Several results on compact metrizable spaces in ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ZF}$$\end{document}

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ZF}$$\end{document}, some are shown to be independent of ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ZF}$$\end{document}. For independence results, distinct models of ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ZF}$$\end{document} and permutation models of ZFA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ZFA}$$\end{document} with transfer theorems of Pincus are applied. New symmetric models of ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ZF}$$\end{document} are constructed in each of which the power set of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} is well-orderable, the Continuum Hypothesis is satisfied but a denumerable 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0, 1]^{\mathbb {R}}$$\end{document}.


The set-theoretic framework
In this paper, the intended context for reasoning and statements of theorems is the Zermelo-Fraenkel set theory ZF without the axiom of choice AC. The system ZF + AC is denoted by ZFC. We recommend [32,33] as a good introduction to ZF. To stress the fact that a result is proved in ZF or in ZF + A (where A is a statement independent of ZF), we shall write at the beginning of the statements of the theorems and propositions (ZF) or (ZF + A), respectively. Apart from models of ZF, we refer to some models of ZFA (or ZF 0 in [15]), that is, we refer also to ZF with an infinite set of atoms (see [15,20,21]). Our theorems proved here in ZF are also provable in ZFA; however, we also mention some theorems of ZF that are not theorems of ZFA.
A well-ordered cardinal number is 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 ordinals α, we define: where, for a set A, H (A) is the Hartogs' number of A, i.e., the least ordinal α which is not equipotent to a subset of A. For each ordinal number α, ω α is an infinite wellordered cardinal number and, as it is customary, it is denoted by ℵ α . One usually uses ℵ α when referring to the cardinality of an infinite well-orderable set, and ω α when referring to the order-type of an infinite well-ordered set. Every well-ordered cardinal number is either a finite ordinal number or an ℵ α for some ordinal α. As usual, if n ∈ ω, then n + 1 = n ∪ {n}. Members of the set N = ω\{0} are called natural numbers. The power set of a set X is denoted by P(X ). A set X is called countable if X is equipotent to a 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 is not finite. An infinite countable set is called denumerable. If X is a set and κ is a non-zero well-ordered cardinal number, then [X ] κ is the family of 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 (well-ordered) cardinal number in κ.
For sets X and Y , -|X | ≤ |Y | means that X is equipotent to a subset of Y ; -|X | = |Y | means that X is equipotent to Y ; and -|X | < |Y | means that |X | ≤ |Y | and |X | = |Y |.
The set of all real numbers is denoted by R and, if it is not stated otherwise, R and every subspace of R are considered with the usual topology and with the metric induced by the standard absolute value on R.

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 The collection (i) Given a real number ε > 0, a subset D of X is called ε-dense or an ε-net in X if X = x∈D B d (x, ε). (ii) X is called totally bounded if, for every real number ε > 0, there exists a 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 1 n -net in X. (iv) (Cf. [24].) d is called strongly totally bounded if X is 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 of ZF.
Definition 2 Let X = X , τ 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 ).
Clearly, in Definition 2 (iii), B Y is a base of Y. In Sect. 5, it is shown that 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 underlying sets.
Definition 3 A collection U of subsets of a space X is called: (i) locally finite if every point of X has a neighbourhood meeting only finitely many members of U; (ii) point-finite if every point of X belongs to at most finitely many members of U; (iii) σ -locally finite (respectively, σ -point-finite) if U is a countable union of locally finite (respectively, point-finite) subfamilies.

Definition 4 A space X is called:
(i) first-countable if every point of X has a countable base of neighbourhoods; (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 : j∈J X j → X i defined by π i (x) = x(i) for each x ∈ j∈J X j . If τ j is a topology on X j , then X = j∈J X j denotes the Tychonoff product of the topological spaces X j = X j , τ j with j ∈ J . If X j = X for every j ∈ J , then X J = j∈J X j . As in [8], for an infinite set J and the unit interval [0, 1] of R, the cube [0, 1] J is called the Tychonoff cube. If J is denumerable, then the Tychonoff cube [0, 1] J is called the Hilbert cube. In [12], all Tychonoff cubes are called Hilbert cubes. In [42], Tychonoff cubes are called cubes.
We recall that if j∈J X j = ∅, then it is said that the family {X j : j ∈ J } has a choice function, and every element of 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 a function f ∈ 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 a 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 each pair i, j of distinct elements of J . If τ j is a topology on X j for every j ∈ J , then j∈J X j denotes the direct sum of the spaces X j = X j , τ j with j ∈ J .
Definition 5 (Cf. [2,26,34].) (i) A space X is said to be Loeb (respectively, weakly Loeb) if the family of all nonempty 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 of the family of all non-empty closed subsets of X is called a (weak) Loeb function of X.
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 set X is called: (i) a cuf set if X is expressible as a countable union of finite sets (cf. [5,6,19]  Definition 7 (Cf. [31].) A topological space X , τ is called a cuf space if X is a cuf set.

