On Iso-dense and Scattered Spaces without AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{AC}$$\end{document}

A topological space is iso-dense if it has a dense set of isolated points, and it is scattered if each of its non-empty subspaces has an isolated point. In ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ZF}$$\end{document} (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice (AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{AC}$$\end{document})), basic properties of iso-dense spaces are investigated. A new permutation model is constructed, in which there exists a discrete weakly Dedekind-finite space having the Cantor set as a remainder; the result is transferable to ZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ZF}$$\end{document}. This settles an open problem posed by Keremedis, Tachtsis and Wajch in 2021. A metrization theorem for a class of quasi-metric spaces is deduced. The statement “Every compact scattered metrizable space is separable” and several other statements about metric iso-dense spaces are shown to be equivalent to the axiom of countable choice for families of finite sets. Results related to the open problem of the set-theoretic strength of the statement “Every non-discrete compact metrizable space contains an infinite compact scattered subspace” are also included.


Set-theoretic Framework and Preliminary Definitions
In this article, the intended context for reasoning and statements of theorems is ZF without any form of the axiom of choice AC. However, we also refer to permutation models of ZFA (cf. [13,15]). We are mainly concerned with iso-dense and scattered spaces in ZF, defined as follows: For a topological space X = X, τ and for Y ⊆ X, let τ | Y = {V ∩ Y : V ∈ τ } and let Y = Y, τ | Y . Then Y is the topological subspace of X such that Y is the underlying set of Y. If this is not misleading, we may denote the topological subspace Y of X by Y .
A topological space X = X, τ is called (quasi-) metrizable if there exists a (quasi-) metric d on X such that τ = τ (d).
For a (quasi-) metric space X = X, d and for Y ⊆ X, let d Y = d Y ×Y and Y = Y, d Y . Then Y is the (quasi-) metric subspace of X such that Y is the underlying set of Y. Given a (quasi-) metric space X = X, d , if not stated otherwise, we also denote by X the topological space X, τ (d) . For every n ∈ N, R n denotes also R n , d e and R n , τ(d e ) where d e is the Euclidean metric on R n .
For a topological space X = X, τ and a set A ⊆ X, we denote by cl X (A) or by cl τ (A) the closure of A in X.
For any topological space X = X, τ , let Iso τ (X) = {x ∈ X : x is an isolated point of X}.
If this is not misleading, as in Definition 1, we use Iso(X) to denote Iso τ (X). By transfinite recursion, we define a decreasing sequence (X (α) ) α∈ON of closed subsets of X as follows: , For α ∈ ON , the set X (α) is called the α-th Cantor-Bendixson derivative of X. The least ordinal α such that X (α+1) = X (α) is denoted by |X| CB and is called the Cantor-Bendixson rank of X.

Definition 2.
A set X is called: if X is not quasi Dedekind-finite; (iii) weakly Dedekind-finite if P(X) is Dedekind-finite; weakly Dedekind-infinite if X is not weakly Dedekind-finite; (iv) a cuf set if X is a countable union of finite sets; (v) amorphous if X is infinite and, for every infinite subset Y of X, the set X \ Y is finite.
It is worth pointing out that quasi Dedekind-finite sets are called H-finite, for instance, in Brot, Cao and Fernández-Bretón [1], and weakly Dedekindfinite sets are called III-finite in Tarski [40] and in Lévy [34], as well as Cfinite, for instance, in Herrlich, Howard and Tachtsis [12]. The terms 'weakly Dedekind-finite' and 'weakly Dedekind-infinite' that we adopt here, and which are frequently used in the literature, were introduced by Degen [5].

Definition 3. (i)
A space X is called a cuf space if its underlying set X is a cuf set.
(ii) A base B of a space X is called a cuf base if B is a cuf set.

