, for every model N with M ⊆ N ⊆ M [ G ] , we have the following: N | = ∀ m ∈ n (2 ℵ m = ℵ m +1 ) . By [32, Theorem 4.2, p. 201], AC is true M [ G ] .In M [ G ] , for k ∈ ω , t ∈ ℓ , and i ∈ ω n , we define the following sets alongwith their canonical names:1. a k,t,i = { j ∈ ω n : ∃ p ∈ G ( p ( k, t, i, j ) = 1) } , a k,t,i = {h ˇ j, p i : j ∈ ω n ∧ p ∈ P ∧ p ( k, t, i, j ) = 1 } .2. A k,t = { a k,t,i : i ∈ ω n } , A k,t = {h a k,t,i , i : i ∈ ω n } . 18. A k = { A k, , A k, , . . . , A k, ( ℓ − } , A k = {h A k, , i , h A k, , i , . . . , h A k, ( ℓ − , i} .4. A = { A k : k ∈ ω } , A = {h A k , i : k ∈ ω } .Now, every permutation φ of ω × ℓ × ω n induces an order-automorphismof h P , ≤i by requiring, for every p ∈ P , the following:(1) dom φ ( p ) = {h φ ( k, t, i ) , j i : h k, t, i, j i ∈ dom( p ) } ,φ ( p )( φ ( k, t, i ) , j ) = p ( k, t, i, j ) . Let G be the group of all order-automorphisms of h P , ≤i induced (as in (1))by all those permutations φ of ω × ℓ × ω n which are defined as follows.For every k ∈ ω , let σ k be a permutation of ℓ = { , , . . . , ℓ − } and alsolet η k be a permutation of ω n . We define(2) φ ( k, t, i ) = h k, σ k ( t ) , η k ( i ) i , for all h k, t, i i ∈ ω × ℓ × ω n . By (2), it follows that for every φ ∈ G such that φ ( k, t, i ) = h k, σ k ( t ) , η k ( i ) i , we have that, for every k ∈ ω and every t ∈ ℓ ,(3) φ ( A k,t ) = A k,σ k ( t ) , and thus, for every k ∈ ω ,(4) φ ( A k ) = A k . It follows that for every φ ∈ G ,(5) φ ( A ) = A . For every finite subset E ⊆ ω × ℓ × ω n , we let fix G ( E ) = { φ ∈ G : ∀ e ∈ E ( φ ( e ) = e ) } and we also let Γ be the filter of subgroups of G generatedby the filter base { fix G ( E ) : E ∈ [ ω × ℓ × ω n ] <ω } . Then Γ is a normal filteron G (see [20, Section 5.2, p. 64] for the definition of the term “normalfilter”). An element x ∈ M is called symmetric if there exists a finite subset E ⊆ ω × ℓ × ω n such that, for every φ ∈ fix G ( E ) , we have φ ( x ) = x ; ifsuch a set E exists, we call E a support of x . An element x ∈ M is called hereditarily symmetric if x and all elements of the transitive closure of x are19ymmetric. Let HS be the set of all hereditarily symmetric names in M . Asin [32, Definition 2.7, p. 189], for τ ∈ HS , let τ G denote the value of thename τ . Let N n,ℓ = { τ G : τ ∈ HS } be the symmetric extension model of M . Then N n,ℓ ⊂ M [ G ] .In view of the observations at the beginning of the proof, we have N n,ℓ | = ∀ m ∈ n (2 ℵ m = ℵ m +1 ) , and thus N n,ℓ | = CH ∧ WO ( P ( R )) . For k ∈ ω , t ∈ ℓ , and i ∈ ω n , the sets a k,t,i , A k,t , A k , and A are all elementsof N n,ℓ . Let us fix k ∈ ω , t ∈ ℓ , and i ∈ ω n . Then E = {h k, t, i i} is a supportof a k,t,i and A k,t . By (4) and (5), we have that, for every φ ∈ G , φ ( A k ) = A k and φ ( A ) = A . Thus, a k,t,i , A k,t , A k , and A all belong to N n,ℓ . For σ, τ ∈ HS ,let op( σ, τ ) be the name for the ordered pair h σ G , τ G i (see [32, Definition 2.16,p. 191]). Let f = {h k, A k i : k ∈ ω } and ˙ f = {h op(ˇ k, A k ) , i : k ∈ ω } . Since,for every φ ∈ G , φ ( ˙ f ) = ˙ f , we deduce that ˙ f is an HS -name for the mapping f (in M [ G ] ). This proves that A is denumerable in N n,ℓ .Now, by making suitable adjustments to the proof that CAC is false inthe Second Cohen Model (see [20, Section 5.4, p. 68]), one may verify that A has no partial choice function in the model N n,ℓ . We invite interested readersto fill in the missing details. Remark . Let us note that Theorem 3.1 provides a class of symmetricmodels satisfying CH ∧ WO ( P ( R )) ∧ ¬ CAC fin . ICMDI and M ( T B, W O ) Since every compact metric space is totally bounded and every infinite sep-arable Hausdorff space is Dedekind-infinite, let us begin our investigationsof the forms of type M ( C, (cid:3) ) with a deeper look at the forms M ( T B, W O ) , M ( T B, S ) and ICMDI . We include a simple proof to the following propo-sition for completeness.
Proposition 4.1. ( ZF ) Let X = h X, d i is a totally bounded metric spacesuch that X is well-orderable. Then X is separable. roof. Since X is well-orderable, so is the set Y = S n ∈ N ( X n ×{ n } ) . Let ≤ be afixed well-ordering in Y . For every m ∈ N , let y m = h x m , k m i ∈ X k m × { k m } be the first element of h Y, ≤i such that X = S { B d ( x m ( i ) , m ) : i ∈ k m } . Theset D = S m ∈ N { x m ( i ) : i ∈ k m } is countable and dense in X . Theorem 4.2. ( ZF ) (i) M ( T B, W O ) → M ( T B, S ) and M ( C, W O ) → M ( C, S ) . None of theseimplications is reversible.(ii) M ( T B, W O ) → M ( C, W O ) → M ( C, S ) → ICMDI .(iii) (Cf. [23, Theorem 7 (i)].)
CAC → M ( T B, S ) → M ( C, S ) .(iv) Neither M ( T B, W O ) nor M ( T B, S ) implies CAC .Proof.