The list of weaker forms of AC
In this subsection, for readers' convenience, we define and denote most of the weaker forms of AC used 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]. binary relation ρ on X if, for each x ∈ X there exists y ∈ X such that xρ y, 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 2
The following are well-known facts in ZF:  [15] and [12,Theorem 4.37]). (iv) CMC ω is equivalent to the following sentence: Every denumerable family of denumerable sets has a multiple choice function. [19] that the following implications are true in ZF and none of the implications are reversible in ZF:
Let us pass to definitions of forms concerning metric and metrizable spaces.
Definition 9 1. CAC(R, C): For every disjoint family A = {A n : n ∈ N} of nonempty subsets of R, if there exists a family {d n : n ∈ N} of metrics such that, for every n ∈ N, A n , d n is a compact metric space, then A has a choice function.

CAC(C, M):
If { X n , d n : n ∈ ω} is a family of non-empty compact metric spaces, then the family {X n : n ∈ ω} has a choice function.

M(T B, W O):
For every totally bounded metric space X , d , the set X is well-

The content of the article in brief
Although mathematicians are aware that a lot of theorems of ZFC that are included in standard textbooks on general topology (e.g., in [8,42]) may fail in ZF and many amazing disasters in topology in ZF have been discovered, new non-trivial results showing significant differences between truth values in ZFC and in ZF of some given propositions can be still surprising. In this paper, we show new results concerning forms of type M(C, ) in ZF. The main aim of our work is to establish in ZF the 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 the Pincus Transfer Theorem (cf. Theorem 7), which will be useful for the transfer of certain ZFA-independence results to ZF. The main new results of the paper are included in Sects. 3-5. Section 6 contains 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 symmetric ZF-models in each of which the conjunction CH ∧ WO(P(R)) ∧ ¬CAC f in is true (see Theorem 8).
In The status of the reverse implication is also unknown (see the discussion in Remark 14(a)). Taking the opportunity, we also fill a gap in [15,16] by proving that WoAm implies CUC (see Proposition 8).
In Sect. 5, among a plethora of results, we show that CAC f in implies neither

A list of several known theorems
We list below some known theorems for future references.  (ii) (ZF) If a T 1 -space X is regular and has a σ -locally finite base, then X is metrizable.

Remark 5
The fact that, in ZFC, a T 1 -space is metrizable if and only if it is regular and has a σ -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 in ZF that every regular T 1space which admits a σ -locally finite base is metrizable. It was established in [14] that M(σ − l. f .) is 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 model of 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 ZFmodels constructed in [4,10], there exists a metrizable space which fails to have a σ -point-finite base. This means that M(σ − l. f ) is unprovable in ZF. In Sect. 4, it is clearly explained that M(C, σ − f .l) is also unprovable in ZF.

Frequently used metrics
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, n∈N is a sequence such that, for each n ∈ N, ρ n is a metric on A n . Let d n (x, y) = min{ρ n (x, y), 1 n } for all x, y ∈ A n . We define a function d : X × X → R as follows:

Proposition 1
The function d, defined by ( * ), has the following properties: Metrics defined by ( * ) were used, for instance, in [26,27,31], as well as in several other papers not cited here.

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 model M of ZFA + AC with an infinite set A of all atoms of M, and a group G of permutations of A. For a set x ∈ M, we denote by TC(x) the transitive closure of x in M. Every permutation φ of A extends uniquely to an ∈-automorphism (usually denoted also by φ) of M. For x ∈ M, we put: We refer the readers to [20,Chapter 4, for the definitions of the concepts of a normal filter and a normal ideal.

Definition 12 (i) The permutation model N determined by M, G and a normal filter
F of subgroups of G is defined by the equality: (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:  .
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], the Basic Fraenkel Model (labeled as N 1 in [15]) and the Mostowski Linearly Ordered Model (labeled as N 3 in [15]). Let us give definitions of these models and recall some of their properties for future references. (i) A is expressed as n∈N A n where {A n : n ∈ N} is a disjoint family such that, for every n ∈ N, A n = a n,x : x ∈ S 0, 1 n and S(0, 1 n ) is the circle of the Euclidean plane R 2 , ρ e of radius 1 n , centered at 0; (ii) 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 7
We need to recall some properties of N cr for applications in this paper. Let us use the notation from Definition 13. In [26, proof of Theorem 2.6], it was proved that {A n : n ∈ N} does not have a multiple choice function in N cr . In [27, proof of Theorem 3.5], it was proved that IDI holds in N cr , so CAC f in also holds in N cr (see Remark 2(i)). Then the Basic Fraenkel Model N 1 is the permutation model determined by M, G and I.

Remark 8
It is known that, in N 1, 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 1 but CAC f in is true in N 1 (see [15, p. 177]). (i) the set A is denumerable and there is a fixed ordering ≤ in A such that A, ≤ is order isomorphic to the set of all rational numbers equipped with the standard linear order; (ii) G is the group of all order-automorphisms of A, ≤ .
Then the Mostowski Linearly Ordered Model N 3 is the permutation model determined by M, G and I.