Definition 4.
A space X is called: (i) first-countable if every point of X has a countable base of open neighborhoods; (ii) second-countable if X has a countable base; (iii) compact if every open cover of X has a finite subcover; (iv) locally compact if every point of X has a compact neighborhood; (v) limit point compact if every infinite subset of X has an accumulation point in X (cf. [19,20]); (vi) dense-in-itself if Iso(X) = ∅; (vii) regular if, for every open set V in X and every x ∈ V , there exists an open set U in X such that x ∈ U and cl X (U ) ⊆ V ; (viii) a T 3 -space if it is a regular T 1 -space; (ix) completely regular if, for every closed set A in X and every x ∈ X \ A, there exists a continuous function f : X → [0, 1] such that f (x) = 0 and A ⊆ f −1 (1); completely regular T 1 -spaces are called T 3 1 2 -spaces or Tychonoff spaces. Definition 5. Let X = X, d be a (quasi-)metric space.
(i) Given a real number ε > 0, a subset D of X is called ε-dense or an ε-net in X if, for every x ∈ X, B d (x, ε) ∩ D = ∅ (equivalently, if X = x∈D B d −1 (x, ε)). (ii) X is called precompact (respectively, totally bounded ) if, for every real number ε > 0, there exists a finite ε-net in X, d −1 (respectively, in X, d ). (iii) d is called precompact (respectively, totally bounded ) if X is precompact (respectively, totally bounded ).
Remark 1. Definition 5 (ii) is based on the notions of precompact and totally bounded quasi-uniformities defined, e.g., in [8,33]. Namely, given a quasimetric d on a set X, the collection is a quasi-uniformity on X called the quasi-uniformity induced by d (cf. [33, p. 504]). The quasi-uniformity U(d) is precompact (resp., totally bounded) in the sense of [8,33] if and only if d is precompact (resp., totally bounded) in the sense of Definition 5. Clearly d is totally bounded if and only if, for every n ∈ ω, there exists a finite set D ⊆ X such that . The notions of a totally bounded and precompact metric are equivalent. We recall that a (Hausdorff) compactification of a space X = X, τ is an ordered pair Y, γ where Y is a (Hausdorff) compact space and γ : X → Y is a homeomorphic embedding such that γ(X) is dense in Y. A compactification Y, γ of X and the space Y are usually denoted by γX. The underlying set of γX is denoted by γX. The subspace γX\X of γX is called the remainder of γX. A space K is said to be a remainder of X if there exists a Hausdorff compactification γX of X such that K is homeomorphic to γX \ X. For compactifications αX and γX of X, we write γX ≤ αX if there exists a continuous mapping f : αX → γX such that f • α = γ. If αX and γX are Hausdorff compactifications of X such that αX ≤ γX and γX ≤ αX, then we write αX ≈ γX and say that the compactifications αX and γX are equivalent. If n ∈ N, then a compactification γX of X is said to be an n-point compactification of X if γX \ X is an n-element set. Definition 6. Let X = X, τ be a non-compact locally compact Hausdorff space and let K(X) be the collection of all compact subsets of X. For an For every non-compact locally compact Hausdorff space X, X(∞) is the unique (up to ≈) one-point Hausdorff compactification of X. Therefore, every one-point Hausdorff compactification of X is called the Alexandroff compactification of X. Chandler's book [4] is a good introduction to Hausdorff compactifications in ZFC. Basic facts about Hausdorff compactifications in ZF can be found in [29]. We recall that if for a Hausdorff space X, there exists a Hausdorff compactification βX of X such that, for every Hausdorff compactification αX of X, αX ≤ βX, then βX is called theČech-Stone compactification of X.
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 [6], 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. We denote by 2 the discrete space with the underlying set 2 = {0, 1}. If J is an infinite set, the space 2 J is called the Cantor cube.
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 every function f ∈ j∈J ([X j ] <ω \ {∅}). If J is infinite, a function 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 nonindexed 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 7. (Cf. [2,24,35].) (i) A space X is said to be Loeb (respectively, weakly Loeb) if the family of all 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 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 [6,42].

A List of Forms Weaker than AC
In this subsection, for the convenience of readers, we define and denote the weaker forms of AC used directly in this paper. For the known forms given in [13], we quote in their statements the form number under which they are recorded in [13].  [18,19,22,23,27].) 14. IDFBI: For every infinite set D, the Cantor cube 2 ω is a remainder of the discrete space D, P(D) . (Cf. [28].) 15. INSHC: Every infinite discrete space has a non-scattered Hausdorff compactification.
Here, we use the same notation and terminology as, for instance, in [31]. However, it is worth pointing out that IQDI has also been called "Hindman's theorem", for instance, in Fernández-Bretón [7] and in Tachtsis [39] recently.
One of the most well-known weak forms of the Axiom of Choice equivalent to CAC fin is Kőnig's Lemma (see [13,Form 10 F,p. 20]).
The form IDFBI has been introduced and investigated in [28] recently. More comments about IDFBI are included in Remark 3. New facts concerning IDFBI-among them, a solution of an open problem posed in [28]-are included in Sect. 2. The form INSHC is new here. That INSHC is essentially weaker than IDFBI is shown in Sect. 2.