It follows from Proposition 4.1 that the implications from (i) are bothtrue. It is known that, in Feferman’s model M in [15], CAC is true but R is not well-orderable (see [15, p. 140]). Then [0 , is a compact, metrizablebut not well-orderable space in M . Hence M ( T B, S ) ∧ ¬ M ( T B, W O ) and M ( C, S ) ∧ ¬ M ( C, W O ) are both true in M . 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 that M ( T B, W O ) does not imply CAC .It was shown in [23, the proof to Theorem 15] that there exists a model M of ZF + ¬ CAC in which it is true that if a metric space X = h X, d i is sequentially bounded (i.e., every sequence of points of X has a Cauchy’ssubsequence), then X is well-orderable and separable. By [23, Theorem 7(vii)], every totally bounded metric space is sequentially bounded. Thisshows that exists a model M of ZF in which M ( T B, W O ) ∧ ¬ CAC is true.Hence (iv) holds.That
ICMDI does not imply M ( C, S ) is shown in Proposition 5.1(iv). Itis unknown whether M ( C, W O ) is equivalent to or weaker than M ( T B, W O ) in ZF .To compare M ( C, S ) with M ( T B, S ) , we recall that it was proved in[24] that the implication M ( T B, S ) → CAC ( R ) holds in ZF ; however, theimplication CAC → M ( T B, S ) of Theorem 2.1 is not reversible in ZF . Onthe other hand, it is known that CAC ( R ) and M ( C, S ) are independent of21ach other in ZF (see, e.g., [23]). The following proposition, together withthe fact that M ( T B, S ) implies M ( C, S ) , shows that M ( T B, S ) is essentiallystronger than M ( C, S ) in ZF . Proposition 4.3. ( ZF ) (i) (Cf. [24, Proposition 8].) If every totally bounded metric space isstrongly totally bounded, then CAC ( R ) holds.(ii) In Cohen’s Original Model M M ( C, S ) istrue, M ( T B, S ) is false and there exists a totally bounded metric spacewhich is not strongly totally bounded.(iii) M ( C, S ) does not imply M ( T B, S ) .Proof. That (i) holds was proved in [24]. It is known that
BPI is true M (see [15, p. 147]). It follows from Theorem 2.7(iii) that M ( C, S ) holds in M . On the other hand, it is known that CAC ( R ) fails in M (see [15, p.147]). Therefore, by (i), it holds in M that there exists a totally boundedmetric space which is not strongly totally bounded.We recall that a topological space X is called limit point compact if everyinfinite subset of X has an accumulation point in X (see, e.g., [23]). Proposition 4.4. ( ZFA ) WoAm implies both M ( T B, W O ) and “everylimit point compact, first-countable T -space is well-orderable”.Proof. Let us assume
WoAm . Consider an arbitrary metric space h X, d i .Suppose that X is not well-orderable. By WoAm , there exists an amorphoussubset B of X . Let ρ = d ↾ B × B . Lemma 1 of [5] states that every metric onan amorphous set has a finite range. Therefore, the set ran( ρ ) = { ρ ( x, y ) : x, y ∈ B } is finite. Since B is infinite, the set ran( ρ ) \ { } is non-empty.If ε = min(ran( ρ ) \ { } ) , then there does not exist a finite ε -net in h B, ρ i because B is infinite and, for every x ∈ B , B ρ ( x, ε ) = { x } . This implies that ρ is not totally bounded. Hence d is not totally bounded.Now, suppose that Y = h Y, τ i is a first-countable, limit point compact T -space. Let C be an infinite subset of Y . Since Y is limit point compact,the set C has an accumulation point in Y . Let y be an accumulation pointof C and let { U n : n ∈ N } be a base of neighborhoods of y in Y . Since Y is a T -space, we can inductively define an increasing sequence ( n k ) k ∈ N of naturalnumbers such that, for every k ∈ N , C ∩ ( U n k \ U n k +1 ) = ∅ . This implies that22 is not amorphous. Hence, no infinite subset of Y is amorphous, so Y iswell-orderable by WoAm . Corollary 4.5. N | = M ( T B, W O ) .Proof. This follows from Proposition 4.4 and the known fact that
WoAm istrue in N (see p. 177 in [15]).To prove that M ( T B, W O ) does not imply WoAm in ZFA , let us usethe model N . In what follows, the notation concerning N is the same asin Definition 2.15. For a, b ∈ A with a < b (where A is the set of atoms of N and ≤ is the fixed linear order on A ), we denote by ( a, b ) the open intervalin the linearly ordered set h A, ≤ ) ; that is, ( a, b ) = { x ∈ A : a < x < b } . Aproof to the following lemma can be found in [18]. Lemma 4.6. (Cf. [18, Lemma 3.17 and its proof].) If X ∈ N , E is asupport of X and there is x ∈ X for which E is not a support, then thereexist a subset Y of X and atoms a, b ∈ A with a < b , such that Y ∈ N , E ∩ ( a, b ) = ∅ and, in N , there exists a bijection f : ( a, b ) → Y having asupport E ′ such that E ∪ { a, b } ⊆ E ′ and E ′ ∩ ( a, b ) = ∅ . Theorem 4.7. N | = M ( T B, W O ) .Proof. We use the notation from Definition 2.15. Suppose that h X, d i is ametric space in N such that X is not well-orderable in N . Then X isinfinite. Let E ∈ [ A ] <ω be a support of both X and d . By Proposition 2.17,there exists x ∈ X such that E is not a support of x .By Lemma 4.6, there exist a, b ∈ A with a < b and ( a, b ) ∩ E = ∅ , suchthat there exists in N an injection f : ( a, b ) → X which has a support E ′ such that E ∪ { a, b } ⊆ E ′ and E ′ ∩ ( a, b ) = ∅ . We put B = ( a, b ) and ρ ( x, y ) = d ( f ( x ) , f ( y )) for all x, y ∈ B . Let us notice that ρ ∈ N because E ′ is also a support of ρ . We prove that ran( ρ ) = { ρ ( x, y ) : x, y ∈ B } is a two-element set. To this aim, we fix b , b ∈ B with b < b and put r = ρ ( b , b ) .Let u, v ∈ B and u = v . To show that ρ ( u, v ) = r , we must consider severalcases regarding the ordering of the elements b , b , u, v . We consider only oneof the possible cases since all the other cases can be treated in much thesame way as the chosen one. So, assume, for example, that b < u < v . Let φ be an order-automorphism of h A, ≤i such that φ ( b ) = u , φ ( b ) = v , and φ is the identity mapping on A \ B . Then φ ∈ fix G ( E ′ ) , so φ ( ρ ) = ρ . Thisimplies that φ ( r ) = ρ ( φ ( b ) , φ ( b )) . Since, in addition φ ( r ) = r , we have23 = ρ ( b , b ) = ρ ( φ ( b ) , φ ( b )) = ρ ( u, v ) . Therefore, ran( ρ ) = { , r } . Sincethe range of ρ is finite, in much the same way, as in the proof to Proposition4.4, we deduce that h B, ρ i is not totally bounded. Hence d is not totallybounded. Corollary 4.8. M ( T B, W O ) does not imply WoAm in ZFA .Proof.
It is known that
WoAm is false in N (see [15, p.183]). Therefore,the conjunction M ( T B, W O ) ∧ ¬ WoAm is true in N by Theorem 4.7.In contrast to Corollary 4.5 and Theorem 4.7, we have the following propo-sition: Proposition 4.9. N cr | = ¬ M ( C, W O ) .Proof. It was shown in [26, the proof to Theorem 2.5] that, in N cr , thereexists a compact metric space X = h X, d i which is not weakly Loeb. Then X cannot be well-orderable in N cr . Remark . In [40], a symmetric model M of ZF was constructed suchthat, in M , there exists a compact metric space h X, d i which is not weaklyLoeb; thus, M ( C, W O ) fails in M .It is obvious that IDI implies
ICMDI in ZF ; however, it seems to bestill an open problem of whether this implication is not reversible in ZF .To solve this problem, first of all, let us notice that the following corollaryfollows directly from Corollary 4.5 and Theorem 4.7: Corollary 4.11. (i) N | = ( ICMDI ∧ ¬
IDI ) .(ii) N | = ( BPI ∧ ICMDI ∧ ¬
IDI ) . To transfer
BPI ∧ ICMDI ∧ ¬
IDI to a model of ZF , let us prove thefollowing lemma: Lemma 4.12.
ICMDI is injectively boundable.Proof.
First, we put Φ( x ) = ¬ ( ∃ y )( y ⊆ x ∧ | y | = ω ) , and Ψ( x ) = “ x is finite” . Ψ( x ) is boundable (see [37] or Note 103, p. 284 in[15]). Now it is not hard to verify that ICMDI is logically equivalent to thestatement Ω , where Ω = ( ∀ x )( | x | − ≤ ω → ( ∀ ρ ∈ P ( x × x × R ))[(Φ( x ) ∧ ( ρ is a metric on x such that h X, ρ i is compact )) → Ψ( x )]) Since Ω is obviously injectively boundable, so is ICMDI . Theorem 4.13.