Remark 9
It is known that the power set of the set of all atoms is Dedekind-finite in N 3, so IDI fails in N 3 (see [15, pp. 182-183]). However, BPI and CAC f in are true in N 3 (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:

Remark 10
If a statement A is satisfied in a permutation model, then to show that there exists a ZF-model in which A is satisfied, we use transfer theorems due to Pincus (cf. [37,38]). Pincus transfer theorems, together with definitions of a boundable formula and an injectively boundable formula that 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].) Let be a conjunction of statements that are either injectively boundable or BPI. If has a permutation model, then has a ZF-model.
In the definition of an injectively boundable formula, the notion of injective cardinality is involved (see [37], [15,Item (3), p. 284]). Let us recall the definition of the latter notion.

Definition 16
For a set x, the injective cardinality of x, denoted by |x| − , is the (wellordered) cardinal number defined as follows: |x| − = sup{κ : κ is a well-ordered cardinal equipotent to a subset of x}. Now, we are in a position to pass to the main body of the paper.

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 a ZF-model in which is satisfied. This is why it is good to give a direct relatively simple description of a ZF-model satisfying . By the proof of Theorem 8 below, we shall obtain an infinite class of symmetric models, each satisfying CH∧WO(P(R))∧¬CAC f in . In Sect. 5, models of this class are applied to a proof that the conjunction Part(R) ∧ ¬M(C, → [0, 1] R ) has a ZF-model (see the forthcoming Theorem 19).
For the convenience of readers, before embarking on the proof of Theorem 8, let us recall in brief the construction of symmetric extension models. Assume that M is a countable transitive model of ZFC and that P, ≤ ∈ M is a poset with a maximum element denoted by 1 P ; such a poset P, ≤ in M is said to be a notion of forcing. Let M P be the (proper) class of all P-names, which are defined by transfinite recursion within M (cf. [32, Definitions 2.5, 2.6, pp. 188-189]). We will denote a P-name byẋ and, following the notation of [32,Definition 2.10,p. 190], for x ∈ M, we will denote byx the canonical name { y, 1 P : y ∈ x} for x.
If φ is an order-automorphism of P, ≤ , then φ can be extended to an automorphism φ of M P defined by recursion, For every x ∈ M,φ(x) =x. We shall henceforth use φ to denote both the automorphism of P, ≤ and the automorphismφ of the P-names.
Let G be a group of order-automorphisms of P, ≤ , and also let Γ be a normal filter on G, that is, Γ is a filter of subgroups of G closed under conjugation (i.e., for all φ ∈ G and x is called hereditarily Γ -symmetric ifẋ is Γ -symmetric and, for every ẏ, p ∈ẋ,ẏ is hereditarily Γ -symmetric. The class of hereditarily Γ -symmetric P-names (which, in view of the above, is defined by transfinite recursion over the rank ofẋ) is denoted by HS Γ .
Let G be a P-generic filter over M, and also let In the proof of Theorem 8 below, we will write "1" instead of "1 P ", "(hereditarily) symmetric" instead of "(hereditarily) Γ -symmetric", and "HS" instead of "HS Γ ".
There is a symmetric model N n, of ZF such that Hence, it is also the case that N n, | CH ∧ WO(P(R)) ∧ ¬CAC f in .
Proof By [32, Theorem 6.18, p. 216], we may fix a countable transitive model M of ZFC + ∀m ∈ n(2 ℵ m = ℵ m+1 ). Our plan is to construct a symmetric extension model N n, of M with the required properties.
We use as our notion of forcing the set P = Fn(ω × ×ω n ×ω n , 2, ω n ) of all partial functions p with | p| < ℵ n , dom( p) ⊆ ω × × ω n × ω n and ran( p) ⊆ 2 = {0, 1}, partially ordered by reverse inclusion, i.e., for p, q ∈ P, p ≤ q if and only if p ⊇ q. The poset P, ≤ has the empty function as its maximum element, which we denote by 1. Furthermore, since ω n is a regular cardinal, it follows from [32, Lemma 6.13, p. 214] that P, ≤ is an ω n -closed poset. Therefore, 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 subsets of ω n . Moreover, by [ In M[G], for k ∈ ω, t ∈ , and i ∈ ω n , we define the following sets, together with their canonical names: Now, every permutation φ of ω × × ω n induces an order-automorphism of P, ≤ as follows: for every p ∈ P, Let G be the group of all order-automorphisms of P, ≤ induced (as in (1)) by all those permutations φ of ω × × ω n which are defined as follows: For every k ∈ ω, let σ k be a permutation of and also let η k be a permutation of ω n . We define for all k, t, i ∈ ω × × ω n . By (2), it follows that, for all φ ∈ G, k ∈ ω, and t ∈ , and thus, for all φ ∈ G and k ∈ ω, Hence, for every φ ∈ G, For every E ∈ [ω × × ω n ] <ω , we let fix G (E) = {φ ∈ G : ∀e ∈ E(φ(e) = e)} and we also let Γ be the filter of subgroups of G generated by the filter base be the symmetric extension model of M. By the definitions of Γ and HS, it is clear that everyẋ ∈ HS has a support in the above sense.
In view of the observations at the beginning of the proof, we have N n, | ∀m ∈ n(2 ℵ m = ℵ m+1 ), and thus N n, | CH ∧ WO(P(R)).
Claim For k ∈ ω, t ∈ , and i ∈ ω n , the sets a k,t,i , A k,t , A k , and A, are all elements of N n, . Moreover, A is denumerable in N n, .
Proof Fix k ∈ ω, t ∈ , and i ∈ ω n . By the definition of G, it easily follows that E = { k, t, i } is a support ofȧ k,t,i andȦ k,t , and since (by (4) and (5)) φ(Ȧ k ) =Ȧ k and φ(Ȧ) =Ȧ for all φ ∈ G, we conclude that a k,t,i , A k,t , A k , and A all belong to the model N n, . Furthermore,ḟ = { op(m,Ȧ m ), 1 : m ∈ ω}, where op(σ, τ ) is the name for the ordered pair σ G , τ G given in [32,Definition 2.16,p. 191], is an HS-name for the mapping f = { m, A m : m ∈ ω} since, for every φ ∈ G, φ fixesḟ (pointwise), and all names inḟ are hereditarily symmetric. Thus, A is denumerable in N n, .