Some Known Results
In this subsection, we quote several known results that we refer to in the sequel. Some of the quoted results have been obtained recently, so they can be unknown to possible readers of this article. Proposition 1. (Cf. [21]). (ZF) A topological space X is scattered if and only if there exists α ∈ ON such that X (α) = ∅. If X is scattered then Moreover, if X is a non-empty scattered compact space, then |X| CB is a successor ordinal. Theorem 1. (Cf., e.g., [3] Several essential applications of Theorem 2(ii), especially to the theory of Hausdorff compactifications in ZF, have been shown in [28] recently. We show some other applications of Theorem 2(ii) in the forthcoming Sects. 3 and 4.

Theorem 3. (Cf. [32]). (ZF)
(a) For every compact Hausdorff, quasi metric space X = X, d the following are equivalent:  [13] stating that every non-empty family of non-empty finite sets has a choice function. It is obvious that OP implies AC fin in ZF. It was shown in [41, p. 196] that AC fin implies NAS in ZF. In consequence, OP implies NAS in ZF. In Feferman's model M2 in [13], NAS is true and OP is false (see [13, p. 148]). In Mathias' model M3 in [13], OP is true and BPI is false (see [13, p. 150]).
Other remarks about Theorem 6 can be found, for instance, in [28].
Vol. 78 (2023) On Iso-dense and Scattered Spaces Page 9 of 30 153 We recall that a family A of subsets of a set X is called stable (equivalently, closed ) under finite unions (respectively, finite intersections) if, for every pair A, B of members of A, A ∪ B ∈ A (respectively, A ∩ B ∈ A). Theorem 7. (Cf. [28]). (ZF) For every locally compact Hausdorff space X, the following conditions are all equivalent: (a) every non-empty second-countable compact Hausdorff space is a remainder of X; . . , 2 n }} such that, for every n ∈ N, the following conditions are satisfied:  [10] has been shown to be unprovable in ZF. In [28], an infinite set D is called dyadically filterbase infinite if 2 ω is a remainder of the discrete space D, P(D) . An equivalent purely set-theoretic definition of a dyadically filterbase infinite set is given in [28] and it can be easily obtained from condition (c) of Theorem 7 applied to discrete spaces. Clearly, IDFBI is equivalent to the sentence "Every infinite set is dyadically filterbase infinite".  [32]); (ii) the discrete space D has a metrizable compactification if and only if D is a cuf set (cf. [28]).

non-empty family of open sets of X such that V n i is stable under finite unions and finite intersections, and, for every
As we have already mentioned at the beginning of Sect. 1.1, in the sequel, we apply not only ZF-models but also permutation models of ZFA. To transfer a statement Φ from a permutation model to a ZF-model, we use the Jech-Sochor First Embedding Theorem (see, e.g., [15,Theorem 6.1]) if Φ is a boundable statement. When Φ has a permutation model but Φ is a conjunction of statements each one of which is equivalent to BPI or to an injectively boundable statement, we use Pincus' transfer results (see [36,37] and [13, Note 3, page 286]) to show that Φ has a ZF-model. The notions of boundable and injectively boundable statements can be found in [36], [15, Problem 1 on page 94] and [13, Note 3, page 284]. Every boundable statement is equivalent to an injectively boundable one (see [36] or [13, Note 3, page 285]). We recommend [15,Chapter 4] as an introduction to permutation models.

