On Iso-Dense and Scattered Spaces in $\Mathbf {ZF} $

$On Iso-Dense and Scattered Spaces in $\Mathbf {ZF} $$

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1 Preliminaries

1.1 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. [10] and [9]). In this article, we are concerned mainly with iso-dense and scattered spaces in ZF, deﬁned as follows:

Deﬁnition 1.1. A topological space X is called:

(i) iso-dense if the set Iso(X) of all isolated points of X is dense in X;

(ii) scattered or dispersed if, for every non-empty subspace Y of X, ∅= 6 Iso(Y ).

Before we pass to the main body of the article, let us establish notation and recall some known definitions in this subsection, make a list of weaker forms of AC in Subsection 1.2, and recall several known results for future references in Subsection 1.3. The content of the article is described in brief in Subsection 1.4. Our main new results are included in Sections 2-4. We denote by ON the class of all (von Neumann) ordinal numbers. The first infinite ordinal number is denoted by ω. Then N = ω \{0}. If X is a set, the power set of X is denoted by P(X), and the set of all finite subsets of X is denoted by [X]<ω. A quasi-metric on a set X is a function d : X × X → [0, +∞) such that, for all x, y, z ∈ X, d(x, y) ≤ d(x, z)+ d(z,y) and d(x, y)=0 if and only if x = y (cf. [5], [12], [26], [31]). If a quasi-metric d on X is such that d(x, y)= d(y, x) for all x, y ∈ X, then d is a metric. A (quasi-)metric space is an ordered pair hX,di where X is a set and d is a (quasi-)metric on X. Let d be a quasi-metric on X. The conjugate of d is the quasi-metric d−1 defined by: d−1(x, y)= d(y, x) for x, y ∈ X.

2 The metric d⋆ associated with d is deﬁned by:

d⋆(x, y) = max{d(x, y),d(y, x)} for x, y ∈ X.

Clearly, d is a metric if and only if d = d−1 = d⋆. The d-ball with centre x ∈ X and radius r ∈ (0, +∞) is the set

Bd(x, r)= {y ∈ X : d(x, y)

The collection

1 τ(d)= V ⊆ X :(∀x ∈ V )(∃n ∈ ω)B x, ⊆ V d 2n is the topology in X induced by d. For a set A ⊆ X, let δd(A)=0 if A = ∅, and let δd(A) = sup{d(x, y): x, y ∈ A} if A =6 ∅. Then δd(A) is the diameter of A in hX,di. A quasi-metric d on X is called strong if τ(d) ⊆ τ(d−1). In the sequel, topological or (quasi-)metric spaces (called spaces in ab- breviation) are denoted by boldface letters, and the underlying sets of the spaces are denoted by lightface letters. For a topological space X = hX, τi and for Y ⊆ X, let τ|Y = {V ∩ Y : V ∈ τ} and let Y = hY, τ|Y i. 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 = hX, τi is called (quasi-)metrizable if there exists a (quasi-)metric d on X such that τ = τ(d). For a (quasi-)metric space X = hX,di and for Y ⊆ X, let dY = d ↾ Y ×Y and Y = hY,dY i. Then Y is the (quasi-)metric subspace of X such that Y is the underlying set of Y. Given a (quasi-)metric space X = hX,di, if not stated otherwise, we also denote by X the topological space hX, τ(d)i. n n n For every n ∈ N, R denotes also hR ,dei and hR , τ(de)i where de is the Euclidean metric on Rn. For a topological space X = hX, τi and a set A ⊆ X, we denote by clX(A) or by clτ (A) the closure of A in X. For any topological space X = hX, τi, let

Isoτ (X)= {x ∈ X : x is an isolated point of X}.

3 If this is not misleading, as in Deﬁnition 1.1, we use Iso(X) to denote Isoτ (X). (α) By transitive recursion, we deﬁne a decreasing sequence (X )α∈ON of closed subsets of X as follows: X(0) = X, X(α+1) = X(α) \ Iso(X(α)), X(α) = X(γ) if α is a limit ordinal. γ\∈α For α ∈ ON, the set X(α) is called the α-th Cantor-Bendixson derivative of (α+1) (α) X. The least ordinal α such that X = X is denoted by |X|CB and is called the Cantor-Bendixson rank of X.

Deﬁnition 1.2. A set X is called:

(i) Dedekind-finite if there is no injection f : ω → X; Dedekind-infinite if X is not Dedekind-finite;

(ii) quasi Dedekind-finite if [X]<ω is Dedekind-finite; quasi Dedekind-infinite if X is not quasi Dedekind-finite;

(iii) weakly Dedekind-finite if P(X) is Dedekind-finite; weakly Dedekind- infinite if P(X) is Dedekind-infinite;

(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.

Deﬁnition 1.3. (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 1.4. 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;

4 (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. [14] and [15]).

Deﬁnition 1.5. Let X = hX,di 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, Bd(x, ε) ∩ D =6 ∅ (equivalently, if X = Bd−1 (x, ε)). ∈ xSD (ii) X is called precompact (respectively, totally bounded) if, for every real number ε > 0, there exists a ﬁnite ε-net in hX,d−1i (respectively, in hX,d⋆i).

(iii) d is called precompact (respectively, totally bounded) if X is precompact (respectively, totally bounded).

Remark 1.6. Deﬁnition 1.5 (ii) is based on the notions of precompact and totally bounded quasi-uniformities deﬁned, e.g., in [5] and [26]. Namely, given a quasi-metric d on a set X, the collection

1 U(d)= U ⊆ X × X : ∃ ∈ hx, yi ∈ X × X : d(x, y) < ⊆ U n ω 2n is a quasi-uniformity on X called the quasi-uniformity induced by d (cf. [26, p. 504]). The quasi-uniformity U(d) is precompact (resp., totally bounded) in the sense of [5] and [26] if and only if d is precompact (resp., totally bounded) in the sense of Definition 1.5. Clearly d is totally bounded if and only if, for every n ∈ ω, there exists a finite set D ⊆ X such that 1 1 −1 X = Bd(x, 2n ) ∩ Bd (x, 2n ) . The notions of a totally bounded and x∈D precompactS metric are equivalent. We recall that a (Hausdorff) compactification of a space X = hX, τi is an ordered pair hY,γi 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 hY,γi 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

5 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 1.7. Let X = hX, τi is a non-compact locally compact Hausdorff space and let K(X) be the collection of all compact subsets of X. For an element ∞ ∈/ X, we define X(∞)= X ∪ {∞},

τ(∞)= τ ∪{X(∞) \ K : K ∈K(X)} and X(∞)= hX(∞), τ(∞)i. Then X(∞) is called the Alexandroﬀ compact- iﬁcation of X.