The conjunction
BPI ∧ ICMDI ∧ ¬
IDI has a ZF -model.Proof. It is known that ¬ IDI is boundable, and hence injectively boundable(see [37, 2A5, p. 772] or [15, p. 285]). By Theorem 2.19, if a conjunction of
BPI with injectively boundable statements has a Fraenkel-Mostowski model,then it has a ZF -model. This, together with Corollary 4.11(ii) and Lemma4.12, completes the proof. Theorem 4.14.
The conjunction ( ¬ BPI ) ∧ ICMDI ∧¬ IDI has a ZF -model.Proof. Let Φ be the conjunction ( ¬ BPI ) ∧ ICMDI ∧ ¬
IDI . It is known that
BPI is false in N (see [15, p. 177]). Hence Φ has a permutation modelby Corollary 4.11(i). Since the statements ¬ BPI , ICMDI and ¬ IDI are allinjectively boundable, Φ has a ZF -model by Theorem 2.19. Corollary 4.15.
ICMDI does not imply
IDI in ZF . Furthermore, BPI isindependent of ZF + ICMDI + ¬ IDI . Proposition 4.16.
ICMDI implies
CAC fin in ZF .Proof. It suffices to apply Corollary 2.2 (iii) in [27] which states that if everyinfinite compact metrizable space has an infinite well-orderable subset, then
CAC fin holds.At this moment, it is unknown whether there is a model of ZF in whichthe conjunction CAC fin ∧ ¬
ICMDI is false.
Remark . In Cohen’s original model M in [15], CUC holds and thereexists a dense Dedekind-finite subset X of the interval [0, 1] of R . The metricspace X = h X, d i , where d ( x, y ) = | x − y | for all x, y ∈ X , is totally boundedbut X is not well-orderable in M . It was remarked in [24] that, since h X, d i is not separable, CUC does not imply M ( T B, S ) in ZF . Now, it is easilyseen that CUC does not imply M ( T B, W O ) in ZF .25t is not stated in [15] nor in [16] that WoAm implies
CUC . Since wehave not seen a solution of the problem of whether this implication is truein other sources, let us notice that it follows from the following propositionthat this implication holds in ZF : Proposition 4.18. (i) ( ZFA ) WoAm → CUC ;(ii) If N is a model of ZFA in which R is well-orderable (in particular, if N is a permutation model), then: N | = (
CAC
W O → CUC ) . (iii) If N is a permutation model, then: N | = ( AC fin → CUC ) . Proof.
Let A = { A n : n ∈ ω } be a disjoint family of non-empty countablesets and let A = S n ∈ ω A n . Clearly, if S n ∈ ω ( A n × ω ) is countable, then A iscountable. Therefore, to show that A is countable, we may assume that, forevery n ∈ ω , the set A n is denumerable. For every n ∈ ω , let B n be the setof all bijections from ω onto A n . Since | ω ω | = | R ω | = | R | and the sets A n areall denumerable, for every n ∈ ω , the set B n is equipotent to R .(i) Let B = S n ∈ ω B n . If B is well-orderable, then there exists a sequence ( f n ) n ∈ ω of bijections f n : ω → A n , so A is countable. Suppose that B is notwell-orderable. Then it follows from WoAm that there exists an amorphoussubset C of B . Since C cannot be partitioned into two infinite subsets, theset { n ∈ ω : C ∩ B n = ∅} is finite. This implies that there exists m ∈ ω such that C ⊆ S n ∈ m +1 B n , so C is equipotent to a subset of R . But this isimpossible because R does not have amorphous subsets. The contradictionobtained completes the proof to (i).(ii) Now, assume that R is well-orderable and CAC
W O holds. Then thesets B n , being equipotent to R , are all well-orderable. Hence, it follows from CAC
W O that there exists f ∈ Q n ∈ ω B n . Then we have a sequence ( f ( n )) n ∈ ω of bijections f ( n ) : ω → A n , so A is countable.(iii) Since AC fin implies AC W O in every permutation model (cf. [13]and Note 2 in [15]), we infer that if AC fin is satisfied in N and N is apermutation model, then CAC
W O holds in N . This, taken together with(ii), implies (iii). 26 emark . ( a ) In Felgner’s Model I (labeled as M in [15]), AC W O holds and
CUC fails (cf. [15, p. 159]). We recall that AC fin ∧ ¬ AC W O is true in Sageev’s Model I (labeled as model M in [15]). Moreover, since IDI ∧ ¬
CUC is true in M (cf. [15, p. 152]), ICMDI does not imply
CUC is ZF .( b ) One should not claim that CAC fin and
CAC
W O are equivalent inevery permutation model. Namely, let N be the permutation model of IDI ∧¬ CUC constructed in [41, the proof to Theorem 4 (iv)]. Since
IDI implies
CAC fin , it follows from Proposition 4.18 that, in this model N , CAC fin istrue but
CAC
W O is false.
Remark . ( a ) It is unknown whether M ( C, S ) implies CUC in ZF orin ZFA . We recall that
BPI implies M ( C, S ) . The problem of whether BPI implies
CUC in ZF or in ZFA is still unsolved. However, if N is apermutation model in which BPI is true, then AC fin is also true in N (see,e.g., [12, Proposition 4.39]); hence, by Proposition 4.18(iii), BPI implies
CUC in every permutation model.( b ) Since M ( C, S ) implies CAC fin (see Theorem 2.2(ii) or Theorem 4.2with Proposition 4.16), it follows from Proposition 4.18(iii) that
CAC fin ∧¬ AC fin is satisfied in every permutation model of M ( C, S ) ∧ ¬ CUC .It still eludes us whether or not
CUC implies M ( C, S ) in ZF . However,we are able to provide a partial solution to this intriguing open problemby proving that the statement UT ( ℵ , ℵ , cuf ) ∧ ¬ ICMDI is a conjunctionof injectively boundable statement and it has a permutation model. It isobvious that ¬ ICMDI is boundable, so also injectively boundable. To showthat UT ( ℵ , ℵ , cuf ) is injectively boundable, we need the following lemmaproved in [17]: Lemma 4.21. ( ZF ) (Cf. [17, Lemma 3.5].) For any ordinal α , if R is acollection of sets such that |R| ≤ ℵ α +1 and, for every x ∈ R , | x | ≤ ℵ α , then | S R| 6≥ ℵ α +2 . Proposition 4.22.