The Content Concerning New Results in Brief
In Sect. 2, we notice that, in ZF, the class of all iso-dense compact Hausdorff spaces is essentially wider than the class of all Hausdorff compact scattered spaces; similarly, the class of all iso-dense compact metrizable spaces is essentially wider than the class of all compact metrizable scattered spaces. A compact Hausdorff iso-dense space may fail to be completely regular in ZF (see Proposition 6). We show that the new form INSHC holds in every model of ZF + BPI, is independent of ZF, does not imply BPI and is strictly weaker than IDFBI (see Theorem 11). We construct a new permutation model to prove that the existence of a dyadically filterbase infinite, weakly Dedekindfinite set is relatively consistent with ZFA, and we then observe that the result is transferable to ZF (see Theorem 12). This solves an open problem posed by us in [28] (see [28,Problem (6) of Section 6]). For further related research and solutions to other open problems from [28], the interested readers are referred to Tachtsis [38].
In Sect. 3, we prove in ZF that if X, d is a quasi-metric T 3 -space such that d is strong and either X, τ (d) is limit point compact or d −1 is precompact, then the space X, τ (d) is metrizable (see Theorem 14). This result and its direct consequence that if X, d is a compact Hausdorff quasi-metric space such that X, τ (d −1 ) is iso-dense, then X, τ (d) is metrizable (see Corollary 2) are new applications of Theorem 2(ii) and adjuncts to Theorem 3. By applying Theorem 2, we show in ZF that if X, d is an iso-dense metric space such that either d is totally bounded or X, τ (d) is limit point compact, then X, τ (d) has a cuf base and can be embedded in a metrizable Tychonoff cube (see Theorem 15).
Section 4 concerns equivalents of CAC fin in terms of scattered or isodense spaces (see Theorems 16 and 18). Among our new equivalents of CAC fin there are, for instance, the following statements: (a) for every iso-dense metric space X, if X is either limit point compact or totally bounded, then X is separable; (b) every totally bounded scattered metric space is countable; (c) every compact metrizable scattered space is countable; (d) every totally bounded, complete scattered metric space is compact. We show that, in ZF, every compact metrizable cuf space is scattered (see Theorem 19). We prove that WOAC fin is equivalent to the statement: for every well-orderable nonempty set S and every family { X s , d s : s ∈ S} of compact scattered metric spaces, the product s∈S X s , τ(d s ) is compact (see Theorem 21). Several other relevant results are included in Sect. 4, too.
Section 5 is related to the problem of the deductive strength of the statement "Every non-empty dense-in-itself compact metrizable space contains an infinite compact scattered subspace". Among other results of Sect. 5, we show the following; (a) each of IDI, WoAm and BPI implies that every infinite compact first-countable Hausdorff space contains a copy of N(∞); (b) every infinite first-countable compact Hausdorff separable space contains a copy of N(∞); (c) every infinite first-countable compact Hausdorff space having an infinite cuf subset contains a copy of D(∞) for some infinite discrete cuf space (see Theorem 22). We prove that the statement "every infinite first-countable Hausdorff compact space contains an infinite metrizable compact scattered subspace" implies neither CAC fin nor IQDI, nor CMC in ZFA (see Theorem 23). It is important to mention here that, recently Keremedis and Tachtsis [26] established that the statement "Every non-empty dense-in-itself compact metrizable space contains an infinite compact scattered subspace" is not provable in ZF. In fact, the latter result of [26] settled the corresponding open problem posed by us in a former, preliminary, unpublished version of this paper, in which it was labeled as 'Problem 6.4' (see arXiv:2101.02825). This completely justifies the motivation of the study in Sect. 5 whose goal is to shed some light on the problem stated at the beginning of this paragraph.   It follows from Theorem 10(ii) that U ∩ D is amorphous but this is impossible because BPI implies NAS by Theorem 6. The contradiction obtained shows that βD \ D is dense-in-itself, so βD is not scattered.
(v) It was shown in [28] that the conjunction BPI ∧ ¬IDFBI has a ZFmodel. This, together, with (iv), implies that there is a model of ZF in which the conjunction INSHC ∧ ¬IDFBI is true. Hence INSHC IDFBI. To prove INSHC BPI, let us use the Feferman's forcing model M2 in [13]. It is known that DC ∧ ¬BPI is true in M2 (see [13, page 148] In [28, the proof of Theorem 5.14], a permutation model has been constructed in which there exists a weakly Dedekind-finite discrete space which has a remainder homeomorphic to N(∞). Now, we are in a position to solve Problem 1(ii) (that is, [28, Problem (6) of Section 6]) by the following theorem:

Theorem 12. It is relatively consistent with ZF that there exists a dyadically filterbase infinite set which is weakly Dedekind-finite.
Proof. Let Φ be the following statement: "There exists a dyadically filterbase infinite set which is weakly Dedekind-finite".
Since Φ is a boundable statement, by the Jech-Sochor First Embedding Theorem (see [15,Theorem 6.1]), it suffices to prove that Φ has a permutation model. To this end, let us modify the model constructed in [28, the proof of Theorem 5.14] to get a new permutation model N in which Φ is true.
In what follows, for an arbitrary non-empty set S and every permutation ψ of S, we denote by supp(ψ) the support of ψ, that is, supp(ψ) = {x ∈ S : ψ(x) = x}.
We start with a model M of ZFA + AC with a denumerable set A of atoms such that A has a denumerable partition 2i−1 , B n+1 2i } is a partition of B n i into two infinite sets. We may thus view B as an infinite binary tree, having A as its root.
Let G be the group of all permutations φ of A which satisfy the following two properties: (c) φ moves only finitely many elements of A.
For every n ∈ N and for every i ∈ {1, 2, . . . , 2 n }, we let We also let For every E ∈ [Q] <ω , we let Then and To argue for (1), let a ∈ A. Since A is a partition of A in M , there exists a unique i ∈ ω such that a ∈ A i . Since the set {B 1 1 2 ). Pick an A j ∈ B 1 2 and an a ∈ A j . Let φ ∈ G be the transposition (a, a ) (i.e. φ interchanges a and a and fixes all other atoms). Then Towards a contradiction, assume that π(a) = b for some b ∈ A \ {a}. Since a ∈ B 1 1 and π fixes B 1 1 , it follows that b = π(a) ∈ π( B 1 1 ) = B 1 1 . But then, since π ∈ G E , we have the following: which is a contradiction. Therefore, (1) holds.
Let N be the permutation model determined by M , G and F. We say that an element In N , the set (P(A)) N = (P(A)) M ∩ N is the power set of A. To prove that A is dyadically filterbase infinite in N , let us show that, in N , the discrete space A, (P(A)) N satisfies condition (c) of Theorem 7. To this aim, for every n ∈ N and for every i ∈ {1, 2, . . . , 2 n }, we let . . , 2 n }}. We notice that any permutation of A in G fixes V pointwise. Hence, V ∈ N and, moreover, V is well-orderable in the model N (see [15, page 47]). Since V is denumerable in M and well-orderable in N , it follows that V is also denumerable in N . Furthermore, in view of the properties of the family B and of the elements of G, and the construction of V, it is easy to see that, if we put X = A and X = A, (P(A)) N , then V has properties (i)-(iv) of condition (c) of Theorem 7. This, together with Theorem 7, proves that A is dyadically filterbase infinite in the model N .
To complete the proof, it remains to show that A is weakly Dedekindfinite in N . By way of contradiction, we assume that A is weakly Dedekindinfinite in N . Thus, it holds in N that there exists a denumerable disjoint family U = {U n : n ∈ ω} of (P(A)) N . Let E ∈ [Q] <ω be a support of U n for all n ∈ ω. By the definitions of G, Q and the supports, as well as the fact that U is infinite and disjoint, it follows that there exist distinct k, m ∈ ω and atoms x ∈ U k , y ∈ U m such that the transposition ψ = (x, y) of A is an element of G E . Since E is a support of U k and ψ ∈ G E , ψ(U k ) = U k , and so y = ψ(x) ∈ ψ(U k ) = U k . This is impossible because y ∈ U m and U k ∩ U m = ∅. The contradiction obtained shows that A is weakly Dedekind-finite in N .
The model constructed in [28, the proof of Theorem 5.14] is (substantially) different from the one we have just introduced in the proof of Theorem 12. By Theorem 5.15 of [28], NAS is false in the model from [28, the proof of Theorem 5.14], whereas it is unknown to us whether or not NAS is true in the model N of the proof of Theorem 12. With regards to the latter model, let us notice that, for every i ∈ ω, no E ∈ [Q] <ω can support A i and, in consequence, A i / ∈ N . This is the chief reason why one should not be tempted to mimic the proof of Theorem 5.15 in [28] to show that NAS fails in N . We also do not know if INSHC holds in N .

A Metrization Theorem for a Class of Quasi-metrizable Spaces
The following theorem is of significant importance because of its consequences that will be shown in the forthcoming results.
Theorem 13. (ZF) Let d be a quasi-metric on a set X such that either d −1 is precompact or the space X = X, τ (d) is limit point compact. Then the set For every n ∈ N, let A n = {x ∈ D : n = n x }. Suppose that n 0 ∈ N is such that A n0 is infinite. Let us show that there exist x 0 ∈ X and x 1 ∈ A n0 such that x 0 = x 1 and which is impossible by the definition of A n0 .
If X is limit point compact, there exists an accumulation point of A n0 in X. In this case, for a fixed accumulation point x 0 of A n0 , we can fix n0 and, since A n0 is infinite, we can fix x 0 ∈ F and x 1 ∈ A n0 such that x 1 = x 0 and x 1 ∈ B d x 0 , 1 n0 . Hence, the assumption that A n0 is infinite leads to a contradiction. Therefore, D = n∈N A n is a cuf set.