Claim
The denumerable family A = {A k : k ∈ ω} has no partial choice function in N n, . Hence, N n, | ¬CAC .
Proof By way of contradiction, we assume that A has an infinite subfamily in N n, , B = {A m : m ∈ W } for some infinite set W ⊆ ω, which has a choice function in N n, , f say. Clearly, B has a canonical HS-name, namelyḂ = { Ȧ m , 1 : m ∈ W }. We leṫ f be an HS-name for f . There exists p ∈ G such that p "ḟ is a choice function forḂ".
Let E ∈ [ω × × ω n ] <ω be a support ofḟ . Since W is infinite and E is finite, there Since |q| < ℵ n , there exists k ∈ ω n such that, for all i ∈ ω n with i ≥ k and for all t ∈ and j ∈ ω n , m 0 , t, i, j / ∈ dom(q). We let σ m 0 be the following -cycle: and we also let η : [0, k) → [k, 2k) be an order-isomorphism. Then η induces a permutation η m 0 of ω n defined by We define a ψ ∈ G by stipulating, for all m, t, i ∈ ω × × ω n , (So, for m = m 0 , σ m and η m are the identity permutations of and ω n , respectivelyrecall the definition of G.) Then the following hold: (a) ψ ∈ fix G (E), and hence ψ(ḟ ) =ḟ (since E is a support ofḟ ), q and ψ(q) are compatible conditions. Thus, q ∪ ψ(q) is a well-defined extension of q, ψ(q), and p.
By (4), (a), (b), and (7), we obtain that and from (c), together with Eqs. (6), (7) and (8), we conclude that But then, (9) However, since t 0 = σ m 0 (t 0 ) (recall that σ m 0 is the cycle (0, 1, . . . , l − 1), so σ m 0 moves every element of ), we have that, for any P-generic filter Q over M, If not, then there exist a P-generic filter Q over M and an x ∈ (Ȧ m 0 ,t 0 ) Q ∩(Ȧ m 0 ,σ m 0 (t 0 ) ) Q . Then x = (ȧ m 0 ,t 0 ,i ) Q and x = (ȧ m 0 ,σ m 0 (t 0 ),i ) Q for some ordinals i, i ∈ ω n . Let Then D ∈ M and it is fairly easy to verify that D is dense in P. Hence, Q ∩ D = ∅. 1 In particular, for the P-generic filter H over M, we have This, together with (ii) and (iii), yields thatḟ H is not a function, contradicting property (i) ofḟ H . Hence, A has no partial choice function in the model N n, , finishing the proof of the claim.
The above arguments complete the proof of the theorem.

M(IC, DI) and M(TB, WO)
Since every compact metric space is totally bounded and every infinite separable Hausdorff space is Dedekind-infinite, let us begin our investigations of the forms of type

M(C, ) with a deeper look at the forms M(T B, W O), M(T B, S) and M(I C, D I ).
We include a simple proof of the following proposition for completeness. Proof It follows from Proposition 3 that the implications from (i) are both true. It is known that, in Feferman's model M2 in [15], CAC is true but R is not well-orderable (see [15, p. 140]). Then [0, 1] is a compact, metrizable but not well-orderable space in 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, proof of Theorem 15] that there exists a model M of ZF + ¬CAC in which it is true that if a metric space X = X , d is sequentially bounded (i.e., every sequence of points of X has a Cauchy's subsequence), then X is wellorderable and separable. By

(ii) In Cohen's Original Model M1 of [15] the following hold: M(C, S) is true but both M(T B, S) and M(T B, ST B) are false. (iii) M(C, S) imples neither M(T B, S) nor M(T B, ST B).
Proof (i) has been established in [24].
(ii)-(iii) It is known that BPI is true M1 (see [15, p. 147]). It follows from Theorem 6(iii) that M(C, S) holds in M1. On the other hand, it is known that CAC(R) fails in M1 (see [15, p. 147 We recall that a topological space X is called limit point compact if every infinite subset of X has an accumulation point in X (see, e.g., [23]).

Proposition 5 (ZFA) WoAm implies both M(T B, W O) and "every limit point compact, first-countable T 1 -space is well-orderable".
Proof Let us assume WoAm. Consider an arbitrary metric space X , d . Suppose that X is not well-orderable. By WoAm, there exists an amorphous subset B of X . Let ρ = d B × B. Lemma 1 of [5] states that every metric on an 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)\{0} is non-empty. If ε = min(ran(ρ)\{0}), then there does not exist a finite ε-net in B, ρ 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 = Y , τ is a first-countable, limit point compact T 1 -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 0 be an accumulation point of C and let {U n : n ∈ N} be a base of neighborhoods of y 0 in Y. Since Y is a T 1 -space, we can inductively define an increasing sequence (n k ) k∈N of natural numbers such that, for every k ∈ N, C ∩ (U n k \U n k+1 ) = ∅. This implies that C is not amorphous. Hence, no infinite subset of Y is amorphous, so Y is well-orderable by WoAm.

Corollary 1 N 1 | M(T B, W O).
Proof This follows from Proposition 5 and the known fact that WoAm is true in N 1 (see p. 177 in [15]).
To prove that M(T B, W O) does not imply WoAm in ZFA, let us use the model N 3. In what follows, the notation concerning N 3 is the same as in Definition 15. For a, b ∈ A with a < b (where A is the set of atoms of N 3 and ≤ is the fixed linear order on A), we denote by (a, b) the open interval in the linearly ordered set A, ≤); that is, (a, b) = {x ∈ A : a < x < b}. A proof of the following lemma can be found in [18]. A with a < b, such that Y ∈ N 3, E ∩ (a, b) = ∅ and, in N 3, there  exists a bijection f : (a, b) → Y having a support E such that E ∪ {a, b} ⊆ E and E ∩ (a, b) = ∅.

Theorem 10 N 3 | M(T B, W O).
Proof We use the notation from Definition 15. Suppose that X , d is a metric space in N 3 such that X is not well-orderable in N 3. Then X is infinite. Let E ∈ [A] <ω be a support of both X and d. By Proposition 2, there exists x ∈ X such that E is not a support of x. By Lemma 1, there exist a, b ∈ A with a < b and (a, b) ∩ E = ∅, such that there exists in N 3 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 3 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 1 , b 2 ∈ B with b 1 < b 2 and put r = ρ(b 1 , b 2 ). Let u, v ∈ B and u = v. To show that ρ(u, v) = r , we must consider several cases regarding the ordering of the elements b 1 , b 2 , u, v. We consider only one of the possible cases since all the other cases can be treated in much the same way as the chosen one. So, assume, for example, . Therefore, ran(ρ) = {0, r }. Since the range of ρ is finite, in much the same way, as in the proof to Proposition 5, we deduce that B, ρ is not totally bounded. Hence d is not totally bounded.

Corollary 2 M(T B, W O) does not imply WoAm in ZFA.
Proof It is known that WoAm is false in N 3 (see [15, p.183]). Therefore, the conjunction M(T B, W O) ∧ ¬WoAm is true in N 3 by Theorem 10.
In contrast to Corollary 1 and Theorem 10, we have the following proposition:

Proposition 6 N cr | ¬M(C, W O).
Proof It was shown in [27, proof of Theorem 3.5] that, in N cr , there exists a compact metric space X = X , d which is not weakly Loeb. Then X cannot be well-orderable in N cr . It is obvious that IDI implies M(I C, D I ) in ZF; however, it seems to be still 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 corollary follows directly from Corollary 1 and Theorem 10:

Remark 11
To transfer BPI ∧ M(I C, D I ) ∧ ¬IDI to a model of ZF, let us prove the following lemma:

Lemma 2 M(I C, D I ) is injectively boundable.
Proof First, we put It is stated neither in [15] nor in [16] that WoAm implies CUC. Since we have not seen a solution to the problem of whether this implication is true in other sources, let us notice that it follows from the following proposition that this implication holds in ZF: N is a model of ZFA in which R is well-orderable (in particular, if N is a  permutation model), then:

(iii) If N is a permutation model, then:
N | (AC f in → CUC).
Proof Let A = {A n : n ∈ ω} be a disjoint family of non-empty countable sets and let A = A. Clearly, if n∈ω (A n × ω) is countable, then A is countable. Therefore, to show that A is countable, we may assume that, for every n ∈ ω, the set A n is denumerable. For every n ∈ ω, let B n be the set of all bijections from ω onto A n .
Since |ω ω | = |R ω | = |R| and the sets A n are all denumerable, for every n ∈ ω, the set B n is equipotent to R.
Then it follows from WoAm that there exists an amorphous subset C of B. Since C cannot be partitioned into two infinite subsets, the set {n ∈ ω : C ∩ B n = ∅} is finite. This implies that there exists m ∈ ω such that C ⊆ n∈m+1 B n , so C is equipotent to a subset of R. But this is impossible because R does not have amorphous subsets. The contradiction obtained completes the proof of (i