The statement UT ( ℵ , ℵ , cuf ) is injectively boundable.Proof. In the light of Lemma 4.21, UT ( ℵ , ℵ , cuf ) is equivalent to the state-ment:(6) ( ∀ x )( | x | 6≥ ℵ → ( ∀ y ) “if y is a countable collection of countable setswhose union is x , then x is a cuf set” ) . x , the statements | x | 6≥ ℵ and | x | − ≤ ℵ are equivalent,it is obvious that (6) is injectively boundable. Thus, UT ( ℵ , ℵ , cuf ) is alsoinjectively boundable. Theorem 4.23. (i) The statement LW ∧ UT ( ℵ , ℵ , cuf ) ∧ ¬ ICMDI hasa permutation model.(ii) The statement UT ( ℵ , ℵ , cuf ) ∧ ¬ ICMDI has a ZF -model.Proof. (i) Let us apply the permutation model N which was constructed in[6, the proof to Theorem 3.3]. To describe N , we start with a model M of ZFA + AC with a set A of atoms such that A has a denumerable partition { A i : i ∈ ω } into denumerable sets, and for each i ∈ ω , A i has a denumerablepartition P i = { A i,j : j ∈ N } into finite sets such that, for every j ∈ ω \ { } , | A i,j | = j . Let Sym ( A ) be the group of all permutations of A and let G = { φ ∈ Sym ( A ) : ( ∀ i ∈ ω )( φ ( A i ) = A i ) and |{ x ∈ A : φ ( x ) = x }| < ℵ } . Let P i = { φ ( P i ) : φ ∈ G } and also let P = S { P i : i ∈ ω } . Let F be thenormal filter of subgroups of G generated by the collection { fix G ( E ) : E ∈ [ P ] <ω } . Then N is the permutation model determined by M , G and F . Wesay that a finite subset E of P is a support of an element x of N if, for every φ ∈ fix G ( E ) , φ ( x ) = x .It was noticed in [6, the proof to Theorem 3.3] that, for every i ∈ ω andevery Q ∈ P i , the following hold:(a) for any φ ∈ G , φ fixes Q if and only if φ fixes Q pointwise;(b) ( ∃ j Q ∈ ω )( Q ⊇ { A i,j : j > j Q } ) .To prove that LW is true in N , we fix a linearly ordered set h Y, ≤i in N and prove that fix G ( Y ) ∈ F . To this aim, we choose a set E ∈ [ P ] <ω suchthat E is a support of both Y and ≤ . To show that fix G ( E ) ⊆ fix G ( Y ) , letus consider any element y ∈ Y and a permutation φ ∈ fix G ( E ) . Supposethat φ ( y ) = y . Then either y < φ ( y ) or φ ( y ) < y . Since every element of G moves only finitely many atoms, there exists k ∈ ω \ { } such that φ k is theindentity mapping on A . Assuming that φ ( y ) < y , for such a k , we obtainthe following: y < φ ( y ) < φ ( y ) < . . . < φ k − ( y ) < φ k ( y ) = y, y < y . Arguing similarly, we deduce that if φ ( y ) < y , then y < y .The contradiction obtained shows that φ ( y ) = y for every y ∈ Y and every φ ∈ fix G ( E ) . Hence fix G ( E ) ⊆ fix G ( Y ) . Since F is a filter and fix G ( E ) ∈ F ,we infer that fix G ( Y ) ∈ F . This, together with Proposition 2.17, implies thatthe set Y is well-orderable in N . Hence N | = LW .Now, let us prove that ICMDI fails in N . First, to find a metric d on A such that h A , d i is a compact metric space in N , we denote by ∞ theunique element of A , and, for every n ∈ N , we denote by ρ n the discretemetric on A ,n +1 . Then, making obvious adjustments in notation, we let d be the metric on A defined by ( ∗ ) in Subsection 2.3. By Proposition 2.8,the metric space h A , d i is compact. Using (a), one can check that { P } is asupport h A , d i and, therefore, h A , d i ∈ N . Moreover, for every n ∈ N , { P } is a support of A ,n . Hence the family A = { A ,n +1 : n ∈ N } is denumerablein N . We notice that if M ⊆ N , then { A ,n +1 : n ∈ M } ∈ N because { P } is a support of A ,n +1 for every n ∈ M . Suppose that A has a partialchoice function in N . Then there exists an infinite set M ⊆ N such that thefamily B = { A ,n +1 : n ∈ M } has a choice function in N . Let f be a choicefunction of B such that f ∈ N . Let D ∈ [ P ] <ω be a support of f . Then D ′ = D ∩ P is also a support of f . Let n ∈ ω and φ i ∈ G with i ∈ n + 1 be such that D ′ = { φ i ( P ) : i ∈ n + 1 } . Since every permutation from G moves only finitely many atoms, there exists n ∈ M such that n ≥ and A ,n ∈ φ i ( P ) for all i ∈ n + 1 .Assume that f ( A ,n ) = x . Since | A ,n | = n ≥ , there exists y ∈ A ,n such that y = x . Let η = ( x , y ) , i.e., η is the permutation of A whichinterchanges x and y , and fixes all other atoms of N . Then η ( A ,n ) = A n and η ∈ fix G ( D ′ ) . Since D ′ is a support of f , we have η ( f ) = f .Therefore, since h A ,n , x i ∈ f , we infer that h η ( A ,n ) , η ( x ) i ∈ η ( f ) = f ,so h A ,n , y i ∈ f and, in consequence, x = y . The contradiction obtainedshows that A does not have a partial choice function in N . This implies thatthe set A is Dedekind-finite, and, thus, ICMDI is false in N .(ii) Let Φ be the statement UT ( ℵ , ℵ , cuf ) ∧ ¬ ICMDI . We have al-ready noticed that ¬ ICMDI is injectively boundable. This, together withProposition 4.22, implies that Φ is a conjunction of injectively boundablestatements. Therefore, (ii) follows from (i) and from Theorem 2.19.Clearly, every ZF - model for UT ( ℵ , ℵ , cuf ) ∧ ¬ ICMDI is also a modelfor UT ( ℵ , ℵ , cuf ) ∧ ¬ M ( C, S ) . This, together with Theorem 4.23, impliesthe following corollary: 29 orollary 4.24. The conjunction UT ( ℵ , ℵ , cuf ) ∧ ¬ M ( C, S ) has a ZF -model.Remark . Let N be the model that we have used in the proof to Theorem4.23. The proof to Theorem 3.3 in [6] shows that UT ( ℵ , cuf, cuf ) is false in N . Hence CUC is also false in N . Since the statement UT ( ℵ , ℵ , cuf ) ∧¬ UT ( ℵ , cuf, cuf ) has a permutation model (for instance, N ), it also has a ZF -model by Proposition 4.22 and Theorem 2.19. M ( C, (cid:3) ) It is known that every separable metrizable space is second-countable in ZF .It is also known, for instance, from Theorem 4.54 of [12] or from [9] that,in ZF , the statement “every second-countable metrizable space is separable”is equivalent to CAC ( R ) . The negation of CAC ( R ) is relatively consistentwith ZF , so it is relatively consistent with ZF that there are non-separablesecond-countable metrizable spaces. On the other hand, by Theorem 2.7(i),it holds in ZF that separability and second-countability are equivalent inthe class of compact metrizable spaces. Theorem 2.1 shows that totallybounded metric spaces are second-countable in ZF + CAC ; in particular, itholds in ZF + CAC that all compact metrizable spaces are second-countable.However, the situation is completely different in ZF . There exist ZF -modelsincluding compact non-separable metric spaces. Namely, it follows from The-orem 2.2 that in every ZF -model satisfying the negation of CAC fin , thereexists an uncountable, non-separable compact metric space whose size is in-comparable to | R | . The following proposition shows (among other facts)that M ( C, ≤ | R | ) and M ( C, S ) are essentially stronger than CAC fin in ZF and, furthermore, IDI is independent of both ZF + M ( C, ≤ | R | ) and ZF + M ( C, S ) . Proposition 5.1. (i) ( ZFA ) M ( C, S ) → M ( C, ≤ | R | ) → CAC fin .(ii) N cr | = ( ¬ M ( C, S )) ∧ ¬ M ( C, ≤ | R | ) .(iii) CAC fin implies neither M ( C, ≤ | R | ) nor M ( C, S ) in ZF .(iv) IDI implies neither M ( C, ≤ | R | ) nor M ( C, S ) in ZF .(v) Neither M ( C, ≤ | R | ) nor M ( C, S ) implies IDI in ZF . roof. (i) It follows directly from Theorem 2.2 that the implications given in(i) are true in ZF ; however, the arguments from [23] are sufficient to showthat these implications are also true in ZFA .(ii) By Proposition 4.9, there exists a compact metric space X = h X, d i in N cr such that the set X is not well-orderable in N cr . Since R is well-orderable in N cr , the set X is not equipotent to a subset of R in N cr . Hence M ( C, ≤ | R | ) fails in N cr . This, together with (i), implies (ii).(iii)–(iv) Let Φ be either CAC fin or IDI . In the light of (i), to prove(iii) and (iv), it suffices to show that the conjunction Φ ∧ ¬ M ( C, ≤ | R | ) hasa ZF -model. It follows from (ii) that the conjunction Φ ∧ ¬ M ( C, ≤ | R | ) hasa permutation model (for instance, N cr ). Therefore, since the statements CAC fin , IDI and ¬ M ( C, ≤ | R | ) are all injectively boundable, Φ ∧ ¬ M ( C, ≤| R | ) has a ZF -model by Theorem 2.19.(v) Let Ψ be either M ( C, ≤ | R | ) or M ( C, S ) . Since BPI is true in N ,it follows from Theorem 2.7 (iii) that Ψ is true in N . It is known that IDI is false in N . Hence, the conjunction Ψ ∧ ¬ IDI has a permutation model.To complete the proof, it suffices to apply Theorem 2.19.