Corollary 1. (ZF)
Let X = X, d be a metric space which is either limit point compact or totally bounded. Then Iso(X) is a cuf set. Furthermore, if Iso(X) is infinite, then X is quasi Dedekind-infinite.
Proof. That Iso(X) is a cuf set follows from Theorem 13. The second assertion is straightforward.
Remark 6. Let X be a compact metrizable space. Then, using Proposition 4(ii), we may deduce that Iso(X) is a cuf set. Namely, suppose that Iso(X) is infinite. Let Y = cl X (Iso(X)). Then Y is a metrizable compactification of the discrete space Iso(X), so Iso(X) is a cuf set by Proposition 4(ii).
Theorem 3(b) improves the well-known result of ZFC that every compact Hausdorff quasi-metrizable space is metrizable (see Corollary in [8, Corollary in 7.1, p. 153] since it establishes that the weaker (than AC) choice principle CAC suffices for the proof. An open problem posed in [32] is whether it can be proved in ZF that every quasi-metrizable compact Hausdorff space is metrizable. Theorem 3 is a partial solution to this problem. Now, we can shed a little more light on it via the following theorem: Theorem 14. (ZF) Let d be a strong quasi-metric on a set X such that X, τ (d) is a T 3 -space. Then the following conditions are satisfied: Proof. (i) Assume that A = n∈ω A n is a dense set in X, τ (d −1 ) such that, for every n ∈ ω, A n is a non-empty finite set. For m, n ∈ ω, we define Since d is strong, in much the same way, as in the proof of Theorem 4.6 in [32], one can show that B = n,m∈ω B n,m is a base of X, τ (d) . Since B is a cuf set, the space X, τ (d) is metrizable by Theorem 2(ii).
(ii) Now, we assume that X, τ (d −1 ) is iso-dense and either X, τ (d) is limit point compact or d −1 is precompact. Let E = Iso τ (d −1 ) (X). Then E is dense in X, τ (d −1 ) . By Theorem 13, the set E is a cuf set. Hence, to conclude the proof, it suffices to apply (i).

Corollary 2. (ZF) Let X, d be a compact Hausdorff quasi-metric space such
Proof. This follows immediately from Proposition 2 and Theorem 14.
Remark 7. In Corollary 2, we cannot omit the assumption that X, τ (d) is Hausdorff. Indeed, there is a quasi-metric d on ω such that τ (d) is the cofinite topology on ω and τ (d −1 ) is the discrete topology on ω (see [32]). Then d is a strong quasi-metric such that ω, τ (d) is a compact T 1 -space which is not metrizable.
Theorem 15. (ZF) Let X, d be an iso-dense metric space such that either d is totally bounded or X, τ (d) is limit point compact. Then X, τ (d) has a cuf base and can be embedded in a metrizable Tychonoff cube.
Proof. It follows from the proof of Theorem 14 that X, τ (d) has a cuf base. Since X, τ (d) is a T 3 -space, to conclude the proof, it suffices to apply Theorem 2(ii).

CAC f in via Iso-dense Metrizable Spaces
It is known that it holds in ZFC that every iso-dense compact metrizable space is separable and every scattered compact metrizable space is countable. In this section, we show that the situation with compact iso-dense metrizable spaces and compact scattered metrizable spaces in ZF is different from the one in ZFC. To begin, let us recall the following lemma proved in [27]: If X = X, d is a metric space and Y is a topological space, then we say that X embeds in Y if the space X, τ (d) embeds in Y.
The following theorem is a characterization of CAC fin in terms of isodense (limit point) compact metrizable spaces and in terms of iso-dense totally bounded metric spaces.

Theorem 16. (ZF)
The following conditions are all equivalent: (i) CAC fin ; (ii) for every iso-dense metric space X, if X is either limit point compact or totally bounded, then X is separable; (iii) for every iso-dense metric space X, if X is either limit point compact or totally bounded, then X embeds in the Hilbert cube [0, 1] N ; (iv) for every iso-dense metric space X, if X is either limit point compact or totally bounded, then | Iso(X)| ≤ |R|; (v) for every iso-dense metric space X, if X is either limit point compact or totally bounded, then the set Iso(X) is countable. In (ii)-(v), the term "iso-dense" can be replaced by "scattered".
Proof. Let X = X, d be an iso-dense (respectively, scattered) metric space such that X is either limit point compact or totally bounded. By Corollary 1, the set Iso(X) is a cuf set. Hence, it follows from CAC fin that Iso(X) is countable. In consequence, (i) implies (ii). Since every separable metrizable space is second-countable, it follows from Lemma 1 that it is true in ZF that (ii) implies (iii). Now, to show that (iii) implies (iv), suppose that X, τ (d) is homeomorphic to a subspace of [0, 1] N . Then Iso(X) is equipotent to a subset of [0, 1] N . Since it holds in ZF that [0, 1] N and R are equipotent, we deduce that Iso(X) is equipotent to a subset of R. Hence, (iii) implies (iv).
It is obvious that, in ZF, every cuf subset of R is countable as a countable union of finite well-ordered sets. Hence, if Iso(X) is equipotent to a subset of R, then Iso(X) is countable as a set equipotent to a cuf set contained in R. This shows that (iv) implies (v).
Finally, suppose that CAC fin fails. Then there exists an uncountable discrete cuf space D. It follows from Proposition 4(i) that the Alexandroff compactification D(∞) of D is metrizable. Since D(∞) is an iso-dense compact mertizable space whose set of all isolated points is uncountable, (v) fails if CAC fin fails. Hence (v) implies (i). Proof. Let X = X, d be an infinite totally bounded metric space. Let α = |X| CB . Then For every γ ∈ α and every x ∈ Iso(X (γ) ), let For every γ ∈ α and every n ∈ N, let We have already shown in the proof of Theorem 13 that, for every γ ∈ α and every n ∈ N, the set A γ,n is finite and Iso(X (γ) ) = n∈N A γ,n .
(i) Suppose that α is uncountable. For every n ∈ N, let Since α is supposed to be uncountable, there exists n 0 ∈ N such that B n0 is infinite. We fix such an n 0 and put By the total boundedness of d, the open cover U of X has a finite subcover. Hence, there exists a non-empty finite subset F of X such that X = x∈F B d x, 1 3n0 . Since B n0 is infinite, there exist γ 1 , γ 2 ∈ B n0 and elements 3n0 , such that x 1 = x 2 . We may assume that γ 1 ≤ γ 2 . Then X (γ2) ⊆ X (γ1) and it follows from the definition of A γ1,n0 that d(x 1 , x 2 ) ≥ 1 n0 . On the other hand, since The contradiction obtained proves that α is countable.
(ii) Now, suppose that the space X is also scattered. Then it follows from Proposition 1 that X (α) = ∅. Hence, X = {A γ,n : γ ∈ α and n ∈ N}. Since α is countable, the set α × N is countable. This implies that the family {A γ,n : γ ∈ α and n ∈ N} is also countable. We have already shown that, for every γ ∈ α and every n ∈ N, the set A γ,n is finite. Hence, X is a cuf set.
It follows immediately from (ii) and Theorem 15 that (iii) holds. Proof. Since CAC fin implies that all cuf sets are countable, it follows from Theorem 17 that (i) implies (ii) and (iii). It is provable in ZF that every totally bounded, complete countable metric space is compact. Hence, in the light of Theorem 17, (i) implies (iv). Assume that CAC fin is false. Then there exists a family {A n : n ∈ ω} of non-empty pairwise disjoint finite sets such that the set D = n∈ω A n is Dedekind-finite (see [13,Form 10 M]). Let D = D, P(D) . By Proposition 4(i), the space D(∞) is metrizable. Let d be any metric which induces the topology of D(∞). Since D(∞) is compact, the metric d is totally bounded. Moreover, D(∞) is scattered but uncountable. For ρ = d D × D, the metric space D, ρ is also totally bounded. Since D is Dedekind-finite and D, ρ is discrete, the metric ρ is complete. Clearly, D, ρ is not compact. All this taken together completes the proof.

Theorem 19. (ZF) Every compact metrizable cuf space is scattered. In particular, every compact metrizable countable space is scattered.
Proof. Our first step is to prove that every non-empty compact metrizable cuf space has an isolated point. To this aim, suppose that X = X, d is a compact metric space such that the set X is a non-empty cuf set. Towards a contradiction, suppose that X is dense-in-itself. We fix a partition {X n : n ∈ ω} of X into non-empty finite sets.
Let S = {{0, 1} n : n ∈ N}. For n ∈ N, s ∈ {0, 1} n and t ∈ {0, 1}, let s t ∈ {0, 1} n+1 be defined by: s t(i) = s(i) for every i ∈ n, and s t(n) = t. Using ideas from [3], let us define by induction (with respect to n) a family {B s : s ∈ S} such that, for every s ∈ S, the following conditions are satisfied: (1) B s is a non-empty open subset of X; To start the induction, for n = 1 = {0} and every s ∈ {0, 1} 1 , we define: Now, suppose that n ∈ N is such that, for every s ∈ n i=1 {0, 1} i , we have defined a non-empty open subset B s of X. For an arbitrary s ∈ {0, 1} n+1 , we consider the set B s n . We put n s = min{m ∈ ω : X m ∩ B s n = ∅}. Since X is dense-in-itself, we have ∅ = {m ∈ ω : X m ∩ (B s n \X ns ) = ∅}, so we can define k s = min{m ∈ ω : X m ∩ (B s n \X ns ) = ∅}. Now, we put Y s,0 = X ns ∩ B s n and Y s,1 = X ks ∩ (B s n \ X ns ). We define In this way, we have inductively defined the required family {B s : s ∈ S}.
We notice that it follows from (2) that, for every f ∈ {0, 1} ω and n ∈ N, ∅ = cl X (B f (n+1) ) ⊆ cl X (B f n ). Thus, by the compactness of X, for every f ∈ {0, 1} ω , the set We define a mapping F : {0, 1} ω → {P(X n ) : n ∈ ω} by putting: It follows from (3) that F is an injection. In consequence, the set {0, 1} ω is equipotent to a subset of the cuf set {P(X n ) : n ∈ ω}. But this is impossible because {0, 1} ω , being equipotent to R, is not a cuf set. The contradiction obtained shows that every non-empty compact metrizable cuf space has an isolated point.
To complete the proof, we let X be any compact metrizable cuf space. We have proved that every non-empty compact subspace of X has an isolated point. Hence X cannot contain non-empty dense-in-itself subspaces. This implies that X is scattered. Now, we can give the following modification of Theorem 1: Theorem 20. (ZF) Let X be a compact Hausdorff, non-scattered space which has a cuf base. If X is weakly Loeb, then |R| ≤ |[X] <ω |. If X is Loeb, then |R| ≤ |X|.
Proof. Without loss of generality, we may assume that X is dense-in-itself because we can replace X with its non-empty dense-in-itself compact subspace. By Theorem 1.13(ii), X is metrizable. It is known that every compact metrizable Loeb space is second-countable (see, e.g., [23]). Hence, if X is Loeb, then |R| ≤ |X| by Theorem 1. Suppose that X is weakly Loeb. Let d be any metric on X which induces the topology of X. It follows from Proposition 3 that X has a dense cuf set. Since X is non-empty and dense-in-itself, every dense subset of X is infinite. Therefore, we can fix a disjoint family {X n : n ∈ ω} of non-empty finite subsets of X such that the set {X n : n ∈ ω} is dense in X. Mimicking the proof of Theorem 19, we can define an injection F : {0, 1} ω → P(X) such that, for every f ∈ {0, 1} ω , the set M f = F (f ) is a non-empty closed subset of X and, for every pair f, g of distinct functions from {0, 1} ω , M f ∩ M g = ∅.
Taking the opportunity, let us give a proof of the following theorem: ) is a non-empty finite set. In consequence, by assigning to any i ∈ I the set Iso(X (βi) i ), we obtain a multiple choice function of {X i : i ∈ I}. (b) Let us consider the family F of all non-empty closed sets of X. By the proof of (a), there exists a family {f γ : γ ∈ α} such that, for every γ ∈ α, f γ is a weak Loeb function of X γ . For every F ∈ F, let γ(F ) = min{γ ∈ α : F ∩ X γ = ∅} and let f (F ) = f γ(F ) (F ∩ X γ ). Then f is a weak Loeb function of X.
(c) (i)→(ii) Let us assume WOAC fin . Suppose that S is a well-orderable non-empty set and, for every s ∈ S, X s = X s , d s is a non-empty scattered totally bounded metric space. To prove that X = s∈S X s is well-orderable, without loss of generality, we may assume that S = α for some non-zero ordinal α, and X i ∩X j = ∅ for every pair i, j of distinct elements of α. In much the same way, as in the proof of Theorem 17, we can define a family {A i,n : i ∈ α, n ∈ N} of non-empty finite sets such that, for every i ∈ α, X i = n∈N A i,n . Now, we can easily define a family {M i : i ∈ α} of subsets of N and a family {F i,n : i ∈ α, n ∈ M i } of pairwise disjoint non-empty finite sets such that, for every i ∈ α, X i = n∈Mi F i,n . The set J = { i, n : i ∈ α, n ∈ M i } is well-orderable, so we can fix an ordinal number γ and a bijection h : γ → J. For every j ∈ γ, let n(j) ∈ ω be equipotent to F h(j) , and let B j = {f ∈ F n(j) h(j) : f is a bijection}. By WOAC fin , there exists ψ ∈ j∈γ B j . Now, we can define a well-ordering ≤ on X = j∈γ F h(j) as follows: for i, j ∈ γ, x ∈ F h(i) , y ∈ F h(j) , we put: x ≤ y if either i ∈ j, or i = j and ψ(i) −1 (x) ⊆ ψ(i) −1 (y).