Remark 14 (a) It is unknown whether M(C, S) implies CUC in ZF or in 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 a permutation model in which BPI is true, then AC f in is also true in N (see, e.g., [12,Proposition 4.39]); hence, by Proposition 8(iii), BPI implies CUC in every permutation model. 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 problem by proving (in Corollary 5) that UT(ℵ 0 , ℵ 0 , cu f ) does not imply M(C, S) in ZF. To achieve our goal, we will first prove (in Theorem 13) that the statement UT(ℵ 0 , ℵ 0 , cu f ) ∧ ¬M(I C, D I ) has a permutation model and that it is transferable to ZF. For the transfer of the latter conjunction to ZF, we will need the following two auxiliary results of Lemma 3 and Proposition 9. (∀x)(|x| ℵ 3 → (∀y) "if y is a countable collection of countable sets whose union is x, then x is a cuf set").
Since, for every set x, the statements |x| ℵ 3 and |x| − ≤ ℵ 2 are equivalent, it is obvious that (10) is injectively boundable. Thus, UT(ℵ 0 , ℵ 0 , cu f ) is also injectively boundable. It was observed in [6, proof of Theorem 3.3] that, for every i ∈ ω and every Q ∈ P i , the following hold: (a) for any φ ∈ G, φ fixes Q if and only if φ fixes Q pointwise; To prove that LW is true in N , we fix a linearly ordered set Y , ≤ in N and prove that fix G (Y ) ∈ F. To this aim, we choose a set E ∈ [P] <ω such that E is a support of both Y and ≤. To show that fix G (E) ⊆ fix G (Y ), let us consider any element y ∈ Y and a permutation φ ∈ fix G (E). Suppose that 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 ∈ N such that φ k is the identity mapping on A. Assuming that y < φ(y), for such a k, we obtain the following: and thus 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).
Since F is a filter and fix G (E) ∈ F, we infer that fix G (Y ) ∈ F. This, together with Proposition 2, implies that the set Y is well-orderable in N . Hence, N | LW.
Now, let us prove that M(I C, D I ) fails in N . First, to find a metric d on A 0 such that A 0 , d is a compact metric space in N , we denote by ∞ the unique element of A 0,1 and, for every n ∈ N, we denote by ρ n the discrete metric on A 0,n+1 . Then, making obvious adjustments in notation, we let d be the metric on A 0 defined by ( * ) in Sect. 2.3. By Proposition 1, the metric space A 0 , d is compact. Using (a), one can check that {P 0 } is a support A 0 , d and, therefore, A 0 , d ∈ N . Moreover, for every n ∈ N, {P 0 } is a support of A 0,n . Hence the family A = {A 0,n+1 : n ∈ N} is denumerable in N . We note that if M ⊆ N, then {A 0,n+1 : n ∈ M} ∈ N because {P 0 } is a support of A 0,n+1 for every n ∈ M. Suppose that A has a partial choice function in N . Then there exists an infinite set M ⊆ N such that the family B = {A 0,n+1 : n ∈ M} has a choice function in N . Let f be a choice function of B such that f ∈ N . Let D ∈ [P] <ω be a support of f . Then D = D ∩ P 0 is also a support of f . Let n ∈ ω and φ i ∈ G with i ∈ n + 1 be such that D = {φ i (P 0 ) : i ∈ n + 1}. Since every permutation from G moves only finitely many atoms, there exists n 0 ∈ M such that n 0 ≥ 2 and A 0,n 0 ∈ φ i (P 0 ) for all i ∈ n + 1.
Assume that f (A 0,n 0 ) = x 0 . Since |A 0,n 0 | = n 0 ≥ 2, there exists y 0 ∈ A 0,n 0 such that y 0 = x 0 . Let η = (x 0 , y 0 ), i.e., η is the permutation of A which interchanges x 0 and y 0 , and fixes all other atoms of N . Then η(A 0,n 0 ) = A n 0 and η ∈ fix G (D ). Since D is a support of f , we have η( f ) = f . Therefore, since A 0,n 0 , x 0 ∈ f , we infer that η(A 0,n 0 ), η(x 0 ) ∈ η( f ) = f , so A 0,n 0 , y 0 ∈ f and, in consequence, x 0 = y 0 . The contradiction obtained shows that A does not have a partial choice function in N . This implies that the set A 0 is Dedekind-finite, and thus M(I C, D I ) is false in N .
(ii) Let be the statement UT(ℵ 0 , ℵ 0 , cu f ) ∧ ¬M(I C, D I ). Since the statement ¬M(I C, D I ) is boundable, it is also injectively boundable, so this, together with Proposition 9, implies that is a conjunction of injectively boundable statements. Therefore, (ii) follows from (i) and from Theorem 7.

Remark 15
Let N be the permutation model of the proof of Theorem 13(i). The proof of Theorem 3.3 in [6] shows that UT(ℵ 0 , cu f , cu f ) is false in N (and hence CUC is also false in N ). Since UT(ℵ 0 , ℵ 0 , cu f ) ∧ ¬UT(ℵ 0 , cu f , cu f ) has a permutation model (for instance, N ), it also has a ZF-model by Proposition 9 and Theorem 7.

The forms of type M(C, )
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 consistent with ZF, so it is relatively consistent with ZF that there are non-separable second-countable metrizable spaces. On the other hand, by Theorem 6(i), it holds in ZF that separability and second-countability are equivalent in the class of compact metrizable spaces. Theorem 1 shows that totally bounded metric spaces are second-countable in ZF + CAC; in particular, it holds in ZF + CAC that all compact metrizable spaces are second-countable. However, the situation is completely different in ZF. There exist ZF-models including compact non-separable metric spaces. Namely, it follows from Theorem 2 that in every ZFmodel satisfying the negation of CAC f in , there exists an uncountable, non-separable compact metric space whose size is incomparable to |R|. Proof (i) It follows directly from Theorem 2 that the implications given in (i) are true in ZF; however, the arguments from [23] are sufficient to show that these implications are also true in ZFA. (ii) By Proposition 6, there exists a compact metric space X = X , d 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 be either CAC f in or IDI. In the light of (i), to prove (iii) and (iv), it suffices to show that the conjunction ∧ ¬M(C, ≤ |R|) has a ZF-model. It follows from (ii) that the conjunction ∧ ¬M(C, ≤ |R|) has a permutation model (for instance, N cr ). Therefore, since the statements CAC f in , IDI and ¬M(C, ≤ |R|) are all injectively boundable, ∧ ¬M(C, ≤ |R|) has a ZF-model by Theorem 7. (v) Let Ψ be either M(C, ≤ |R|) or M(C, S). Since BPI is true in N 3, it follows from Theorem 6 (iii) that Ψ is true in N 3. It is known that IDI is false in N 3. Hence, the conjunction Ψ ∧ ¬IDI has a permutation model. To complete the proof, it suffices to apply Theorem 7.

Theorem 14 (ZF)
Proof Let X = X , d be a compact metric space.
(→) We assume both CAC f in and M(C, σ − l. f ). By our hypothesis, X has a base B = {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 notice that the collection V of all open sets V of X such that V meets only finitely many 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 f in , implies that the family B is countable, so X is second-countable. Hence, by Theorem 6(i), X is separable as required.
(←) By Proposition 10, M(C, S) implies CAC f in . To conclude the proof to (i), it suffices to notice that M(C, S) implies M(C, 2) and M(C, 2) trivially implies that every compact metric space has a σ -locally finite base.
(ii) (→) Now, we assume both CAC f in and M(C, ST B). Since X is strongly totally bounded, it follows that it admits a sequence (D n ) n∈N such that, for every n ∈ N, D n is a 1 n -net of X. By CAC f in , the set D = {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 metric space is strongly totally bounded. Hence M(C, S) implies M(C, ST B). Proposition 10 completes the proof. Proof (i) (→) We assume CAC(R, C) and M(C, ≤ |R|). We fix a compact metric space X = X , d and prove that X is separable. For every n ∈ N, let X n = X n , d n where d n is the metric on X n defined by: 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, for every n ∈ N, the set is non-empty. Let k n = min M n for every n ∈ N. To prove that X is strongly totally bounded, for every n ∈ N, we consider the set C n defined as follows: We claim that for every n ∈ N, C n is a closed subset of X k n . To this end, we fix y 0 ∈ X k n \C n . Then, since X is infinite, there exists and d k n (y 0 , z 0 ) = max{d(y 0 (i), z 0 (i)) : i ∈ k n } < ε.
For every i ∈ k n , we have: Hence, for every i ∈ k n , the following inequalities hold: Therefore, ε < d k n (z 0 , y 0 ). The contradiction obtained shows that B d kn (y 0 , ε) ∩ C n = ∅. Hence, for every n ∈ N, the non-empty set C n is compact 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 ∈ n∈N C n . Then, for every n ∈ N, the set D n = { f (n)(i) : i ∈ k n } is a 1 n -net in X. This shows that X is strongly totally bounded. It is easily seen that the set D = n∈N D n is countable and dense in X. for every family {X n : n ∈ ω} of pairwise disjoint compact spaces, if the direct sum X = n∈ω X n is metrizable, then it is separable.
We include a sketch of a ZF-proof to the following lemma for completeness. We use this lemma in our ZF-proof that M(C, → [0, 1] N ) and M(C, S) are equivalent.