Theorem 5.2. ( ZF ) (i) ( CAC fin ∧ M ( C, σ − l.f )) ↔ M ( C, S ) .(ii) ( CAC fin ∧ M ( C, ST B )) ↔ M ( C, S ) .Proof. Let X = h X, d i be a compact metric space.( → ) We assume both CAC fin and M ( C, σ − l.f ) . By our hypothesis, X has a base B = S {B n : n ∈ N } such that, for every n ∈ N , the family B n is locally finite. In ZF , to check that if A is a locally finite family in X , then it follows from the compactness of X that A is finite, we noticethat the collection V of all open sets V of X such that V meets only finitelymany members of A is an open cover of X , so V has a finite subcover.In consequence, X meets only finitely many members of A , so A is finite.Therefore, for every n ∈ N , the family B n is finite. This, together with CAC fin , implies that the family B is countable, so X is second-countable.Hence, by Theorem 2.7(i), X is separable as required.( ← ) By Proposition 5.1, M ( C, S ) implies CAC fin . To conclude the proofto (i), it suffices to notice that M ( C, S ) implies M ( C, and M ( C, triviallyimplies that every compact metric space has a σ - locally finite base.31ii) ( → ) Now, we assume both CAC fin and M ( C, ST B ) . Since X isstrongly totally bounded, it follows that it admits a sequence ( D n ) n ∈ N suchthat, for every n ∈ N , D n is a n -net of X . By CAC fin , the set D = S { D n : n ∈ N } is countable. Since, D is dense in X , it follows that X is separable.( ← ) It is straightforward to check that every separable compact met-ric space is strongly totally bounded. Hence M ( C, S ) implies M ( C, ST B ) .Proposition 5.1 completes the proof. Remark . By Proposition 5.1, there exists a model M of ZF in which CAC fin holds and M ( C, S ) fails. By Theorem 5.2(i), M ( C, σ − l.f ) fails in M . This, together with Theorem 2.5(i), implies that M ( C, σ − l.f. ) inde-pendent of ZF . Theorem 5.4. ( ZF ) (i) ( CAC ( R , C ) ∧ M ( C, ≤ | R | )) ↔ M ( C, S ) . (ii) CAC ( R ) does not imply M ( C, ≤ | R | ) .Proof. (i) ( → ) We assume CAC ( R , C ) and M ( C, ≤ | R | ) . We fix a compactmetric space X = h X, d i and prove that X is separable. For every n ∈ N , let X n = h X n , d n i where d n is the metric on X n defined by: d n ( x, y ) = max { d ( x ( i ) , y ( i )) : i ∈ n } . Then X n is compact for every n ∈ N . By M ( C, ≤ | R | ) , | X | ≤ | R | . Therefore,since | X N | ≤ | R | , there exists a family { ψ n : n ∈ N } such that, for every n ∈ N , ψ n : X n → R is an injection. The metric d is totally bounded, so, forevery n ∈ N , the set M n = { m ∈ N : ∃ y ∈ X m , ∀ x ∈ X, d ( x, { y ( i ) : i ∈ n } ) < n } is non-empty. Let k n = min M n for every n ∈ N . To prove that X is stronglytotally bounded, for every n ∈ N , we consider the set C n defined as follows: C n = { y ∈ X k n : ∀ x ∈ X ( d ( x, { y ( i ) : i ∈ k n } ) < n ) } . We claim that for every n ∈ N , C n is a closed subset of X k n . To this end,we fix y ∈ X k n \ C n . Then, since X is infinite, there exists x ∈ X suchthat B d ( x , n ) ∩ { y ( i ) : i ∈ n } = ∅ . Let r = d ( x , { y ( i ) : i ∈ n } ) and32 = r − n . Then ε > . To show that B d kn ( y , ε ) ∩ C n = ∅ , suppose that z ∈ B d kn ( y , ε ) ∩ C n . Then d ( x , { z ( i ) : i ∈ k n } ) = max { d ( x , z ( i )) : i ∈ k n } < n and d k n ( y , z ) = max { d ( y ( i ) , z ( i )) : i ∈ k n } < ε. For every i ∈ k n , we have: r ≤ d ( x , y ( i )) ≤ d ( x , z ( i )) + d ( z ( i ) , y ( i )) ≤ d ( x , z ( i )) + d k n ( z , y ) . Hence, for every i ∈ k n , the following inequalities hold: r − d k n ( z , y ) ≤ d ( x , z ( i )) < n . In consequence, ε < d k n ( z , y ) . The contradiction obtained shows that B d kn ( y , ε ) ∩ C n = ∅ . Hence, for every n ∈ N , the non-empty set C n iscompact in the metric space X k n . Therefore, it follows from CAC ( R , C ) that the family { ψ n ( C n ) : n ∈ N } has a choice function. This implies that { C n : n ∈ N } has a choice function, so we can fix f ∈ Q n ∈ N C n . Then, for every n ∈ N , the set D n = { f ( n )( i ) : i ∈ k n } is a n -net in X . This shows that X isstrongly totally bounded. It is easily seen that the set D = S n ∈ N D n is count-able and dense in X . Hence CAC ( R , C ) ∧ M ( C, ≤ | R | ) implies M ( C, S ) .( ← ) By Proposition 5.1(i), M ( C, S ) implies M ( C, ≤ | R | ) . Assuming M ( C, S ) , we prove that CAC ( R , C ) holds. To this aim, we fix a disjointfamily A = { A n : n ∈ N } of non-empty subsets of R such that there existsa family { ρ n : n ∈ N } of metrics such that, for every n ∈ N , h A n , ρ n i is acompact metric space. Let A = S n ∈ N A n , let ∞ / ∈ A and X = A ∪ {∞} . Let d be the metric on X defined by ( ∗ ) in Subsection 2.3. Then, by Proposition2.8, X = h X, d i is a compact metric space. It follows from M ( C, S ) that X is separable. Let H = { x n : n ∈ N } be a dense set in X . For every n ∈ N ,let m n = min { m ∈ N : x m ∈ A n } and let h ( n ) = x m n . Then h is a choicefunction of A . Hence M ( C, S ) implies CAC ( R , C ) .(ii) It was shown in [23] that CAC ( R ) and M ( C, S ) are independentof each other. Since M ( C, S ) implies M ( C, ≤ | R | ) (see Proposition 5.1(i)),while CAC ( R ) implies CAC ( R , C ) but not M ( C, S ) the conclusion followsfrom (i). 33 orollary 5.5. In every permutation model, M ( C, ≤ | R | ) and M ( C, S ) areequivalent.Proof. Let N be a permutation model. Since R is well-orderable in N (seeSubsection 2.4), CAC ( R ) is true in N . This, together with Theorem 5.4(i),completes the proof. Remark . (i) The proof to Corollary 5.5 shows that M ( C, ≤ | R | ) and M ( C, S ) are equivalent in every model of ZFA in which R is well-orderable.(ii) In much the same way, as in the proof to Theorem 5.4(ii)( ← ), onecan show that, for every family { X n : n ∈ ω } of pairwise disjoint compactspaces, if the direct sum X = L n ∈ ω X n is metrizable, then it is separable.We include a sketch of a ZF -proof to the following lemma for complete-ness. We use this lemma in our ZF -proof that M ( C, ֒ → [0 , N ) and M ( C, S ) are equivalent. Lemma 5.7. ( ZF ) Suppose that B is a base of a non-empty metrizable space X = h X, τ i . Then there exists a homeomorphic embedding of X into the cube [0 , B×B .Proof.