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 [3] is a good introduction to Hausdorff compactifications in ZFC. Basic facts about Hausdorff compactifications in ZF can be found in [23]. If X is a space which has the Čech-Stone compactification, then, as usual, βX stands for the Čech-Stone compactification of X. Given a collection {Xj : j ∈ J} of sets, for every i ∈ J, we denote by πi the projection πi : Xj → Xi defined by πi(x) = x(i) for each x ∈ Xj. ∈ ∈ jQJ jQJ If τj is a topology in Xj, then X = Xj denotes the Tychonoff product ∈ jQJ of the topological spaces Xj = hXj, τji with j ∈ J. If Xj = X for every J j ∈ J, then X = Xj. As in [4], for an infinite set J and the unit interval ∈ jQJ [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 2J is called the Cantor cube. We recall that if Xj =6 ∅, then it is said that the family {Xj : j ∈ J} ∈ jQJ has a choice function, and every element of Xj is called a choice function ∈ jQJ of the family {Xj : j ∈ J}. A multiple choice function of {Xj : j ∈ J} is every

6 <ω function f ∈ ([Xj] \ {∅}). If J is infinite, a function f is called partial ∈ jQJ (multiple) choice function of {Xj : j ∈ J} if there exists an infinite subset I of J such that f is a (multiple) choice function of {Xj : 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 {Xj : j ∈ J} be a disjoint family of sets, that is, Xi ∩ Xj = ∅ for each pair i, j of distinct elements of J. If τj is a topology in Xj for every j ∈ J, then Xj denotes the direct sum of the spaces Xj = hXj, τji with j ∈ J. ∈ jLJ Definition 1.8. (Cf. [1], [27] and [19].)

(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 [4] and [30].

1.2 The list of forms weaker than AC In this subsection, for readers’ convenience, we deﬁne and denote the weaker forms of AC used directly in this paper. For the known forms given in [9], we quote in their statements the form number under which they are recorded in [9].

Definition 1.9. 1. IQDI: Every infinite set is quasi Dedekind-infinite.

2. IWDI ([9, Form 82]): Every inﬁnite set is weakly Dedekind-inﬁnite.

3. IDI ( [9, Form 9]): Every inﬁnite set is Dedekind-inﬁnite.

4. CAC ([9, Form 8]): Every denumerable family of non-empty sets has a choice function.

7 5. CACfin ( [9, Form 10]): Every denumerable family of non-empty ﬁnite sets has a choice function.

6. WOACfin ([9, Form 122]): Every well-orderable non-empty family of non-empty ﬁnite sets has a choice function.

7. MC (the Axiom of Multiple Choice, [9, Form 67]): Every non-empty family of non-empty sets has a multiple choice function.

8. CMC (the Countable Axiom of Multiple Choice, [9, Form 126]): Every denumerable family of non-empty sets has a multiple choice function.

9. WoAm ([9, Form 133]): Every inﬁnite set is either well-orderable or has an amorphous subset.

10. DC (the Principle of Dependent Choice, [9, Form 43]): For every non- empty set A and every binary relation S on A such that (∀x ∈ A)(∃y ∈ A)(hx, yi ∈ S), there exists a ∈ Aω such that:

(∀n ∈ ω)(ha(n), a(n + 1)i ∈ S).

11. BPI (the Boolean Prime Ideal Principle, [9, Form 14]): Every Boolean algebra has a prime ideal.

12. NAS ( [9, Form 64]): There are no amorphous sets.

13. M(C,S): Every compact metrizable space is separable. (Cf. [13], [14], [17], [18] and [21].)

14. IDFBI: For every inﬁnite set D, the Cantor cube 2ω is a remainder of the discrete space hD, P(D)i. (Cf. [22].)

15. INSHC: Every infinite discrete space has a non-scattered Hausdorff compactification.

The form IDFBI has been introduced and investigated in [22] recently. More comments about IDFBI are included in Remark 1.22. New facts concerning IDFBI (among them, a solution of an open problem posed in [22]), are included in Section 2. The form INSHC is new here. That INSHC is essentially weaker than IDFBI is shown in Section 2.

8 1.3 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 1.10. (Cf. [16].) (ZF) A topological space X is scattered if and only if there exists α ∈ ON such that X(α) = ∅. If X is scattered then

(α) |X|CB = min{α ∈ ON : X = ∅}.

Moreover, if X is a non-empty scattered compact space, then |X|CB is a successor ordinal.

Theorem 1.11. (Cf., e.g., [2].) (ZF) Every non-empty dense-in-itself compact second-countable Hausdorﬀ space is of size at least |R|.

Proposition 1.12. (Cf. [26, Proposition 2.1.11] and [25].) (ZF) If d is a quasi-metric on X such that hX, τ(d)i is compact, then d is strong.

Theorem 1.13. (Cf. [6].) (ZF)

(i) (Cf. [6].) (Urysohn’s Metrization Theorem) Every second-countable T3-space is metrizable.

(ii) (Cf. [22].) Every T3-space which has a cuf base can be embedded in a metrizable Tychonoﬀ cube and that, it is metrizable.

(iii) (Cf. [22].) A T3-space X is embeddable in a compact metrizable Ty- chonoﬀ cube if and only if X is embeddable in the Hilbert cube [0, 1]ω.

Several essential applications of Theorem 1.13(ii), especially to the theory of Hausdorﬀ compactiﬁcations in ZF, have been shown in [22] recently. We show some other applications of Theorem 1.13(ii) in the forthcoming Sections 3 and 4.

Theorem 1.14. (Cf. [25].) (ZF)

(a) For every compact Hausdorﬀ, quasi metric space X = hX,di the following are equivalent:

(i) X is Loeb;

9 (ii) hX,d−1i is separable; (iii) X and hX,d−1i are both separable; (iv) X is second-countable.

In particular, every compact, Hausdorﬀ, quasi-metrizable Loeb space is metrizable.

(b) CAC implies that every compact, Hausdorﬀ quasi-metrizable space is metrizable.

Proposition 1.15. (Cf. [22].) (ZF) Every weakly Loeb regular space which admits a cuf base has a dense cuf set.

Theorem 1.16. (Cf. [27].) (ZF) Let κ be an inﬁnite cardinal number of von Neumann, {Xi : i ∈ κ} be a family of compact spaces, {fi : i ∈ κ} be a collection of functions such that for every i ∈ κ, fi is a Loeb function of Xi. Then the Tychonoﬀ product X = Xi is compact. ∈ iQκ Theorem 1.17. (Cf., e.g., [8, Theorem 4.37] and [9, Forms: 14, 14 A, 14 J].) (ZF) The following statements are equivalent to BPI:

(i) For every non-empty set X, every ﬁlter in P(X) can be enlarged to an ultraﬁlter in P(X).

(ii) Every product of compact Hausdorﬀ spaces is compact.

Theorem 1.18. (Cf. [9] and [22].) (ZF) BPI implies NAS but this implication is not reversible.

Theorem 1.19. (Cf. [22].) (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;

(b) the Cantor cube 2ω is a remainder of X;

n N n (c) there exists a family V = {Vi : n ∈ , i ∈ {1,..., 2 }} such that, for every n ∈ N, the following conditions are satisﬁed:

10 n n (i) for every i ∈ {1,..., 2 }, Vi is a non-empty family of open sets n of X such that Vi is stable under finite unions and finite inter- n sections, and, for every U ∈ Vi , the set clX U is non-compact; n n (ii) for every i ∈ {1,..., 2 } and for any U, V ∈ Vi , clX(U) \ V is compact; (iii) for every pair i, j of distinct elements of {1,..., 2n}, for any W ∈ n n n+1 n+1 Vi and G ∈ Vj , there exist U ∈ V2i−1,V ∈ V2i with clX(U ∪ V ) \ W compact and clX((U ∪ V ) ∩ G) compact; 2n n n (iv) if, for every i ∈{1,..., 2 }, Vi ∈Vi , then X \ Vi is compact. iS=1 Theorem 1.20. (Cf. [22].) (ZF) For a set D, let D = hD, P(D)i. Then the following statements hold: (i) If D is a cuf space, then every non-empty second-countable compact Hausdorff space is a remainder of a metrizable compactification of D. In particular, all non-empty second countable compact Hausdorff spaces are remainders of metrizable compactifications of N.

(ii) If D is weakly Dedekind-inﬁnite, then every non-empty second-countable compact Hausdorﬀ space is a remainder of D. Theorem 1.21. (Cf. [22].) (ZF) (i) IDFBI implies NAS but this implication cannot be reversed.

(ii) The statement “All non-empty metrizable compact spaces are remainders of metrizable compactifications of N” is equivalent to M(C,S) and, thus, it implies CACfin. Remark 1.22. In ZFC, an archetype of Theorem 1.19 is included in [7, Theo- rem 2.1]; however, in [22], Theorem 2.1 of [7] has been shown to be unprovable in ZF. In [22], an infinite set D is called dyadically filterbase infinite if 2ω is a remainder of the discrete space hD, P(D)i. An equivalent purely set- theoretic definition of a dyadically filterbase infinite set is given in [22] and it can be easily obtained from condition (c) of Theorem 1.19 applied to discrete spaces. Clearly, IDFBI is equivalent to the sentence “Every infinite set is dyadically filterbase infinite”. Theorem 1.23. (Cf. [23].) (ZF)

11 (i) For every non-empty compact Hausdorﬀ space K, there exists a Dede- kind-inﬁnite discrete space D such that K is a remainder of D.

(ii) If D is an infinite set, then the Alexandroff compactification of the discrete space D = hD, P(D)i is the unique (up to the equivalence) Hausdorff compactification of D if and only if D is amorphous. Proposition 1.24. (ZF) Let D be an infinite set and let D = hD, P(D)i. Then: (i) D(∞) is metrizable if and only if D is a cuf set (cf. [25]); (ii) the discrete space D has a metrizable compactification if and only if D is a cuf set (cf. [22]). As we have already mentioned at the beginning of Subsection 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., [10, 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 [28], [29] and [9, Note 3, page 286]) to show that Φ has a ZF-model. The notions of boundable and injectively boundable statements can be found in [28], [10, Problem 1 on page 94] and [9, Note 3, page 284]. Every boundable statement is equivalent to an injectively boundable one (see [28] or [9, Note 3, page 285]). We recommend [10, Chapter 4] as an introduction to permutation models.

1.4 The content concerning new results in brief In Section 2, we notice that, in ZF, the class of all iso-dense compact Haus- dorff 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 2.2). 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 2.3). We construct a new permutation model to prove that a dyadically filterbase infinite set can be weakly

12 Dedekind-ﬁnite (see the proof to Theorem 2.6. This solves an open problem posed by us in [22].

In Section 3, we prove in ZF that if hX,di is a quasi-metric T3-space such that d is strong and either hX, τ(d)i is limit point compact or d−1 is precompact, then the space hX, τ(d)i is metrizable (see Theorem 3.4). This result and its direct consequence that if hX,di is compact Hausdorﬀ quasi- metric space such that hX, τ(d−1)i is iso-dense, then hX, τ(d)i is metrizable (see Corollary 3.5) are new applications of Theorem 1.13(ii) and adjuncts to Theorem 1.14. By applying Theorem 1.13, we show in ZF that if hX,di is an iso-dense metric space such that either d is totally bounded or hX, τ(d)i is limit point compact, then hX, τ(d)i has a cuf base and can be embedded in a metrizable Tychonoﬀ cube (see Theorem 3.7).

Section 4 concerns equivalents of CACfin in terms of scattered or iso- dense spaces (see Theorems 4.2 and 4.4). Among our new equivalents of CACfin there are, for instance, the following sentences: (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 4.5). We prove that WOACfin is equivalent to the sentence: for every well- orderable non-empty set S and every family {hXs,dsi : s ∈ S} of compact scattered metric spaces, the product hXs, τ(ds)i is compact (see Theorem s∈S 4.7). Several other relevant results areQ included in Section 4, too. Section 5 concerns the problem of whether it is provable in ZF that every non-empty dense-in-itself compact metrizable space contains an infinite compact scattered subspace. Among other results of Section 4, 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 5.2). We prove that the sentence “every infinite first- countable Hausdorff compact space contains an infinite metrizable compact scattered subspace” implies neither CACfin nor IQDI, nor CMC in ZFA (see Theorem 5.4). Section 6 contains a shortlist of new open problems strictly relevant to

13 the topic of this paper.

2 From compact Hausdorﬀ iso-dense spaces that are not scattered to INSHC

Since every isolated point of an open subspace of a topological space X is an isolated point of X, it is obvious that the following proposition holds in ZF:

Proposition 2.1. (ZF) Every scattered space is iso-dense.

Let us notice that a compact Hausdorff space is iso-dense if and only if it is a Hausdorff compactification of a discrete space. Every iso-dense locally compact Hausdorff space which satisfies condition (c) of Theorem 1.19 has an iso-dense non-scattered Hausdorff compactification. In particular, for every dyadically filterbase infinite set D, the discrete space hD, P(D)i has a non-scattered Hausdorff compactification. It follows from Theorem 1.20(i) that every denumerable discrete space has non-scattered metrizable Haus- dorff compactifications. Thus, in ZF, the class of all (compact) metrizable scattered spaces is essentially smaller than the class of all iso-dense (compact) metrizable spaces, and the class of all (compact Hausdorff) scattered spaces is essentially smaller than the class of all (compact Hausdorff) iso-dense spaces. It was proved in [16] that it holds in ZF that every compact Hausdorff scattered space is zero-dimensional, so completely regular. Let us show that a compact Hausdorff iso-dense space may fail to be completely regular in ZF.

Proposition 2.2. There exists a model M of ZF in which there is a compact Hausdorﬀ iso-dense space which is not completely regular.

Proof. Let M be any model of ZF in which there exists a compact Hausdorff, not completely regular space Z (cf, e.g., [6]) and let us work inside M. By Theorem 1.23(i), it holds in M that there exists a Hausdorff compactification γD of a discrete space D such that γD \ D is homeomorphic to Z. Then, in M, γD is a non-scattered, iso-dense compact Hausdorff space which is not completely regular.

Let us shed more light on the forms INSHC and IDFBI.

Theorem 2.3. (ZF)

14 (i) Every compact Hausdorﬀ space is a subspace of a compact Hausdorﬀ iso-dense space.

(ii) Every compact second-countable Hausdorﬀ space is a subspace of a compact second-countable iso-dense space.

(iii) DC → IWDI → IDFBI → INSHC → NAS;

(iv) BPI → INSHC;

(v) INSHC 9 IDFBI and INSHC 9 BPI.

Proof. To show that (i) holds, it suffices to apply Theorem 1.23. It follows directly from Theorem 1.20(i) that (ii) holds. (iii) It is known that DC implies IWDI (see, e.eg., [9, pages 326 and 339]). It has been noticed in [22] that, by Theorem 1.20(ii), IWDI implies IDFBI. The implications IDFBI → INSHC → NAS can be deduced from Theorems 1.19 and 1.23(ii). To prove (iv), we assume BPI and consider any infinite set D. Let D = hD, P(D)i. In the light of Theorem 1.17(i) and [23, Theorem 3.27], it follows from BPI that there exists the Čech-Stone compactification βD of D. Suppose that βD \D has an isolated point y0. Then there exist disjoint open subsets U, V of βD such that y0 ∈ U, (βD \ D) \{y0} ⊆ V and U ∩ V = ∅. Then the subspace U ∪{y0} of βD is the Čech-Stone compactification of the subspace U ∩ D of D. It follows from Theorem 1.23(ii) that U ∩ D is amorphous but this is impossible because BPI implies NAS by Theorem 1.18. The contradiction obtained shows that βD \ D is dense-in-itself, so βD is not scattered. (v) It was shown in [22] that the conjunction BPI ∧ ¬IDFBI has a ZF- model. This, together, with (iv), implies that there is a model of ZF in which the conjunction INSHC ∧ ¬IDFBI is true. Hence INSHC 9 IDFBI. To prove INSHC 9 BPI, let us use the Feferman’s forcing model M2 in [9]. It is known that DC ∧ ¬BPI is true in M2 (see [9, page 148]). To complete the proof, it suffices to notice that it follows from (iii) that INSHC is also true in M2. Remark 2.4. (a) We do not know if the conjunction NAS ∧ ¬INSHC has a ZF-model. (b) It is worth noticing that it follows from Theorem 1.23(ii) that it holds in ZF that NAS is equivalent to the following statements:

15 (i) Every infinite discrete space has a two-point Hausdorff compactification.

(ii) For every natural number n, every infinite discrete space has an n-point Hausdorff compactification. (c) In view of Theorem 2.3(iii), it holds in ZF that INSHC follows from every form of [9] which implies IWDI. In particular, the implication CMC → INSHC holds in ZF (see [9, page 339]). (d) It is known that BPI implies CACfin (see, e.g., [9, pages 325 and 354]). Since BPI implies INSHC by Theorem 2.3(ii), it is worth noticing that the conjunction INSHC ∧ ¬CACfin has a ZF-model. To see this, let us notice that, in the Second Fraenkel Model N 2 in [9], the conjunction IWDI ∧ ¬CACfin is true (see [9, page 179]). Since IWDI ∧ ¬CACfin is a conjunction of two injectively boundable statements and it has a permutation model, it also has a ZF-model by Pincus’ transfer theorems. This, together with Theorem 2.3(iii), implies that INSHC ∧ ¬CACfin has a ZF-model. This is also an alternative proof that INSHC ∧ ¬BPI has a ZF-model. It has been shown in the proof to Theorem 2.3 that INSHC ∧ ¬IDFBI has a ZF. Thus, by Theorem 2.3(iii), INSHC ∧ ¬IWDI has a ZF-model. Let us recall the following open problems posed by us in [22]: Problem 2.5. (i) Is there a ZF-model for IDFBI ∧ ¬IWDI? (See [22, Problem (3) of Section 6].)

(ii) Is there a model of ZF in which a weakly Dedekind-finite set can be dyadically filterbase infinite? (See [22, Problem (6) of Section 6].) In [22, the proof to 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 2.5(ii) (that is, Problem (6) from Section 6 in [22]) by the following theorem: Theorem 2.6. 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 [10, Theorem 6.1]), it suffices to prove that Φ has a permutation

16 model. To this end, let us modify the model constructed in [22, the proof to 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) =6 x}. We start with a model M of ZFA + AC with a denumerable set A of atoms such that A has a denumerable partition A = {Ai : i ∈ ω} into inﬁnite sets. In M, we let n N n B = {Bi : n ∈ , i ∈{1, 2,..., 2 }} be a family with the following two properties:

1 1 (a) For n =1, {B1, B2} is a partition of A = {Ai : i ∈ ω} into two inﬁnite sets. N n n+1 n+1 (b) For every n ∈ and for every i ∈ {1, 2,..., 2 }, {B2i−1, B2i } is a n partition of Bi 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:

<ω (d) (∀i ∈ ω)(∃j ∈ ω)(∃F ∈ [Aj] )(φ[supp(φ ↾ Ai)] = F ).

For every n ∈ N and for every i ∈{1, 2,..., 2n}, we let

n n Qi = { {φ(Z): Z ∈ Bi } : φ ∈ G}. [ We also let n N n Q = {Qi : n ∈ , i ∈{1, 2,..., 2 }}. [ For every E ∈ [Q]<ω, we let

GE = {φ ∈ G : ∀Q ∈ E(φ(Q)= Q)}.

′ <ω Then GE is a subgroup of G. Furthermore, since for all E, E ∈ [Q] , <ω GE∪E′ ⊆ GE ∩ GE′ , the collection {GE : E ∈ [Q] } is a base for a ﬁlter in the set of all subgroups of G. Let F be the ﬁlter of subgroups of G generated by

17 <ω {GE : E ∈ [Q] }. To check that F is a normal ﬁlter on G, we need to show that F has the following two properties:

(1) ∀a ∈ A({π ∈ G : π(a)= a}∈F) and

(2) (∀π ∈ G)(∀H ∈F)(πHπ−1 ∈F).

To argue for (1), let a ∈ A. Since A is a partition of A in M, there exists 1 1 a unique i ∈ ω such that a ∈ Ai. Since the set {B1, B2} is a partition of A, 1 1 1 either Ai ∈ B1 or Ai ∈ B2. Suppose that Ai ∈ B1 (the argument is similar if 1 1 ′ Ai ∈ B2). Pick an Aj ∈ B2 and an a ∈ Aj. Let φ ∈ G be the transposition (a, a′) (i.e. φ interchanges a and a′ and ﬁxes all other atoms). Then

1 1 ′ {φ(Z): Z ∈ B2} =( (B2 \{Aj})) ∪ ((Aj \{a }) ∪{a}). [ [ 1 1 1 1 <ω <ω Let E = { B1, {φ(Z): Z ∈ B2}}. Then, E ∈ [Q1 ∪ Q2] ⊂ [Q] , <ω so E ∈ [Q] .S Furthermore,S GE ⊆ {π ∈ G : π(a) = a}. Indeed, let π ∈ GE. Towards a contradiction, assume that π(a) = b for some b ∈ A \{a}. Since 1 1 1 1 a ∈ B1 and π ﬁxes B1, it follows that b = π(a) ∈ π( B1)= B1. But then,S since π ∈ GE, weS have the following: S S

1 1 a ∈ {φ(Z): Z ∈ B2}} → π(a) ∈ π( {φ(Z): Z ∈ B2}}) [ 1 [ 1 ′ → b ∈ {φ(Z): Z ∈ B2}} =( (B2 \{Aj})) ∪ ((Aj \{a }) ∪{a}), [ [ which is a contradiction. Therefore, (1) holds. To argue for (2), let π ∈ G and H ∈ F. There exists E ∈ [Q]<ω such <ω that GE ⊆ H. By the deﬁnition of Q, we have π[E] ∈ [Q] . We assert that −1 Gπ[E] ⊆ πHπ . Let ρ ∈ Gπ[E]. For every T ∈ E we have the following:

ρ(πT )= πT → π−1ρπ(T )= T ;

Hence, since GE ⊆ H, we have:

−1 −1 −1 π ρπ ∈ GE → ρ ∈ πGEπ ⊆ πHπ .

18 −1 Therefore, ρ ∈ πHπ . Since ρ is an arbitrary element of Gπ[E], we conclude −1 −1 that Gπ[E] ⊆ πHπ . Thus, πHπ ∈ F, so (2) holds. This completes the proof that F is a normal filter on G. Let N be the permutation model determined by M, G and F. We say <ω that an element x ∈N has support E ∈ [Q] if, for all φ ∈ GE, φ(x)= x. In N , the set (P(A))N = P(A) ∩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 hA, (P(A))N i satisfies condition (c) of Theorem 1.19. To this aim, for every n ∈ N and for every i ∈{1, 2,..., 2n}, we let

n n <ω n <ω Vi = R : R ∈ [Qi ] \ {∅} ∪ C : C ∈ [Qi ] \ {∅} , n\ o n[ o and we also let n N n V = {Vi : n ∈ , i ∈{1, 2,..., 2 }}. 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 [10, page 47]). Since V is denumerable in the ground model M, 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 = hA, (P(A))N i, then V has properties (i)-(iv) of condition c of Theorem 1.19. This, together with Theorem 1.19, proves that A is dyadically filterbase infinite in the model N . To complete the proof, it remains to show that A is weakly Dedefind-finite in N . By way of contradiction, we assume that A is weakly Dedekind-infinite in N . Thus, it holds in N that there exists a denumerable disjoint family N <ω U = {Un : n ∈ ω} of (P(A)) . Let E ∈ [Q] be a support of Un for all n ∈ ω. It is not hard to verify now that there exist a pair k, m of distinct elements of ω and a pair x, y of atoms, such that x ∈ Uk and y ∈ Um. The transposition ψ = (x, y) from the group G is an element of GE. It follows that ψ(Uk)= Uk, and so y = ψ(x) ∈ ψ(Uk)= Uk. This is impossible because y ∈ Um and Uk ∩Um = ∅. The contradiction obtained shows that A is weakly Dedekind-finite in N . The model constructed in [22, the proof to Theorem 5.14] is different from that we have just introduced in the proof to Theorem 2.6. By Theorem 5.15 of [22], NAS is false in the model from [22, the proof to Theorem 5.14]. Let N be the model from the proof to Theorem 2.6 above. We do not know if NAS holds N . Let us notice that, for every i ∈ ω, no E ∈ [Q]<ω

19 can support Ai and, in consequence, Ai ∈N/ . This is why we cannot mimic the proof to Theorem 5.15 in [22] to show that NAS fails in N . We also do not know if INSHC holds in N .

3 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 3.1. (ZF) Let d be a quasi-metric on a set X such that either d−1 is precompact or the space X = hX, τ(d)i is limit point compact. Then Isoτ(d−1)(X) is a cuf set.

Proof. Let D = Isoτ(d−1)(X). For every x ∈ D, let

1 n = min n ∈ N : B −1 x, = {x} . x d n

For every n ∈ N, let An = {x ∈ D : n = nx}. Suppose that n0 ∈ N is such that An0 is inﬁnite. Let us show that there exist x0 ∈ X and x1 ∈ An0 such 1 that x0 =6 x1 and x1 ∈ Bd x0, n . If x0, x1 are such points, we notice that 0 − 1 1 1 −1 d (x1, x0) = d(x0, x1) < n , so x0 ∈ Bd x1, n which is impossible by 0 0 the deﬁnition of An0 .

If X is limit point compact, there exists an accumulation point of An0 in X. In this case, for a fixed accumulation point x0 of An0 , we can fix 1 −1 x1 ∈ An0 such that x1 =6 x0 and x1 ∈ Bd x0, n . Assuming that d is 0 1 precompact, we can fix a finite set F ⊆ X such that X = Bd x, n ∈ 0 xSF and, since An0 is infinite, we can fix x0 ∈ F and x1 ∈ An0 such that x1 =6 x0 and x ∈ B x , 1 . Hence, the assumption that A is infinite leads to a 1 d 0 n0 n0 contradiction. Therefore, D = An is a cuf set. ∈N nS Corollary 3.2. (ZF) Let X = hX,di 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.

20 Proof. That Iso(X) is a cuf set follows from Theorem 3.1. The second asser- tion is straightforward.

Remark 3.3. Let X be a compact metrizable space. Then, using Proposition 1.24(ii), we may deduce that Iso(X) is a cuf set. Namely, suppose that Iso(X) is infinite. Let Y = clX(Iso(X)). Then Y is a metrizable compactification of the discrete space Iso(X), so Iso(X) is a cuf set by Proposition 1.24(ii). Theorem 1.14(b) improves the well-known result of ZFC that every compact Hausdorff quasi-metrizable space is metrizable (see Corollary in [5, 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 [25] is whether it can be proved in ZF that every quasi-metrizable compact Haus- dorff space is metrizable. Theorem 1.14 is a partial solution to this problem. Now, we can shed a little more light on it by the following theorem:

Theorem 3.4. (ZF) Let d be a strong quasi-metric on a set X such that hX, τ(d)i is a T3-space. Then the following conditions are satisﬁed:

(i) if hX, τ(d−1)i has a dense cuf set, then hX, τ(d)i is metrizable;

(ii) if hX, τ(d−1)i is iso-dense and either hX, τ(d)i is limit point compact or d−1 is precompact, then the space hX, τ(d)i is metrizable.

−1 Proof. (i) Assume that A = An is a dense set in hX, τ(d )i such that, ∈ nSω for every n ∈ ω, An is a non-empty finite set. For n, m ∈ ω, we define 1 Bm,n = Bd x, m+1 : x ∈ An . Since d is strong, in much the same way, as in the proof to Theorem 4.6 in [25], one can show that B = Bn,m is a ∈ n,mS ω base of hX, τ(d)i. Since B is a cuf set, the space hX, τ(d)i is metrizable by Theorem 1.13(ii). (ii) Now, we assume that hX, τ(d−1)i is iso-dense and either hX, τ(d)i is −1 limit point compact or d is precompact. Let E = Isoτ(d−1)(X). Then E is dense in hX, τ(d−1)i. By Theorem 3.1, the set E is a cuf set. Hence, to conclude the proof, it suffices to apply (i).

Corollary 3.5. (ZF) Let hX,di be a compact Hausdorﬀ quasi-metric space such that hX, τ(d−1)i is iso-dense. Then hX, τ(d)i is metrizable.

Proof. This follows immediately from Proposition 1.12 and Theorem 3.4.

21 Remark 3.6. In Corollary 3.5, we cannot omit the assumption that hX, τ(d)i 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 [25]). Then d is a strong quasi-metric such that hω, τ(d)i is a compact T1-space which is not metrizable.

Theorem 3.7. (ZF) Let hX,di be an iso-dense metric space such that either d is totally bounded or hX, τ(d)i is limit point compact. Then hX, τ(d)i has a cuf base and can be embedded in a metrizable Tychonoﬀ cube.

Proof. It follows from the proof to Theorem 3.4 that hX, τ(d)i has a cuf base. Since hX, τ(d)i is a T3-space, to conclude the proof, it suﬃces to apply Theorem 1.13(ii).

4 CACfin 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 of compact iso-dense metrizable spaces and compact scattered metrizable spaces in ZF is diﬀerent than in ZFC. To begin, let us recall the following lemma proved in [21]:

Lemma 4.1. (ZF). Let X be a non-empty metrizable space and let B be a base of X. Then X embeds in [0, 1]B×B.

If X = hX,di is a metric space and Y is a topological space, then we say that X embeds in Y if the space hX, τ(d)i embeds in Y. The following theorem is a characterization of CACfin in terms of iso- dense (limit point) compact metrizable spaces and in terms of iso-dense totally bounded metric spaces.

Theorem 4.2. (ZF) The following conditions are all equivalent:

(i) CACfin; (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;

22 (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 with “scattered”.

Proof. Let X = hX,di be an iso-dense (respectively, scattered) metric space such that X is either limit point compact or totally bounded. By Corollary 3.2, the set Iso(X) is a cuf set. Hence, it follows from CACfin that Iso(X) is countable. In consequence, (i) implies (ii). Since every separable metrizable space is second-countable, it follows from Lemma 4.1 that it is true in ZF that (ii) implies (iii). Now, to show that (iii) implies (iv), suppose that hX, τ(d)i 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 CACfin fails. Then there exists an uncountable discrete cuf space D. It follows from Proposition 1.24(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 D(∞) is uncountable, (v) fails if CACfin fails. Hence (v) implies (i).

Theorem 4.3. (ZF) (i) For every totally bounded metric space X, the Cantor-Bendixson rank |X|CB of X is a countable ordinal. (ii) Every totally bounded scattered metric space is a cuf space. (iii) Every totally bounded scattered metric space has a cuf base.

Proof. Let X = hX,di be an inﬁnite totally bounded metric space. Let α = |X|CB. Then X = Iso(X(γ)) ∪ X(α). γ[∈α

23 For every γ ∈ α and every x ∈ Iso(X(γ)), let 1 n(x, γ) = min n ∈ N : B x, ∩ X(γ) = {x} . d n For every γ ∈ α and every n ∈ N, let

(γ) Aγ,n = {x ∈ Iso(X ): n(x, γ)= n}.

We have already shown in the proof to Theorem 3.1 that, for every γ ∈ α (γ) and every n ∈ N, the set Aγ,n is ﬁnite and Iso(X )= Aγ,n. ∈N nS (i) Suppose that α is uncountable. For every n ∈ N, let

Bn = {γ ∈ α : Aγ,n =6 ∅}. N Since α is supposed to be uncountable, there exists n0 ∈ such that Bn0 is infinite. We fix such an n0 and put 1 U = Bd x, : x ∈ X . 3n0 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 = B x, 1 . Since B is infinite, there exist γ ,γ ∈ B and d 3n0 n0 1 2 n0 ∈ xSF elements x ∈ F , x ∈ A ∩ B x , 1 and x ∈ A ∩ B x , 1 , 0 1 γ1,n0 d 0 3n0 2 γ2,n0 d 0 3n0 (γ2) (γ1) such that x1 =6 x2. We may assume that γ1 ≤ γ2. Then X ⊆ X and it follows from the definition of A that d(x , x ) ≥ 1 . On the other hand, γ1,n0 1 2 n0 since x , x ∈ B (x , 1 ), we have d(x , x ) ≤ 2 < 1 . The contradiction 1 2 d 0 3n0 1 2 3n0 n0 obtained proves that α is countable. (ii) Now, suppose that the space X is also scattered. Then it follows from (α) Proposition 1.10 that X = ∅. Hence X = {Aγ,n : γ ∈ α and n ∈ N}. Since α is countable, the set α × N is countable.S This implies that the family {Aγ,n : γ ∈ α and n ∈ N} is also countable. We have already shown that, for every γ ∈ α and every n ∈ N, the set Aγ,n is finite. Hence X is a cuf set. It follows immediately from (ii) and Theorem 3.7 that (iii) holds. Theorem 4.4. (ZF) The following conditions are all equivalent:

(i) CACfin;

24 (ii) every totally bounded scattered metric space is countable; (iii) every compact metrizable scattered space is countable; (iv) every totally bounded, complete scattered metric space is compact.

Proof. Since CACfin implies that all cuf sets are countable, it follows from Theorem 4.3 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 4.3, (i) implies (iv). Assume that CACfin is false. Then there exists a family {An : n ∈ ω} of non-empty pairwise disjoint finite sets such that the set D = An is n∈ω Dedekind-finite (see [9, Form [10M]]). Let D = hD, P(D)i. By PropositionS 1.24(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 hD, ρi is also totally bounded. Since D is Dedekind-finite and hD, ρi is discrete, the metric ρ is complete. Clearly, hD, ρi is not compact. All this taken together completes the proof. Theorem 4.5. (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 = hX,di 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 {Xn : 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 a t ∈ {S0, 1}n+1 be defined by: s a t(i) = s(i) for every i ∈ n, and s a t(n)= t. Using ideas from [2], let us define by induction (with respect to n) a family {Bs : s ∈ S} such that, for every s ∈ S, the following conditions are satisfied:

(1) Bs is a non-empty open subset of X;

(2) for every t ∈{0, 1}, Bsat ⊆ Bs;

(3) clX(Bsa0) ∩ clX(Bsa1)= ∅. To start the induction, for n =1= {0} and every s ∈{0, 1}1, we deﬁne:

d(X0,X1) Bs = Bd x, : x ∈ Xs(0) . [ 3 25 n Now, suppose that n ∈ N is such that, for every s ∈ {0, 1}i, we have i=1 S n+1 defined a non-empty open subset Bs of X. For an arbitrary s ∈{0, 1} , we consider the set Bs↾n. We put ns = min{m ∈ ω : Xm ∩ Bs↾n =6 ∅}. Since X is dense-in-itself, we have ∅= 6 {m ∈ ω : Xm ∩(Bs↾n \Xns ) =6 ∅}, so we can define ks = min{m ∈ ω : Xm ∩ (Bs↾n \ Xns ) =6 ∅}. Now, we put Ys,0 = Xns ∩ Bs↾n and Ys,1 = Xks ∩ (Bs↾n \ Xns ). We define

d(Ys,0,Ys,1) Bs = Bd y, : y ∈ Ys,s(n) . [ 3

In this way, we have inductively deﬁned the required family {Bs : s ∈ S}. We notice that it follows from (2) that, for every f ∈{0, 1}ω and n ∈ N, ∅= 6 clX(Bf↾(n+1)) ⊆ clX(Bf↾n). Thus, by the compactness of X, for every f ∈{0, 1}ω, the set

Mf = {clX(Bf↾n): n ∈ N} \ ω is non-empty. For every f ∈ {0, 1} , let mf = min{n ∈ ω : Xn ∩ Mf =6 ∅}. ω We deﬁne a mapping F : {0, 1} → {P(Xn): n ∈ ω} by putting: S ω F (f)= Xmf ∩ Mf for every f ∈{0, 1} .

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(Xn): n ∈ ω}. But this is impossible because {0, 1}ω, being equipotentS 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 1.11:

Theorem 4.6. (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|.

26 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., [18]). Hence, if X is Loeb, then |R|≤|X| by Theorem 1.11. Suppose that X is weakly Loeb. Let d be any metric on X which induces the topology of X. It follows from Proposition 1.15 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 {Xn : n ∈ ω} of non-empty finite subsets of X such that the set {Xn : n ∈ ω} is dense in X. Mimicking the proof to Theorem 4.5, we canS define an injection F : {0, 1}ω → P(X) such that, for every f ∈{0, 1}ω, the set Mf = F (f) is a non-empty closed subset of X and, for every pair ω f,g of distinct functions from {0, 1} , Mf ∩ Mg = ∅. Let ψ be a weak Loeb function for X. Then ψ ◦ F is an injection from {0, 1}ω into [X]<ω. Hence |R| = |{0, 1}ω|≤|[X]<ω|. Taking the opportunity, let us give a proof to the following theorem: Theorem 4.7. (ZF)

(a) For every non-empty set I and every family {hXi, τii : i ∈ I} of denumerable metrizable compact spaces, the family {Xi : i ∈ I} has a multiple choice function.

(b) For a non-zero von Neumann ordinal α, let {Xγ : γ ∈ α} be a family of pairwise disjoint non-empty countable, compact metrizable spaces. Then the direct sum X = Xγ is weakly Loeb. ∈ γLα (c) The following conditions are all equivalent:

(i) WOACfin;

(ii) for every well-orderable set S and every family {hXs,dsi : s ∈ S} of scattered totally bounded metric spaces, the union Xs is well- s∈S orderable. S

(iii) for every well-orderable non-empty set S and every family {hXs,dsi : s ∈ S} of compact scattered metric spaces, the product hXs, τ(ds)i s∈S is compact. Q

Proof. (a) Let Xi = hXi, τii be a denumerable metrizable compact space for every i ∈ I with I =6 ∅. For every i ∈ I, let αi = |Xi|CB. We ﬁx i ∈ I and

27 observe that, since Xi is scattered by Theorem 4.5, it follows from Proposition (αi) 1.10 that Xi = ∅ and αi is a successor ordinal. Let βi ∈ ON be such that (βi) (βi) (βi) αi = βi +1. Then Xi = Iso(Xi ). If the set Iso(Xi ) were inﬁnite, it (βi) would have an accumulation point in Xi by the compactness of Xi. Hence (βi) Iso(Xi ) is a non-empty ﬁnite set. In consequence, by assigning to any i ∈ I (βi) the set Iso(Xi ), we obtain a multiple choice function of {Xi : 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 {fγ : γ ∈ α} such that, for every γ ∈ α, fγ is a weak Loeb function of Xγ. For every F ∈F, let γ(F ) = min{γ ∈ α : F ∩ Xγ =6 ∅} and let f(F )= fγ(F )(F ∩ Xγ). Then f is a weak Loeb function of X.

(c) (i)→(ii) Let us assume WOACfin. Suppose that S is a well-orderable non-empty set and, for every s ∈ S, Xs = hXs,dsi is a non-empty scattered totally bounded metric space. To prove that X = Xs is well-orderable, s∈S without loss of generality, we may assume that S = αSfor some non-zero von Neumann ordinal α, and Xi ∩ Xj = ∅ for every pair i, j of distinct elements of α. In much the same way, as in the proof to Theorem 4.3, we can define a family {Ai,n : i ∈ α, n ∈ N} of non-empty finite sets such that, for every i ∈ α, Xi = Ai,n. Now, we can easily define a family {Mi : i ∈ α} ∈N nS of subsets of N and a family {Fi,n : i ∈ α, n ∈ Mi} of pairwise disjoint non-empty finite sets such that, for every i ∈ α, Xi = Fi,n. The set ∈ nSMi J = {hi, ni : i ∈ α, n ∈ Mi} is well-orderable, so we can fix a von Neumann ordinal number γ and a bijection h : γ → J. For every j ∈ γ, let n(j) ∈ ω n(j) be equipotent to Fh(j), and let Bj = {f ∈ Fh(j) : f is a bijection}. By WOACfin, there exists ψ ∈ Bj. Now, we can define a well-ordering ≤ ∈ jQγ on X = Fh(j) as follows: for i, j ∈ γ, x ∈ Fh(i),y ∈ Fh(j), we put x ≤ y if ∈ jSγ either i ∈ j or i = j and ψ(i)−1(x) ⊆ ψ(i)−1(y). Hence (i) implies (ii) (ii)→(iii) Let S be a non-empty well-orderable set and, for every s ∈ S, let hXs,dsi be a compact scattered metric space. Clearly, we may assume that S is a von Neumann cardinal. We notice that if X = Xs is well- ∈ sSS orderable, then we can define a family {fs : s ∈ S} such that, for every s ∈ S, fs is a Loeb function of Xs = hXs, τ(ds)i and, therefore, by Theorem 1.16, the product Xs is compact. Hence (ii) implies (iii). ∈ sQS 28 (iv)→(i) Now, let S be a well-orderable set and let {As : s ∈ S} be a family of non-empty finite sets. Take an element ∞ ∈/ As and put ∈ sSS Ys = As ∪ {∞} for s ∈ S. Let ρs be the discrete metric on Ys and let Ys = hYs, P(Ys)i for every s ∈ S. Assuming (iii), we obtain that the space Y = Ys is compact. In much the same way, as in the proof of Kelley’s s∈S theoremQ that the Tychonoff Product Theorem implies AC (see [11]), one can show that the compactnes of Y implies that As =6 ∅. Hence (iii) implies s∈S (i). Q

5 Alexadroff compactifications of infinite discrete cuf spaces as subspaces

In this section, we shed some light on the problem of whether it is provable in ZF that every non-empty dense-in-itself compact metrizable space contains an infinite compact scattered subspace. Unfortunately, we are still unable to give a satisfactiory solution to this problem. Certainly, N(∞) or, more generally, for every infinite discrete cuf space D, D(∞) is a simple example of an infinite compact metrizable scattered space. So, let us search for conditions on a non-discrete compact metrizable space to contain a copy of D(∞) for some infinite discrete cuf space D. Let us start with the following simple proposition:

Proposition 5.1. (ZF) Let X be a non-discrete ﬁrst-countable Loeb T3- space. Then X contains a copy of N(∞). In particular, every non-discrete metrizable Loeb space contains a copy of N(∞).

Proof. Let x0 be an accumulation point of X and let f be a Loeb function of X. Since X is a ﬁrst-countable T3-space and x0 is an accumulation point of X, there exists a base {Un : n ∈ N} of open neighborhoods of x0 such that clX(Un+1) ⊂ Un for every n ∈ N. Let xn = f(clX(Un) \ Un+1) for every n ∈ N. Then the subspace {x2n : n ∈ ω} of X is a copy of N(∞).

Theorem 5.2. (ZF) (i) IQDI implies that every infinite compact first-countable Hausdorff space contains a copy of D(∞) for some infinite discrete cuf space D.

29 (ii) Each of IDI, WoAm and BPI implies that every infinite compact first- countable Hausdorff space contains a copy of N(∞). (iii) Every infinite first-countable compact Hausdorff separable space contains a copy of N(∞). Every infinite first-countable compact Hausdorff space having an infinite cuf subset contains a copy of D(∞) for some infinite discrete cuf space. Proof. Let X = hX, τi be an infinite compact first-countable Hausdorff space. For a point x0 ∈ X, let {Un : n ∈ ω} be a base of neighborhoods of x0 such that clX(Un+1) ⊆ Un for every n ∈ ω.

(i) Assuming IQDI, we can fix a disjoint family {Fn : n ∈ ω} of non- empty finite subsets of X. Since X is compact, the set F = Fn has ∈ nSω an accumulation point. Let x0 be an accumulation point of F . We may assume that x0 ∈/ F . Let n0 = m0 = min{n ∈ ω : U0 ∩ Fn =6 ∅}, M0 = ω and Y0 = Fn0 ∩ U0. Since X is a T1-space, Y0 is a finite set and x0 ∈/ Y0, the set M1 = {n ∈ ω : Un ∩ Y0 = ∅} is non-empty. Let m1 = min M1, n1 = min{n ∈ ω : Um1 ∩ Fn =6 ∅} and Y1 = Um1 ∩ Fn1 . Suppose that k ∈ ω \{0} is such that we have already defined nk, mk ∈ ω such that Yk =

Umk ∩ Fnk =6 ∅. We put Mk+1 = {n ∈ ω : Un ∩ Yk = ∅}, mk+1 = min Mk+1, nk+1 = min{n ∈ ω : Umk+1 ∩ Fn =6 ∅} and Yk+1 = Umk+1 ∩ Fnk+1 . This terminates our inductive definition. Let D = Yk and Y = D ∪{x0}. k∈ω Then D is a discrete cuf subspace of X, the subspaceS Y of X is compact and x0 is a unique accumulation point of Y. Hence Y is a copy of D(∞). (ii) If IDI holds or X is well-orderable, then we can fix a disjoint family {Fn : n ∈ ω} of singletons of X and in much the same way, as in the proof to (i), we can find a copy of N(∞) in X. By Proposition 4.4 of [21], WoAm implies that every first-countable limit point compact T1-space is well-orderable. Hence, WoAm implies that X contains a copy of N(∞). Now, let us assume BPI. Let x0 be an accumulation point of X. Without loss of generality, we may aassume that clX(Un+1) =6 Un for every n ∈ ω. Let Gn = {x0} ∪ (clX(Un) \ Un+1) for every n ∈ ω. Then the subspaces Gn of X are compact. By Theorem 1.17(ii), the product G = Gn is compact. n∈ω −1 Q Therefore, since the family G = {πn (clX(Un) \ Un+1): n ∈ ω} has the finite intersection property and consists of closed subsets of the compact space G, −1 there exists f ∈ πn (clX(Un) \ Un+1). Then the subspace {x0}∪{f(2n): n∈ω n ∈ ω} of X is aT copy of N(∞). This completes the proof to (ii).

30 (iii) This can be deduced from the proofs to (i) and (ii). Corollary 5.3. (ZF) (i) Each of IQDI, WoAm and BPI implies that every infinite compact Hausdorff first-countable space contains an infinite metrizable compact scattered subspace. (ii) Every infinite compact Hausdorff first-countable space which has an infinite cuf subset contains an infinite compact metrizable scattered subspace. Let us recall a few known facts about the following permutation models in [9]: the Basic Fraenkel Model N 1, the Second Fraenkel Model N 2 and the Mostowski Linearly Ordered Model N 3. It is known that WoAm is true in N 1, IWDI (and hence the stronger IQDI, which is implied by CMC) is false in both N 1 and N 3, CACfin is true in N 1 but it is false in N 2, MC is true in N 2, and BPI (and hence CACfin) is true in N 3. All this, taken together with Theorem 5.2 and its proof, implies the following theorem: Theorem 5.4. (i) It is true in N 1 that every first-countable limit point compact T1-space is well-orderable. (ii) The sentence “every infinite first-countable Hausdorff compact space contains a copy of N(∞)” is true in N 1 and in N 3. (iii) The sentence “every infinite first-countable Hausdorff compact space contains an infinite metrizable compact scattered subspace” is true in N 2. (iv) The sentence “every infinite first-countable Hausdorff compact space contains an infinite metrizable compact scattered subspace” implies neither CACfin nor IQDI, nor CMC in ZFA. Remark 5.5. In [21], it was shown that M(C,S) ∧ ¬IDI has a ZF-model. Using similar arguments, one can show that M(C,S) ∧ ¬IQDI also has a ZF-model. This, together with Corollary 5.3, proves that the sentence “every infinite compact metrizable space contains an infinite compact scattered subspace” does not imply IQDI in ZF. This can be also deduced from Theorem 5.4. (ii) Suppose that {An : n ∈ ω} is a disjoint family of non-empty finite sets which does not have any partial choice function. Let D = An and ∈ nSω 31 D = hD, P(D)i. Then D(∞) is a compact metrizable scattered space which does not contain a copy of N(∞). Let us finish our results with the following proposition:

Proposition 5.6. (ZF) A metrizable space X contains an inﬁnite compact scattered subspace if and only if X contains a copy of D(∞) for some inﬁnite discrete cuf space D.

Proof. (→) Suppose that X is a metrizable space which has an infinite compact scattered subspace Y. Then Iso(Y ) is a dense discrete subspace of Y, so Y is a metrizable compactification of the discrete subspace Iso(Y ) of Y. We deduce from Proposition 1.24(ii) that Iso(Y ) is an infinite cuf set. Thus, by Theorem 5.2(iii), Y contains a copy of D(∞) for some infinite discrete cuf space D. (←) This is trivial because the one point Hausdorff compactification of an infinite discrete space is a scattered space.

6 A shortlist of new open problems

Problem 6.1. Find, if possible, a ZF-model for NAS ∧ ¬INSHC. Problem 6.2. Find, if possible, a ZF-model for INSHC ∧ ¬BPI ∧ ¬IDFBI. Problem 6.3. Make a list of the forms from [9] that are true in the model N constructed in the proof to Theorem 2.6. Problem 6.4. Is it provable in ZF that every non-empty dense-in-itself compact metrizable space contains an inﬁnite compact scattered subspace? Problem 6.5. Is it provable in ZF that if X is a compact Hausdorﬀ non- scattered weakly Loeb space which has a cuf base, then |R|≤|X|?

Declarations

The authors declare no conflict of interest and no specific financial support for this work.

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