Lemma 4 (ZF)
Suppose that B is a base of a non-empty metrizable space X = X , τ . Then there exists a homeomorphic embedding of X into the cube [0, 1] B×B .
Proof We may assume that X consists of at least two points. Let d be a metric on X such that τ = τ (d) and let we obtain a continous function from X into  Proof Let X = X , τ be an infinite compact metrizable space and let d be a metric on X such that τ = τ (d).
(i) (→) We assume M(C, → [0, 1] N ) and show that X is separable. By our hypothesis, X is homeomorphic to a compact subspace Y of the Hilbert cube [0, 1] N . Since [0, 1] N is second-countable, it follows from Theorem 6(i) that Y is separable. Hence X is separable. In consequence, It is straightforward to verify that B is a base for X of size |B| ≤ |R × N| ≤ |R|, so |B × B| ≤ |R|. This, together with Lemma 4, implies that X is embeddable into  To shed more light on Questions 1 (iii)-(iv), let us prove the following Theorems 17 and 18. Since 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 of non-empty subsets of R such that R = P. Assuming (b), we show that |P| ≤ |R|. For P ∈ P, let f P : R → {0, 1} be the characteristic function of P and let f (x) = 0 for each x ∈ R. We put We claim that the subspace X of [0, 1] R is compact. To see this, let us consider an arbitrary family U of open subsets of [0, 1] R such that X ⊆ U. There exists U 0 ∈ U such that f ∈ U 0 . There exist ε ∈ (0, 1) and a non-empty finite subset J of R such that the set is a subset of U 0 where, for each j ∈ R and x ∈ [0, 1] R , π j (x) = x( j). Since J is finite, there exists a finite set P J ⊆ P such that J ⊆ P J . We notice that, for every P ∈ P\P J and every j ∈ J , f P ( j) = 0. Hence f P ∈ U 0 for every P ∈ P\P J . This implies that there exists a finite set W such that W ⊆ U and X ⊆ W. Hence X is compact as claimed. If P ∈ P and j ∈ P, then Hence f is the unique 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, it follows that |P| ≤ |R| as required. (c) → (a) We assume Part(R) and fix a compact subspace X of the cube [0, 1] R . It is well known that [0, 1] R is separable in ZF (cf., e.g., [25]). Fix a countable dense subset D of [0, 1] R . For every y ∈ D, let Since |[R] <ω | = |R × N| = |R| in ZF, it follows that B = {V y : y ∈ D} is equipotent to R. It is a routine work to verify that B is a base for [0, 1] R . Define an equivalence relation ∼ on B by requiring: 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) such that the family P 0 = A is pairwise disjoint. Let P 1 = P 0 ∪ {R\ P 0 } and P = P 1 \{∅}. 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 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 can show that |P| ≤ |R|. Then | A| ≤ |R|.
(←) Now, we consider an arbitrary compact metrizable subspace X of the cube [0, 1] R such that X has a unique accumulation point. Let x 0 be the accumulation point of X and let d be a metric on X which induces the topology of X. For every x ∈ X \{x 0 } let n x = min n ∈ N : B d 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 is finite. Let us apply the base B of [0, 1] R given in the proof of part (ii) that (c) implies (a). Let ∼ be the equivalence relation on B given by (12). Let : n ∈ N} is a base of X such that |G| ≤ |R|. The following theorem leads to a partial answer to Question 1(iv). (ii) It is obvious that M(C, W (R)) implies M(C, B(R)). Assume M(C, S) and let Y = Y , τ be an infinite compact metrizable separable space. Since Y is secondcountable 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, there exists a disjoint family {U n : n ∈ ω} such that, for each n ∈ ω, U n ∈ τ . For J ∈ P(ω), we put ψ(J ) = {U n : n ∈ J } to obtain an injection ψ : → → τ . Hence |R| = |P(ω)| ≤ |τ |. To see that M(C, B(R)) → CAC f in , we assume M(C, B(R)), fix a disjoint family A = {A n : n ∈ N} of non-empty finite sets and show that A has a choice function. To this aim, we put A = 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 Sect. 2.3. By our hypothesis, the space X = X , d 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 can define A n = {ψ({x}) : x ∈ A n } and a n = min A n . For each n ∈ N, there is a unique x n ∈ A n such that ψ({x n }) = a n . This shows that A has a choice function.
(iii) Now, we assume both CAC(R) and M(C, B(R)). Let us consider an arbitrary compact metric space X = X , ρ . By our hypothesis, X has a base B of size ≤ |R|. Since, |[R] <ω | ≤ |R|, it follows that |[B] <ω | ≤ |R|. For every n ∈ N, let 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 every n ∈ 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 the family F 0 = {F n : n ∈ N} is countable. Furthermore, it is a routine work to verify that F 0 is a base of X. Hence X is second-countable. By Theorem 6 (i), X is separable. This completes the proof to (iii). That (iv) holds follows directly from (ii) and (iii).
Our proof of the following theorem emphasizes the usefulness of Theorems 8 and 18:

Remark 19 (i) To show that Part(R)
is not provable in ZF, let us recall that, in [11], a ZF-model Γ was constructed such that, in Γ , there exists a family F = {F n : n ∈ N} of two-element sets such that F is a partition of R but F does not have a choice function. Then, in Γ , there does not exist an injection ψ : 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 17(ii) that it is not provable in ZF that every compact metrizable subspace of the cube [0, 1] R has a base of size ≤ |R|. We do not know if M(C, → [0, 1] R ) implies every compact metrizable subspace of the cube [0, 1] R has a base of size ≤ |R|. (iii) It is not provable in ZFA that every compact metrizable space with a unique accumulation point embeds in [0, 1] R . Indeed, in the Second Fraenkel model N 2 of [15], there exists a disjoint family of two-element sets A = {A n : n ∈ N} whose union has no denumerable subset. Let A = A, ∞ / ∈ A, X = 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 Sect. 2.3. Let X = X , τ (d) . Then X is a compact metrizable space having ∞ as its unique accumulation point. Since, in N 2, the set [0, 1] R is well-orderable, while A has no choice function, it follows that X does not embed in the Tychonoff cube [0, 1] R . This shows that the statement "There exists a compact metrizable space with a unique accumulation point which is not embeddable in [0, 1] R " has a permutation model.

The list of open problems
For the convenience of readers, we summarize the open problems mentioned in Sects. 4 and 5.