We may assume that X consists of at least two points. Let d be ametric on X such that τ = τ ( d ) and let W = {h U, V i ∈ B × B : ∅ 6 = cl X U ⊆ V = X } . For every W = h U, V i ∈ W , by defining f W ( x ) = d ( x, cl X ( U )) d ( x, cl X ( U )) + d ( x, X \ V ) whenever x ∈ X, we obtain a continous function from X into [0 , . Let h : X → [0 , W be theevaluation mapping defined by: h ( x )( W ) = f W ( x ) for all x ∈ X and W ∈ W .Then h is a homeomorphic embedding of X into [0 , W . To complete theproof, it suffices to notice that [0 , W is homeomorphic to a subspace of [0 , B×B . Theorem 5.8. ( ZF ) (i) M ( C, ֒ → [0 , N ) ↔ M ( C, S ) .(ii) M ( C, ≤ | R | ) → M ( C, ֒ → [0 , R ) . roof. Let X = h X, τ i be an infinite compact metrizable space and let d bea metric on X such that τ = τ ( d ) .(i) ( → ) We assume M ( C, ֒ → [0 , N ) and show that X is separable. Byour hypothesis, X is homeomorphic to a compact subspace Y of the Hilbertcube [0 , N . Since [0 , N is second-countable, it follows from Theorem 2.7(i)that Y is separable. Hence X is separable. In consequence, M ( C, ֒ → [0 , N ) implies M ( C, S ) .( ← ) If M ( C, S ) holds, then every compact metrizable space is second-countable. Since, by Lemma 5.7, every second-countable metrizable space isembeddable in the Hilbert cube [0 , N , M ( C, S ) implies M ( C, ֒ → [0 , N ) .(ii) Now, suppose that M ( C, ≤ | R | ) holds. Then | X | ≤ | R | . Since | [ R ] <ω | = | R | , we infer that | [ X ] <ω | ≤ | R | . For every n ∈ N , let k n = min { m ∈ N : [ x ∈ A B d ( x, n ) = X for some A ∈ [ X ] m } and E n = { A ∈ [ X ] k m : [ x ∈ A B d ( x, n ) = X } . Let B = { B d ( x, n ) : x ∈ A, A ∈ E n , n ∈ N } . It is straightforward to verifythat B is a base for X of size |B| ≤ | R × N | ≤ | R | , so |B × B| ≤ | R | . This,together with Lemma 5.7, implies that X is embeddable into [0 , R . Hence M ( C, ≤ | R | ) implies M ( C, ֒ → [0 , R ) .In view of Theorem 5.8, one may ask the following questions: Question . (i) Does M ( C, ֒ → [0 , R ) imply M ( C, ֒ → [0 , N ) ?(ii) Does M ( C, ֒ → [0 , R ) imply M ( C, ≤ | R | ) ?(iii) Does CAC fin imply M ( C, ֒ → [0 , R ) ?(iv) Does M ( C, ֒ → [0 , R ) imply CAC fin ? Remark . ( a ) Regarding Question 5.9 (i)-(ii), we notice that the answeris in the affirmative in permutation models. Indeed, let N be a permutationmodel. It is known that R and P ( R ) are well-orderable in N (see Subsection2.4). Therefore, assuming that M ( C, ֒ → [0 , R ) holds in N and workinginside N , we deduce that, given a compact metrizable space X in N , X embeds in [0 , R . Hence X is a well-orderable space, so X is Loeb. Since35 is a compact metrizable Loeb space, by Theorem 2.7(i), X is second-countable. Therefore, by Lemma 5.7, X embeds in [0 , N and, consequently, | X | ≤ | R | .( b ) Regarding Question 5.9(iii), we note that CAC fin holds in the per-mutation model N cr . To show that M ( C, ֒ → [0 , R ) fails in N cr , we noticethat, by Proposition 4.9, there exists a compact metric space X = h X, d i in N cr such that X is not well-orderable in N cr . Since [0 , R , being equipotentto the well-orderable set P ( R ) of N cr , is well-orderable in N cr , it is true in N cr that X is not embeddable in [0 , R . This explains why M ( C, ֒ → [0 , R ) fails in N cr . Therefore, since the conjunction CAC fin ∧ ¬ M ( C, ֒ → [0 , R ) has a permutation model, it also has a ZF -model by Theorem 2.19.To give more light to Questions 5.9 (iii)-(iv), let us prove the followingTheorems 5.11 and 5.12. Theorem 5.11. ( ZF ) (i) M ( C, |B Y | ≤ |B| ) → M ([0 , , |B Y | ≤ |B| ) → IDI ( R ) .(ii) The following are equivalent:(a) Every compact (0-dimesional) subspace of the Tychonoff cube [0 , R has a base of size ≤ | R | ;(b) every compact (0-dimensional) subspace of the Tychonoff cube [0 , R with a unique accumulation point has a base of size ≤ | R | ;(c) Part ( R ) .(iii) Every compact metrizable subspace of the Tychonoff cube [0 , R with aunique accumulation point has a base of size ≤ | R | iff for every denu-merable family A of finite subsets of P ( R ) such that S A is pairwisedisjoint, | S A| ≤ | R | .Proof. (i) Assume that IDI ( R ) is false. By a well-known result of N. Brunner(cf. [15, Form [13 A]]), there exists a Dedekind-finite dense subset of theinterval (0 , with its usual topology. Then B = { ( x, y ) : x, y ∈ D, x < y } ∪ { [0 , x ) : x ∈ D } ∪ { ( x,
1] : x ∈ D } is a base for the usual topology of [0 , . Clearly, the set B is Dedekind-finite.Let us consider the compact subspace Y of [0 , where(7) Y = { } ∪ { n : n ∈ N } . {{ n } : n ∈ N } ⊆ B Y , the set B Y is Dedekind-infinite. Therefore,if |B Y | ≤ |B| , then B is Dedekind-infinite but this is impossible. Hence M ([0 , , |B Y | ≤ |B| ) implies IDI ( R ) . It is clear that M ( C, |B Y | ≤ |B| ) implies M ([0 , , |B Y | ≤ |B| ) . This completes the proof to (i).(ii) It is obvious that ( a ) implies ( b ) . ( b ) → ( c ) Fix a partition P of R . That is, P is a disjoint family ofnon-empty subsets of R such that R = S P . Assuming ( b ) , we show that |P| ≤ | R | . For P ∈ P , let f P : R → { , } be the characteristic function of P and let f ( x ) = 0 for each x ∈ R . We put X = { f } ∪ { f P : P ∈ P} . We claim that the subspace X of [0 , R is compact. To see this, let us consideran arbitrary family U of open subsets of [0 , R such that X ⊆ S U . Thereexists U ∈ U such that f ∈ U . There exist ε ∈ (0 , and a non-empty finitesubset J of R such that the set V = \ { π − j ([0 , ε )) : j ∈ J } is a subset of U where, for each j ∈ R and x ∈ [0 , R , π j ( x ) = x ( j ) . Since J is finite, there exists a finite set P J ⊆ P such that J ⊆ S P J . We noticethat, for every P ∈ P \ P J and every j ∈ J , f P ( j ) = 0 . Hence f P ∈ U forevery P ∈ P \ P J . This implies that there exists a finite set W such that W ⊆ U and X ⊆ S W . Hence X is compact as claimed. If P ∈ P and j ∈ P ,then π − j (( , ∩ X = { f P } , so f P is an isolated point of X . Hence f is theunique accumulation point of X . The space X is also 0-dimensional. By ( b ) , X has a base B equipotent to a subset of R . Since {{ f P } : P ∈ P} ⊆ B , itfollows that |P| ≤ | R | as required. ( c ) → ( a ) We assume
Part ( R ) and fix a compact subspace X of the cube [0 , R . It is well known that [0 , R is separable in ZF (cf., e.g., [25]). Fix acountable dense subset D of [0 , R . For every y ∈ D , let V y = { \ { π − i (( y ( i ) − /m, y ( i ) + 1 /m )) : i ∈ F } : ∅ 6 = F ∈ [ R ] <ω , m ∈ N } . Since | [ R ] <ω | = | R × N | = | R | in ZF , it follows that B = S {V y : y ∈ D } isequipotent to R . It is a routine work to verify that B is a base for [0 , R .Define an equivalence relation ∼ on B by requiring:(8) O ∼ Q iff O ∩ X = Q ∩ X .37learly(9) B X = { P ∩ X : [ P ] ∈ B / ∼} is a base for X of size |B / ∼ | . Since |B / ∼ | ≤ | R | , it follows that |B X | ≤ | R | as required.(iii) ( → ) Fix family A = { A n : n ∈ N } of finite subsets of P ( R ) suchthat the family P = S A is pairwise disjoint. Let P = P ∪ { R \ S P } and P = P \ {∅} . Then P is a partition of R . Let f, f P with P ∈ P and X be defined as in the proof of (ii) that ( b ) implies ( c ) . Since P is a cuf set,the space X has a σ -locally finite base. This, together with Theorem 2.5(ii),implies that X is metrizable. Suppose X has a base B such that |B| ≤ | R | .In much the same way, as in the proof that ( b ) implies ( c ) in (ii), we canshow that |P| ≤ | R | . Then | S A| ≤ | R | .( ← ) Now, we consider an arbitrary compact metrizable subspace X ofthe cube [0 , R such that X has a unique accumulation point. Let x bethe accumulation point of X and let d be a metric on X which induces thetopology of X . For every x ∈ X \{ x } let n x = min { n ∈ N : B d ( x, n ) = { x }} . For every n ∈ N , let E n = { x ∈ X : n x = n } .Without loss of generality, we may assume that, for every n ∈ N , E n = ∅ .Since X is compact, it follows easily that, for every n ∈ N , the set E n isfinite. Let us apply the base B of [0 , R given in the proof of part (ii) that ( c ) implies ( a ) . Let ∼ be the equivalence relation on B given by (8). Let C X = { [ P ] ∈ B / ∼ : | P ∩ ( X \ { x } ) | = 1 } . For every n ∈ N , let C n = { [ P ] ∈ C X : P ∩ E n = ∅} . Clearly, for every n ∈ N , | C n | = | E n | . Let C = S { C n : n ∈ N } . Thereexists a bijection ψ : B → R . For every n ∈ N . we put A n = {{ ψ ( U ) : U ∈ H } : H ∈ C n } . Then, for every n ∈ N , A n is a finite subset of P ( R ) . Let A = { A n : n ∈ N } . Then S A is pairwise disjoint. Suppose that | S A| ≤ | R | .Then |C| ≤ | R | . This implies that the family W = { P ∩ ( X \ { x } ) : [ P ] ∈ C} is of size ≤ | R | . The family G = W ∪ { B d ( x , n ) : n ∈ N } is a base of X suchthat |G| ≤ | R | . 38he following theorem leads to a partial answer to Question 5.9(iv). Theorem 5.12. ( ZF ) (i) ( M ( C, ֒ → [0 , R ) ∧ Part ( R )) ↔ M ( C, B ( R )) .(ii) M ( C, S ) → M ( C, W ( R )) → M ( C, B ( R )) → CAC fin .(iii) ( CAC ( R ) ∧ M ( C, B ( R )) → M ( C, S ) .(iv) CAC ( R ) → ( M ( C, S ) ↔ M ( C, W ( R )) ↔ M ( C, B ( R )) .Proof. (i) This follows from Theorem 5.11(ii) and Lemma 5.7.(ii) It is obvious that M ( C, W ( R )) implies M ( C, B ( R )) . Assume M ( C, S ) and let Y = h Y, τ i be an infinite compact metrizable separable space. Since Y is second-countable and | R ω | = | R | , it follows that | τ | ≤ | R | . To show that | R | ≤ | τ | , we notice that, since X is infinite and X is second-countable, thereexists a disjoint family { U n : n ∈ ω } such that, for each n ∈ ω , U n ∈ τ . For J ∈ P ( ω ) , we put ψ ( J ) = S { U n : n ∈ J } to obtain an injection ψ : ω → τ .Hence | R | = |P ( ω ) | ≤ | τ | .To see that M ( C, B ( R )) → CAC fin , we assume M ( C, B ( R )) , fix a dis-joint family A = { A n : n ∈ N } of non-empty finite sets and show that A hasa choice function. To this aim, we put A = S A , take an element ∞ / ∈ A and X = A ∪ {∞} . For each n ∈ N , let ρ n be the discrete metric on A n . Let d be the metric on X defined by ( ∗ ) in Subsection 2.3. By our hypothesis, thespace X = h X, d i has a base B of size ≤ | R | . Let ψ : B → R be an injection.Since {{ x } : x ∈ A } ⊆ B and the sets A n are finite, for each n ∈ N , we candefine A ⋆n = { ψ ( { x } ) : x ∈ A n } and a ⋆n = min A ⋆n . For each n ∈ N , there isa unique x n ∈ A n such that ψ ( { x n } ) = a ⋆n . This shows that A has a choicefunction.(iii) Now, we assume both CAC ( R ) and M ( C, B ( R )) . Let us consideran arbitrary compact metric space X = h X, ρ i . By our hypothesis, X has abase B of size ≤ | R | . Since, | [ R ] <ω | ≤ | R | , it follows that | [ B ] <ω | ≤ | R | . Forevery n ∈ N , let A n = {F ∈ [ B ] <ω : [ F = X ∧ ∀ F ∈ F ( δ ρ ( F ) ≤ n ) } . Since X is compact, ρ is totally bounded. Therefore, A n = ∅ for every n ∈ N . By CAC ( R ) , we can fix a sequence ( F n ) n ∈ N such that, for every39 ∈ N , F n ∈ A n . Since | [ B ] <ω | ≤ | R | , we can also fix a sequence ( ≤ n ) n ∈ N such that, for every n ∈ N , ≤ n is a well-ordering on F n . This implies that thefamily F = S {F n : n ∈ N } is countable. Furthermore, it is a routine workto verify that F is a base of X . Hence X is second-countable. By Theorem2.7 (i), X is separable. This completes the proof to (iii).That (iv) holds follows directly from (ii) and (iii).Our proof to the following theorem emphasizes the usefulness of Theorems3.1 and 5.12: Theorem 5.13. ( a ) The following implications are true in ZF : WO ( P ( R )) → WO ( R ) → Part ( R ) → ( M ( C, ֒ → [0 , R ) → CAC fin ) . ( b ) There exists a symmetric model of ZF + CH + WO ( P ( R )) in which M ( C, ֒ → [0 , R ) is false. Hence, M ( C, ֒ → [0 , R ) does not follow from Part ( R ) in ZF .Proof. It is obvious that the first two implications of ( a ) are true in ZF .Thus, it follows directly from Theorem 5.12 (i)–(ii) that ( a ) holds. To prove( b ), let us notice that, in the light of Theorem 3.1, we can fix a symmetricmodel M of ZF + CH + WO ( P ( R )) + ¬ CAC fin . It follows from ( a ) that Part ( R ) is true in M but M ( C, ֒ → [0 , R ) fails in M . Remark . (i) To show that Part ( R ) is not provable in ZF , let us recallthat, in [11], a ZF -model Γ was constructed such that, in Γ , there exists afamily F = { F n : n ∈ N } of two-element sets such that S F is a partition of R but F does not have a choice function. Then, in Γ , there does not existan injection ψ : S F → R (otherwise, F would have a choice function in Γ ).Hence Part ( R ) fails in Γ .(ii) Since Part ( R ) is independent of ZF , it follows from Theorem 5.11(ii)that it is not provable in ZF that every compact metrizable subspace of thecube [0 , R has a base of size ≤ | R | . We do not know if M ( C, ֒ → [0 , R ) implies every compact metrizable subspace of the cube [0 , R has a base ofsize ≤ | R | .(iii) It is not provable in ZFA that every compact metrizable space with aunique accumulation point embeds in [0 , R . Indeed, in the Second Fraenkelmodel N A = { A n : n ∈ N } whose union has no denumerable subset. Let A = S A , ∞ / ∈ A ,40 = A ∪ {∞} and, for every n ∈ N , let ρ n be the discrete metric on A n . Let d be the metric on X defined by ( ∗ ) in Subsection 2.3. Let X = h X, τ ( d ) i .Then X is a compact metrizable space having ∞ as its unique accumulationpoint. Since, in N , the set [0 , R is well-orderable, while A has no choicefunction, it follows that X does not embed in the Tychonoff cube [0 , R .This shows that the statement “There exists a compact metrizable spacewith a unique accumulation point which is not embeddable in [0 , R ” has apermutation model. For readers’ convenience, let us repeat the open problems mentioned in Sec-tions 4 and 5.1. Is M ( C, W O ) equivalent to or weaker than M ( T B, W O ) in ZF ?2. Does M ( C, S ) imply CUC in ZF ?3. Does BPI imply
CUC in ZF ?4. Does CUC imply M ( C, S ) in ZF ?5. Does M ( C, ֒ → [0 , R ) imply CAC fin in ZF ? (Cf. Question 5.9(iv).) References [1] R. H. Bing,
Metrization of topological Spaces , Canadian J. Math. (1951), 175–186.[2] N. Brunner, Products of compact spaces in the least permutationmodel , Z. Math. Logik Grundlagen Math. (1985), 441–448.[3] P. J. Collins and A. W. Roscoe, Criteria for metrisability , Proc. Amer.Math. Soc. (1984), 631–640.[4] S. M. Corson, The independence of Stone’s theorem from the Booleanprime ideal theorem , https://arxiv.org/pdf/2001.06513.pdf.415] De la Cruz, E. J. Hall, P. Howard, K. Keremedis and J. E. Rubin,
Metric spaces and the axiom of choice , Math. Logic Quart. (2003),455–466.[6] De la Cruz, E. J. Hall, P. Howard, K. Keremedis, and J. E. Rubin, Unions and the axiom of choice,
Math. Logic Quart. (2008), 652–665.[7] E. K. van Douwen, Horrors of topology without AC: A nonnormalorderable space , Proc. Amer. Math. Soc. (1985), 101–105.[8] R. Engelking, General Topology , Sigma Series in Pure Mathematics6, Heldermann, Berlin, 1989.[9] C. Good and I. Tree,
Continuing horrors of topology without choice ,Topology Appl. (1995), 79–90.[10] C. Good, I. Tree, W. Watson, On Stone’s theorem and the axiom ofchoice , Proc. Amer. Math. Soc. (1998), 1211–1218.[11] E.J. Hall, K. Keremedis and E. Tachtsis,
The existence of free ultra-filters on ω does not imply the extension of filters on ω to ultrafilters ,Math. Logic Quart. (2013), 258–267.[12] H. Herrlich, Axiom of Choice , Lecture Notes in Mathematics 1875,Springer, New York, 2006.[13] P. Howard,
Limitations on the Fraenkel-Mostowski method of inde-pendence proofs , J. Symb. Logic (1973), 416–422.[14] P. Howard, K. Keremedis, J. E. Rubin and A. Stanley, Paracompact-ness of metric spaces and the axiom of multiple choice , Math. LogicQuart. (2000), 219–232.[15] P. Howard and J. E. Rubin, Consequences of the axiom of choice ,Math. Surveys and Monographs 59, A.M.S., Providence R.I., 1998.4216] P. Howard and J. E. Rubin, Other forms added to the ones from [15],https://cgraph.inters.co/.[17] P. Howard and J. Solski,
The Strength of the ∆ -system Lemma , NotreDame J. Formal Logic (1993), no. 1, 100–106.[18] P. Howard, D. I. Saveliev, and E. Tachtsis, On the set-theoreticstrength of the existence of disjoint cofinal sets in posets without max-imal elements , Math. Logic Quart. (2016), no. 3, 155–176.[19] P. Howard and E. Tachtsis, On metrizability and compactness of cer-tain products without the axiom of choice , submitted.[20] T. Jech,
The Axiom of Choice , North-Holland Publishing Co., 1973.[21] T. Jech,
Set Theory. The Third Millennium Edition, revised andexpanded,
Springer Monographs in Mathematics, Springer, Berlin,2003.[22] K. Keremedis,
Consequences of the failure of the axiom of choice inthe theory of Lindelöf metric spaces , Math. Logic Quart. (2004),no. 2, 141–151.[23] K. Keremedis, On sequentially compact and related notions of com-pactness of metric spaces in ZF , Bull. Pol. Acad. Sci. Math. (2016), 29–46.[24] K. Keremedis, Some notions of separability of metric spaces in ZF and their relation to compactness , Bull. Pol. Acad. Sci. Math. (2016), 109–136.[25] K. Keremedis, Clopen ultrafilters of ω and the cardinality of the Stonespace S ( ω ) in ZF , Topology Proc. (2018), 1–17.[26] K. Keremedis and E. Tachtsis, On Loeb and weakly Loeb Hausdorffspaces , Sci. Math. Jpn. Online (2001), 15–19.4327] K. Keremedis and E. Tachtsis, Compact metric spaces and weak formsof the axiom of choice , Math. Logic Quart. (2001), 117–128.[28] K. Keremedis and E. Tachtsis, Countable sums and products ofmetrizable spaces in ZF , Math. Logic Quart. (2005), 95–103.[29] K. Keremedis and E. Tachtsis, Countable compact Hausdorff spacesneed not be metrizable in ZF , Proc. Amer. Math. Soc. (2007),1205-1211.[30] K. Keremedis and E. Wajch, On Loeb and sequential spaces in ZF ,Topology Appl. (2020), 101279.[31] K. Keremedis and E. Wajch, Cuf products and cuf sums of(quasi)-metrizable spaces in ZF , submitted, preprint available athttp://arxiv.org/abs/2004.13097[32] K. Kunen, Set Theory. An Introduction to Independence Proofs ,North-Holland, Amsterdam, 1983.[33] K. Kunen,
The Foundations of Mathematics , Individual Authors andCollege Publications, London, 2009.[34] P. A. Loeb,
A new proof of the Tychonoff theorem , Amer. Math.Monthly (1965), 711–717.[35] J. Nagata, On a necessary and sufficient condition on metrizability ,J. Inst. Polytech., Osaka City University (1950), 93–100.[36] J. Nagata, Modern General Topology , North-Holland, 1985.[37] D. Pincus,
Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods