On Iso-dense and Scattered Spaces without $\textbf{AC}$

Abstract

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 $\textbf{ZF}$ (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice ($\textbf{AC}$)), 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 $\textbf{ZF}$. 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.

1 Preliminaries

1.1 Set-theoretic Framework and Preliminary Definitions

In this article, the intended context for reasoning and statements of theorems is $\textbf{ZF}$ without any form of the axiom of choice $\textbf{AC}$. However, we also refer to permutation models of $\textbf{ZFA}$ (cf. [13, 15]). We are mainly concerned with iso-dense and scattered spaces in $\textbf{ZF}$, defined as follows:

Definition 1

A topological space ${\textbf{X}}$ is called:

• (i) iso-dense if the set ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ of all isolated points of ${\textbf{X}}$ is dense in ${\textbf{X}}$;

• (ii) scattered or dispersed if, for every non-empty subspace ${\textbf{Y}}$ of ${\textbf{X}}$, $\emptyset \ne {{\,\mathrm{\textrm{Iso}}\,}}(Y)$.

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 $\textbf{AC}$ in Sect. 2, and recall several known results for future references in Sect. 3. The content of the article is described in brief in Sect. 4. Our main new results are included in Sects. 2–4.

We denote by ON the class of all (von Neumann) ordinal numbers. The first infinite ordinal number is denoted by $\omega$. Then ${\mathbb {N}}=\omega \setminus \{0\}$. An ordinal $\alpha$ is called an initial ordinal or, equivalently, a well-ordered cardinal if there does not exist $\gamma \in \alpha$ such that the sets $\gamma$ and $\alpha$ are equipotent.

A set X is called finite (respectively, countable) if X is equipotent to an element of $\omega$ (respectively, to a subset of $\omega$). Infinite (i.e., not finite) countable sets are called denumerable.

If X is a set, the power set of X is denoted by ${\mathcal {P}}(X)$, and the set of all finite subsets of X is denoted by $[X]^{<\omega }$.

A quasi-metric on a set X is a function $d: X\times X\rightarrow [0, +\infty )$ such that, for all $x,y,z\in X$, $d(x,y)\le d(x,z)+d(z,y)$ and $d(x,y)=0$ if and only if $x=y$ (cf. [8, 17, 33, 43]). If a quasi-metric d on X is such that $d(x,y)=d(y,x)$ for all $x,y\in X$, then d is a metric. A (quasi-)metric space is an ordered pair $\langle X, d\rangle$ 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:

$$\begin{aligned} d^{-1}(x, y)=d(y, x) \text { for } x, y\in X. \end{aligned}$$

The metric $d^{\star }$ associated with d is defined by:

$$\begin{aligned} d^{\star }(x,y)=\max \{d(x,y), d(y,x)\} \text { for } x,y\in X. \end{aligned}$$

Clearly, d is a metric if and only if $d=d^{-1}=d^{\star }$.

The d-ball with centre $x\in X$ and radius $r\in (0, +\infty )$ is the set

$$\begin{aligned} B_{d}(x, r)=\{ y\in X: d(x, y)<r\}. \end{aligned}$$

The collection

$$\begin{aligned} \tau (d)=\left\{ V\subseteq X: (\forall x\in V)(\exists \varepsilon >0) B_{d}\left( x, \varepsilon \right) \subseteq V\right\} \end{aligned}$$

is the topology on X induced by d. For a set $A\subseteq X$, let $\delta _d(A)=0$ if $A=\emptyset$, and let $\delta _d(A)=\sup \{d(x,y): x,y\in A\}$ if $A\ne \emptyset$. Then $\delta _d(A)$ is the diameter of A in $\langle X, d\rangle$.

A quasi-metric d on X is called strong if $\tau (d)\subseteq \tau (d^{-1})$.

In the sequel, topological or (quasi-)metric spaces (called spaces in abbreviation) are denoted by boldface letters, and the underlying sets of the spaces are denoted by lightface letters.

For a topological space ${\textbf{X}}=\langle X, \tau \rangle$ and for $Y\subseteq X$, let $\tau |_Y=\{V\cap Y: V\in \tau \}$ and let ${\textbf{Y}}=\langle Y, \tau |_Y\rangle$. Then ${\textbf{Y}}$ is the topological subspace of ${\textbf{X}}$ such that Y is the underlying set of ${\textbf{Y}}$. If this is not misleading, we may denote the topological subspace ${\textbf{Y}}$ of ${\textbf{X}}$ by Y.

A topological space ${\textbf{X}}=\langle X, \tau \rangle$ is called (quasi-) metrizable if there exists a (quasi-) metric d on X such that $\tau =\tau (d)$.

For a (quasi-) metric space ${\textbf{X}}=\langle X, d\rangle$ and for $Y\subseteq X$, let $d_Y=d\upharpoonright Y\times Y$ and ${\textbf{Y}}=\langle Y, d_Y\rangle$. Then ${\textbf{Y}}$ is the (quasi-) metric subspace of ${\textbf{X}}$ such that Y is the underlying set of ${\textbf{Y}}$. Given a (quasi-) metric space ${\textbf{X}}=\langle X, d\rangle$, if not stated otherwise, we also denote by ${\textbf{X}}$ the topological space $\langle X, \tau (d)\rangle$. For every $n\in {\mathbb {N}}$, ${\mathbb {R}}^n$ denotes also $\langle {\mathbb {R}}^{n}, d_e\rangle$ and $\langle {\mathbb {R}}^n, \tau (d_e)\rangle$ where $d_e$ is the Euclidean metric on ${\mathbb {R}}^n$.

For a topological space ${\textbf{X}}=\langle X, \tau \rangle$ and a set $A\subseteq X$, we denote by ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(A)$ or by ${{\,\mathrm{\textrm{cl}}\,}}_{\tau }(A)$ the closure of A in ${\textbf{X}}$.

For any topological space ${\textbf{X}}=\langle X, \tau \rangle$, let

$$\begin{aligned} {{\,\mathrm{\textrm{Iso}}\,}}_{\tau }(X)=\{x\in X: x\text { is an isolated point of } {\textbf{X}}\}. \end{aligned}$$

If this is not misleading, as in Definition 1, we use ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ to denote ${{\,\mathrm{\textrm{Iso}}\,}}_{\tau }(X)$. By transfinite recursion, we define a decreasing sequence $(X^{({\alpha })})_{\alpha \in ON}$ of closed subsets of ${\textbf{X}}$ as follows:

$$\begin{aligned} X^{(0)}&= X,\\ X^{({\alpha +1})}&=X^{({\alpha })}\setminus {{\,\mathrm{\textrm{Iso}}\,}}(X^{({\alpha })}),\\ X^{({\alpha })}&=\bigcap \limits _{\gamma \in \alpha }X^{({\gamma })}\text { if } \alpha \text { is a limit ordinal}. \end{aligned}$$

For $\alpha \in ON$, the set $X^{({\alpha })}$ is called the $\alpha$-th Cantor-Bendixson derivative of ${\textbf{X}}$. The least ordinal $\alpha$ such that $X^{({\alpha +1})}=X^{({\alpha })}$ is denoted by $|{\textbf{X}}|_{{{\,\mathrm{\textrm{CB}}\,}}}$ and is called the Cantor-Bendixson rank of ${\textbf{X}}$.

Definition 2

A set X is called:

• (i) Dedekind-finite if there is no injection $f:\omega \rightarrow X$; Dedekind-infinite if X is not Dedekind-finite;

• (ii) quasi Dedekind-finite if $[X]^{<\omega }$ is Dedekind-finite; quasi Dedekind-infinite if X is not quasi Dedekind-finite;

• (iii) weakly Dedekind-finite if ${\mathcal {P}}(X)$ 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 $X\setminus Y$ 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 Dedekind-finite sets are called III-finite in Tarski [40] and in Lévy [34], as well as C-finite, 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 ${\textbf{X}}$ is called a cuf space if its underlying set X is a cuf set.

• (ii) A base ${\mathcal {B}}$ of a space ${\textbf{X}}$ is called a cuf base if ${\mathcal {B}}$ is a cuf set.

Definition 4

A space ${\textbf{X}}$ is called:

• (i) first-countable if every point of ${\textbf{X}}$ has a countable base of open neighborhoods;

• (ii) second-countable if ${\textbf{X}}$ has a countable base;

• (iii) compact if every open cover of ${\textbf{X}}$ has a finite subcover;

• (iv) locally compact if every point of ${\textbf{X}}$ has a compact neighborhood;

• (v) limit point compact if every infinite subset of ${\textbf{X}}$ has an accumulation point in ${\textbf{X}}$ (cf. [19, 20]);

• (vi) dense-in-itself if ${{\,\mathrm{\textrm{Iso}}\,}}(X)=\emptyset$;

• (vii) regular if, for every open set V in ${\textbf{X}}$ and every $x\in V$, there exists an open set U in ${\textbf{X}}$ such that $x\in U$ and ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(U)\subseteq V$;

• (viii) a $T_3$-space if it is a regular $T_1$-space;

• (ix) completely regular if, for every closed set A in ${\textbf{X}}$ and every $x\in X\setminus A$, there exists a continuous function $f: X\rightarrow [0, 1]$ such that $f(x)=0$ and $A\subseteq f^{-1}(1)$; completely regular $T_1$-spaces are called $T_{3\frac{1}{2}}$-spaces or Tychonoff spaces.

Definition 5

Let ${\textbf{X}}=\langle X, d\rangle$ be a (quasi-)metric space.

• (i) Given a real number $\varepsilon >0$, a subset D of X is called $\varepsilon$-dense or an $\varepsilon$-net in ${\textbf{X}}$ if, for every $x\in X$, $B_d(x, \varepsilon )\cap D\ne \emptyset$ (equivalently, if $X=\bigcup \nolimits _{x\in D}B_{d^{-1}}(x, \varepsilon )$).

• (ii) ${\textbf{X}}$ is called precompact (respectively, totally bounded) if, for every real number $\varepsilon >0$, there exists a finite $\varepsilon$-net in $\langle X, d^{-1}\rangle$ (respectively, in $\langle X, d^{\star }\rangle$).

• (iii) d is called precompact (respectively, totally bounded) if ${\textbf{X}}$ 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 quasi-metric d on a set X, the collection

$$\begin{aligned} {\mathcal {U}}(d)=\left\{ U\subseteq X\times X: (\exists n\in \omega ) \left\{ \langle x,y\rangle \in X\times X: d(x,y)<\frac{1}{2^n}\right\} \subseteq U\right\} \end{aligned}$$

is a quasi-uniformity on X called the quasi-uniformity induced by d (cf. [33, p. 504]). The quasi-uniformity ${\mathcal {U}}(d)$ 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 $n\in \omega$, there exists a finite set $D\subseteq X$ such that $X=\bigcup \nolimits _{x\in D}\left( B_d(x, \frac{1}{2^n})\cap B_{d^{-1}}(x, \frac{1}{2^n})\right)$. The notions of a totally bounded and precompact metric are equivalent.

We recall that a (Hausdorff) compactification of a space ${\textbf{X}}=\langle X, \tau \rangle$ is an ordered pair $\langle {\textbf{Y}},\gamma \rangle$ where ${\textbf{Y}}$ is a (Hausdorff) compact space and $\gamma : {\textbf{X}}\rightarrow {\textbf{Y}}$ is a homeomorphic embedding such that $\gamma (X)$ is dense in ${\textbf{Y}}$. A compactification $\langle {\textbf{Y}}, \gamma \rangle$ of ${\textbf{X}}$ and the space ${\textbf{Y}}$ are usually denoted by $\gamma {\textbf{X}}$. The underlying set of $\gamma {\textbf{X}}$ is denoted by $\gamma X$. The subspace $\gamma X{\setminus } X$ of $\gamma {\textbf{X}}$ is called the remainder of $\gamma {\textbf{X}}$. A space ${\textbf{K}}$ is said to be a remainder of ${\textbf{X}}$ if there exists a Hausdorff compactification $\gamma {\textbf{X}}$ of ${\textbf{X}}$ such that ${\textbf{K}}$ is homeomorphic to $\gamma X \setminus X$. For compactifications $\alpha {\textbf{X}}$ and $\gamma {\textbf{X}}$ of ${\textbf{X}}$, we write $\gamma {\textbf{X}}\le \alpha {\textbf{X}}$ if there exists a continuous mapping $f:\alpha {\textbf{X}}\rightarrow \gamma {\textbf{X}}$ such that $f\circ \alpha =\gamma$. If $\alpha {\textbf{X}}$ and $\gamma {\textbf{X}}$ are Hausdorff compactifications of ${\textbf{X}}$ such that $\alpha {\textbf{X}}\le \gamma {\textbf{X}}$ and $\gamma {\textbf{X}}\le \alpha {\textbf{X}}$, then we write $\alpha {\textbf{X}}\approx \gamma {\textbf{X}}$ and say that the compactifications $\alpha {\textbf{X}}$ and $\gamma {\textbf{X}}$ are equivalent. If $n\in {\mathbb {N}}$, then a compactification $\gamma {\textbf{X}}$ of ${\textbf{X}}$ is said to be an n-point compactification of ${\textbf{X}}$ if $\gamma X\setminus X$ is an n-element set.

Definition 6

Let ${\textbf{X}}=\langle X, \tau \rangle$ be a non-compact locally compact Hausdorff space and let ${\mathcal {K}}(X)$ be the collection of all compact subsets of ${\textbf{X}}$. For an element $\infty \notin X$, we define $X(\infty )=X\cup \{\infty \}$,

$$\begin{aligned} \tau (\infty )=\tau \cup \{X(\infty )\setminus K: K\in {\mathcal {K}}(X)\} \end{aligned}$$

and ${\textbf{X}}(\infty )=\langle X(\infty ), \tau (\infty )\rangle$. Then ${\textbf{X}}(\infty )$ is called the Alexandroff compactification of ${\textbf{X}}$.

For every non-compact locally compact Hausdorff space ${\textbf{X}}$, ${\textbf{X}}(\infty )$ is the unique (up to $\approx$) one-point Hausdorff compactification of ${\textbf{X}}$. Therefore, every one-point Hausdorff compactification of ${\textbf{X}}$ is called the Alexandroff compactification of ${\textbf{X}}$. Chandler’s book [4] is a good introduction to Hausdorff compactifications in $\textbf{ZFC}$. Basic facts about Hausdorff compactifications in $\textbf{ZF}$ can be found in [29]. We recall that if for a Hausdorff space ${\textbf{X}}$, there exists a Hausdorff compactification $\beta {\textbf{X}}$ of ${\textbf{X}}$ such that, for every Hausdorff compactification $\alpha {\textbf{X}}$ of ${\textbf{X}}$, $\alpha {\textbf{X}}\le \beta {\textbf{X}}$, then $\beta {\textbf{X}}$ is called the Čech-Stone compactification of ${\textbf{X}}$.

Given a collection $\{X_j: j\in J\}$ of sets, for every $i\in J$, we denote by $\pi _i$ the projection $\pi _i:\prod \nolimits _{j\in J}X_j\rightarrow X_i$ defined by $\pi _i(x)=x(i)$ for each $x\in \prod \nolimits _{j\in J}X_j$. If $\tau _j$ is a topology on $X_j$, then ${\textbf{X}}=\prod \nolimits _{j\in J}{\textbf{X}}_j$ denotes the Tychonoff product of the topological spaces ${\textbf{X}}_j=\langle X_j, \tau _j\rangle$ with $j\in J$. If ${\textbf{X}}_j={\textbf{X}}$ for every $j\in J$, then ${\textbf{X}}^{J}=\prod \nolimits _{j\in J}{\textbf{X}}_j$. As in [6], for an infinite set J and the unit interval [0, 1] of ${\mathbb {R}}$, 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 ${\textbf{2}}$ the discrete space with the underlying set $2=\{0, 1\}$. If J is an infinite set, the space ${\textbf{2}}^J$ is called the Cantor cube.

We recall that if $\prod \nolimits _{j\in J}X_j\ne \emptyset$, then it is said that the family $\{X_j: j\in J\}$ has a choice function, and every element of $\prod \nolimits _{j\in J}X_j$ is called a choice function of the family $\{X_j: j\in J\}$. A multiple choice function of $\{X_j: j\in J\}$ is every function $f\in \prod \nolimits _{j\in J}([X_j]^{<\omega }\setminus \{\emptyset \})$. If J is infinite, a function f is called a partial (multiple) choice function of $\{X_j: j\in J\}$ if there exists an infinite subset I of J such that f is a (multiple) choice function of $\{X_j: j\in I\}$. Given a non-indexed family ${\mathcal {A}}$, we treat ${\mathcal {A}}$ as an indexed family ${\mathcal {A}}=\{x: x\in {\mathcal {A}}\}$ to speak about a (partial) choice function and a (partial) multiple choice function of ${\mathcal {A}}$.

Let $\{X_j: j\in J\}$ be a disjoint family of sets, that is, $X_i\cap X_j=\emptyset$ for each pair i, j of distinct elements of J. If $\tau _j$ is a topology on $X_j$ for every $j\in J$, then $\bigoplus \nolimits _{j\in J}{\textbf{X}}_j$ denotes the direct sum of the spaces ${\textbf{X}}_j=\langle X_j, \tau _j\rangle$ with $j\in J$.

Definition 7

(Cf. [2, 24, 35].)

• (i) A space ${\textbf{X}}$ is said to be Loeb (respectively, weakly Loeb) if the family of all non-empty closed subsets of ${\textbf{X}}$ has a choice function (respectively, a multiple choice function).

• (ii) If ${\textbf{X}}$ is a (weakly) Loeb space, then every (multiple) choice function of the family of all non-empty closed subsets of ${\textbf{X}}$ is called a (weak) Loeb function of ${\textbf{X}}$.

Other topological notions used in this article but not defined here are standard. They can be found, for instance, in [6, 42].

1.2 A List of Forms Weaker than $\textbf{AC}$

In this subsection, for the convenience of readers, we define and denote the weaker forms of $\textbf{AC}$ 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].

Definition 8

1. $\textbf{IQDI}$: Every infinite set is quasi Dedekind-infinite. (Cf., e.g., [28].)

• 2. $\textbf{IWDI}$ ( [13, Form 82]): Every infinite set is weakly Dedekind-infinite.

• 3. $\textbf{IDI}$ ( [13, Form 9]): Every infinite set is Dedekind-infinite.

• 4. $\textbf{CAC}$ ( [13, Form 8]): Every denumerable family of non-empty sets has a choice function.

• 5. $\textbf{CAC}_{fin}$ ( [13, Form 10]): Every denumerable family of non-empty finite sets has a choice function.

• 6. $\textbf{WOAC}_{fin}$ ( [13, Form 122]): Every well-orderable non-empty family of non-empty finite sets has a choice function.

• 7. $\textbf{MC}$ (the Axiom of Multiple Choice, [13, Form 67]): Every non-empty family of non-empty sets has a multiple choice function.

• 8. $\textbf{CMC}$ (the Countable Axiom of Multiple Choice, [13, Form 126]): Every denumerable family of non-empty sets has a multiple choice function.

• 9. $\textbf{WoAm}$ ( [13, Form 133]): Every infinite set is either well-orderable or has an amorphous subset.

• 10. $\textbf{DC}$ (the Principle of Dependent Choice, [13, Form 43]): For every non-empty set A and every binary relation S on A such that $(\forall x\in A)(\exists y\in A)(\langle x, y\rangle \in S)$, there exists $a\in A^{\omega }$ such that:

  $$\begin{aligned} (\forall n\in \omega )(\langle a(n), a(n+1)\rangle \in S). \end{aligned}$$

• 11. $\textbf{BPI}$ (the Boolean Prime Ideal Principle, [13, Form 14]): Every Boolean algebra has a prime ideal.

• 12. $\textbf{NAS}$ ( [13, Form 64]): There are no amorphous sets.

• 13. ${\textbf{M}}(C, S)$: Every compact metrizable space is separable. (Cf. [18, 19, 22, 23, 27].)

• 14. $\textbf{IDFBI}$: For every infinite set D, the Cantor cube ${\textbf{2}}^{\omega }$ is a remainder of the discrete space $\langle D, {\mathcal {P}}(D)\rangle$. (Cf. [28].)

• 15. $\textbf{INSHC}$: 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 $\textbf{IQDI}$ 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 $\textbf{CAC}_{fin}$ is Kőnig’s Lemma (see [13, Form 10 F, p. 20]).

The form $\textbf{IDFBI}$ has been introduced and investigated in [28] recently. More comments about $\textbf{IDFBI}$ are included in Remark 3. New facts concerning $\textbf{IDFBI}$—among them, a solution of an open problem posed in [28]—are included in Sect. 2. The form $\textbf{INSHC}$ is new here. That $\textbf{INSHC}$ is essentially weaker than $\textbf{IDFBI}$ is shown in Sect. 2.

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

(Cf. [21]). $(\textbf{ZF})$ A topological space ${\textbf{X}}$ is scattered if and only if there exists $\alpha \in ON$ such that $X^{({\alpha })}=\emptyset$. If ${\textbf{X}}$ is scattered then

$$\begin{aligned} |{\textbf{X}}|_{{{\,\mathrm{\textrm{CB}}\,}}}=\min \{\alpha \in ON: X^{({\alpha })}=\emptyset \}. \end{aligned}$$

Moreover, if ${\textbf{X}}$ is a non-empty scattered compact space, then $|{\textbf{X}}|_{{{\,\mathrm{\textrm{CB}}\,}}}$ is a successor ordinal.

Theorem 1

(Cf., e.g., [3]). $(\textbf{ZF})$ Every non-empty dense-in-itself compact second-countable Hausdorff space is of size at least $|{\mathbb {R}}|$.

Proposition 2

(Cf. [33, Proposition 2.1.11] and [32]). $(\textbf{ZF})$ If d is a quasi-metric on X such that $\langle X, \tau (d)\rangle$ is compact, then d is strong.

Theorem 2

(Cf. [9]). $(\textbf{ZF})$

• (i) (Cf. [9].) (Urysohn’s Metrization Theorem) Every second-countable $T_3$-space is metrizable.

• (ii) (Cf. [28].) Every $T_3$-space which has a cuf base can be embedded in a metrizable Tychonoff cube and, therefore, it is metrizable.

• (iii) (Cf. [28].) A $T_3$-space ${\textbf{X}}$ is embeddable in a compact metrizable Tychonoff cube if and only if ${\textbf{X}}$ is embeddable in the Hilbert cube $\mathbf [0, 1]^{\omega }$.

Several essential applications of Theorem 2(ii), especially to the theory of Hausdorff compactifications in $\textbf{ZF}$, 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]). $(\textbf{ZF})$

• (a) For every compact Hausdorff, quasi metric space ${\textbf{X}}=\langle X,d\rangle$ the following are equivalent:

  • (i) ${\textbf{X}}$ is Loeb;

  • (ii) $\langle X, d^{-1}\rangle$ is separable;

  • (iii) ${\textbf{X}}$ and $\langle X,d^{-1}\rangle$ are both separable;

  • (iv) ${\textbf{X}}$ is second-countable.

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

• (b) $\textbf{CAC}$ implies that every compact, Hausdorff quasi-metrizable space is also metrizable.

Proposition 3

(Cf. [28]). $(\textbf{ZF})$ Every weakly Loeb regular space which admits a cuf base has a dense cuf set.

Theorem 4

(Cf. [35]). $(\textbf{ZF})$ Let $\kappa$ be an infinite well-ordered cardinal, $\{{\textbf{X}}_{i}:i\in \kappa \}$ be a family of compact spaces, $\{f_{i}:i\in \kappa \}$ be a collection of functions such that for every $i\in \kappa ,f_{i}$ is a Loeb function of ${\textbf{X}}_{i}$. Then the Tychonoff product ${\textbf{X}}=\prod \nolimits _{i\in \kappa }{\textbf{X}}_{i}$ is compact.

Theorem 5

(Cf., e.g., [11, Theorem 4.37] and [13, Forms: 14, 14A, 14J]) $(\textbf{ZF})$ The following statements are equivalent to $\textbf{BPI}$:

• (i) For every non-empty set X, every filter in ${\mathcal {P}}(X)$ can be enlarged to an ultrafilter in ${\mathcal {P}}(X)$.

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

Theorem 6

(Cf. [14]) $(\textbf{ZF})$ $\textbf{BPI}$ implies $\textbf{NAS}$ but this implication is not reversible.

Remark 2

Theorem 6 deserves more comments. Namely, let us denote by $\textbf{OP}$ Form 30 of [13] (Ordering Principle) stating that every set can be linearly ordered. Regarding Theorem 6, one can easily check in [14] that it is known that the implications $\textbf{BPI}\rightarrow \textbf{OP}\rightarrow \textbf{NAS}$ hold in $\textbf{ZF}$ but the implications are not reversible in $\textbf{ZF}$. Indeed, that $\textbf{BPI}$ implies $\textbf{OP}$ in $\textbf{ZF}$ is established in [15, Example 2.3.2, p. 19] (also see [15, Historical remarks, top of p. 30]). Let $\textbf{AC}_{fin}$ denote Form 62 of [13] stating that every non-empty family of non-empty finite sets has a choice function. It is obvious that $\textbf{OP}$ implies $\textbf{AC}_{fin}$ in $\textbf{ZF}$. It was shown in [41, p. 196] that $\textbf{AC}_{fin}$ implies $\textbf{NAS}$ in $\textbf{ZF}$. In consequence, $\textbf{OP}$ implies $\textbf{NAS}$ in $\textbf{ZF}$. In Feferman’s model ${\mathcal {M}}2$ in [13], $\textbf{NAS}$ is true and $\textbf{OP}$ is false (see [13, p. 148]). In Mathias’ model ${\mathcal {M}}3$ in [13], $\textbf{OP}$ is true and $\textbf{BPI}$ is false (see [13, p. 150]).

Other remarks about Theorem 6 can be found, for instance, in [28].

We recall that a family ${\mathcal {A}}$ 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 ${\mathcal {A}}$, $A\cup B\in {\mathcal {A}}$ (respectively, $A\cap B\in {\mathcal {A}}$).

Theorem 7

(Cf. [28]). $(\textbf{ZF})$ For every locally compact Hausdorff space ${\textbf{X}}$, the following conditions are all equivalent:

• (a) every non-empty second-countable compact Hausdorff space is a remainder of ${\textbf{X}}$;

• (b) the Cantor cube ${\textbf{2}}^{\omega }$ is a remainder of ${\textbf{X}}$;

• (c) there exists a family ${\mathcal {V}}=\{{\mathcal {V}}^n_i: n\in {\mathbb {N}}, i\in \{1,\ldots , 2^n\}\}$ such that, for every $n\in {\mathbb {N}}$, the following conditions are satisfied:

  • (i) for every $i\in \{1,\ldots , 2^n\}$, ${\mathcal {V}}_i^n$ is a non-empty family of open sets of ${\textbf{X}}$ such that ${\mathcal {V}}_i^n$ is stable under finite unions and finite intersections, and, for every $U\in V_i^n$, the set ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}U$ is non-compact;

  • (ii) for every $i\in \{1,\ldots , 2^n\}$ and for any $U, V\in {\mathcal {V}}_i^n$, ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(U){\setminus } V$ is compact;

  • (iii) for every pair i, j of distinct elements of $\{1,\ldots , 2^n\}$, for any $W\in {\mathcal {V}}_{i}^n$ and $G\in {\mathcal {V}}_j^n$, there exist $U\in {\mathcal {V}}_{2i-1}^{n+1}, V\in {\mathcal {V}}_{2i}^{n+1}$ with ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(U\cup V){\setminus } W$ compact and ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(( U\cup V)\cap G)$ compact;

  • (iv) if, for every $i\in \{1,\ldots , 2^n\}$, $V_i\in {\mathcal {V}}_i^n$, then $X\setminus \bigcup \nolimits _{i=1}^{2^n}V_i$ is compact.

Theorem 8

(Cf. [28]). $(\textbf{ZF})$ For a set D, let ${\textbf{D}}=\langle D, {\mathcal {P}}(D)\rangle$. Then the following statements hold:

• (i) If ${\textbf{D}}$ is an infinite cuf space, then every non-empty second-countable compact Hausdorff space is a remainder of a metrizable compactification of ${\textbf{D}}$. In particular, all non-empty second-countable compact Hausdorff spaces are remainders of metrizable compactifications of ${\mathbb {N}}$.

• (ii) If D is weakly Dedekind-infinite, then every non-empty second-countable compact Hausdorff space is a remainder of ${\textbf{D}}$.

Theorem 9

(Cf. [28]). $(\textbf{ZF})$

• (i) $\textbf{IDFBI}$ implies $\textbf{NAS}$ but this implication cannot be reversed.

• (ii) The statement “All non-empty metrizable compact spaces are remainders of metrizable compactifications of ${\mathbb {N}}$” is equivalent to ${\textbf{M}}(C, S)$ and, thus, it implies $\textbf{CAC}_{fin}$.

Remark 3

In $\textbf{ZFC}$, an archetype of Theorem 7 is included in [10, Theorem 2.1]; however, in [28], Theorem 2.1 of [10] has been shown to be unprovable in $\textbf{ZF}$. In [28], an infinite set D is called dyadically filterbase infinite if ${\textbf{2}}^{\omega }$ is a remainder of the discrete space $\langle D, {\mathcal {P}}(D)\rangle$. 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, $\textbf{IDFBI}$ is equivalent to the sentence “Every infinite set is dyadically filterbase infinite”.

Theorem 10

(Cf. [29]). $(\textbf{ZF})$

• (i) For every non-empty compact Hausdorff space ${\textbf{K}}$, there exists a Dedekind-infinite discrete space ${\textbf{D}}$ such that ${\textbf{K}}$ is a remainder of ${\textbf{D}}$.

• (ii) If D is an infinite set, then the Alexandroff compactification of the discrete space ${\textbf{D}}=\langle D, {\mathcal {P}}(D)\rangle$ is the unique (up to the equivalence) Hausdorff compactification of ${\textbf{D}}$ if and only if D is amorphous.

Proposition 4

$(\textbf{ZF})$ Let D be an infinite set and let ${\textbf{D}}=\langle D, {\mathcal {P}}(D)\rangle$. Then:

• (i) ${\textbf{D}}(\infty )$ is metrizable if and only if D is a cuf set (cf. [32]);

• (ii) the discrete space ${\textbf{D}}$ has a metrizable compactification if and only if D is a cuf set (cf. [28]).

As we have already mentioned at the beginning of Sect. 1.1, in the sequel, we apply not only $\textbf{ZF}$-models but also permutation models of $\textbf{ZFA}$. To transfer a statement $\mathbf {\Phi }$ from a permutation model to a $\textbf{ZF}$-model, we use the Jech-Sochor First Embedding Theorem (see, e.g., [15, Theorem 6.1]) if $\mathbf {\Phi }$ is a boundable statement. When $\mathbf {\Phi }$ has a permutation model but $\mathbf {\Phi }$ is a conjunction of statements each one of which is equivalent to $\textbf{BPI}$ or to an injectively boundable statement, we use Pincus’ transfer results (see [36, 37] and [13, Note 3, page 286]) to show that $\mathbf {\Phi }$ has a $\textbf{ZF}$-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.

1.4 The Content Concerning New Results in Brief

In Sect. 2, we notice that, in $\textbf{ZF}$, 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 $\textbf{ZF}$ (see Proposition 6). We show that the new form $\textbf{INSHC}$ holds in every model of $\mathbf {ZF+BPI}$, is independent of $\textbf{ZF}$, does not imply $\textbf{BPI}$ and is strictly weaker than $\textbf{IDFBI}$ (see Theorem 11). We construct a new permutation model to prove that the existence of a dyadically filterbase infinite, weakly Dedekind-finite set is relatively consistent with $\textbf{ZFA}$, and we then observe that the result is transferable to $\textbf{ZF}$ (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 $\textbf{ZF}$ that if $\langle X, d\rangle$ is a quasi-metric $T_3$-space such that d is strong and either $\langle X, \tau (d)\rangle$ is limit point compact or $d^{-1}$ is precompact, then the space $\langle X, \tau (d)\rangle$ is metrizable (see Theorem 14). This result and its direct consequence that if $\langle X, d\rangle$ is a compact Hausdorff quasi-metric space such that $\langle X, \tau (d^{-1})\rangle$ is iso-dense, then $\langle X, \tau (d)\rangle$ is metrizable (see Corollary 2) are new applications of Theorem 2(ii) and adjuncts to Theorem 3. By applying Theorem 2, we show in $\textbf{ZF}$ that if $\langle X, d\rangle$ is an iso-dense metric space such that either d is totally bounded or $\langle X, \tau (d)\rangle$ is limit point compact, then $\langle X, \tau (d)\rangle$ has a cuf base and can be embedded in a metrizable Tychonoff cube (see Theorem 15).

Section 4 concerns equivalents of $\textbf{CAC}_{fin}$ in terms of scattered or iso-dense spaces (see Theorems 16 and 18). Among our new equivalents of $\textbf{CAC}_{fin}$ there are, for instance, the following statements: (a) for every iso-dense metric space ${\textbf{X}}$, if ${\textbf{X}}$ is either limit point compact or totally bounded, then ${\textbf{X}}$ 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 $\textbf{ZF}$, every compact metrizable cuf space is scattered (see Theorem 19). We prove that $\textbf{WOAC}_{fin}$ is equivalent to the statement: for every well-orderable non-empty set S and every family $\{\langle X_s, d_s\rangle : s\in S\}$ of compact scattered metric spaces, the product $\prod \nolimits _{s\in S}\langle X_s, \tau (d_s)\rangle$ 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 $\textbf{IDI}$, $\textbf{WoAm}$ and $\textbf{BPI}$ implies that every infinite compact first-countable Hausdorff space contains a copy of ${\mathbb {N}}(\infty )$; (b) every infinite first-countable compact Hausdorff separable space contains a copy of ${\mathbb {N}}(\infty )$; (c) every infinite first-countable compact Hausdorff space having an infinite cuf subset contains a copy of ${\textbf{D}}(\infty )$ 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 $\textbf{CAC}_{fin}$ nor $\textbf{IQDI}$, nor $\textbf{CMC}$ in $\textbf{ZFA}$ (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 $\textbf{ZF}$. 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.

Section 6 contains a shortlist of new open problems strictly relevant to the topic of this paper.

2 From Compact Hausdorff Iso-Dense Spaces that are not Scattered to $\textbf{INSHC}$

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

Proposition 5

$(\textbf{ZF})$ 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 7 has an iso-dense non-scattered Hausdorff compactification. In particular, for every dyadically filterbase infinite set D, the discrete space $\langle D, {\mathcal {P}}(D)\rangle$ has a non-scattered Hausdorff compactification. It follows from Theorem 8(i) that every denumerable discrete space has non-scattered metrizable Hausdorff compactifications. Thus, in $\textbf{ZF}$, 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.

Let us recall that a topological space ${\textbf{X}}$ is zero-dimensional if it has a base consisting of clopen sets (that is, of closed and open sets). Every zero-dimensional space is completely regular.

Remark 4

It was shown in [21, Theorem 7] that it holds in $\textbf{ZF}$ that every compact Hausdorff scattered space is zero-dimensional, so completely regular. In consequence, in every model of $\textbf{ZF}$, the following statement is true: Every compact, Hausdorff non-completely regular space is not scattered.

Let us prove that a compact Hausdorff iso-dense space may fail to be completely regular in $\textbf{ZF}$.

Proposition 6

There exists a model ${\mathcal {M}}$ of $\textbf{ZF}$ in which there is a compact Hausdorff iso-dense space which is not completely regular, and thus not scattered.

Proof

Let ${\mathcal {M}}$ be any model of $\textbf{ZF}$ in which there exists a compact Hausdorff, not completely regular space ${\textbf{Z}}$ (see, for example, [9]) and let us work inside ${\mathcal {M}}$. By Theorem 10(i), it holds in ${\mathcal {M}}$ that there exists a Hausdorff compactification $\gamma {\textbf{D}}$ of a discrete space ${\textbf{D}}$ such that $\gamma D\setminus D$ is homeomorphic to ${\textbf{Z}}$. Then, in ${\mathcal {M}}$, the compact Hausdorff space $\gamma {\textbf{D}}$ is iso-dense but not completely regular. By Remark 4, $\gamma {\textbf{D}}$ is not scattered in ${\mathcal {M}}$ either. $\square$

Let us shed more light on the forms $\textbf{INSHC}$ and $\textbf{IDFBI}$.

Theorem 11

$(\textbf{ZF})$

• (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) $\textbf{DC}\rightarrow \textbf{IWDI}\rightarrow \textbf{IDFBI}\rightarrow \textbf{INSHC}\rightarrow \textbf{NAS}$;

• (iv) $\textbf{BPI}\rightarrow \textbf{INSHC}$;

• (v) $\textbf{INSHC}\nrightarrow \textbf{IDFBI}$ and $\textbf{INSHC}\nrightarrow \textbf{BPI}$.

Proof

To show that (i) holds, it suffices to apply Theorem 10. It follows directly from Theorem 8(i) that (ii) holds.

(iii) It is known that $\textbf{DC}$ implies $\textbf{IWDI}$ (see, for example, [13, pages 326 and 339]). It has been noticed in [28] that, by Theorem 8(ii), $\textbf{IWDI}$ implies $\textbf{IDFBI}$. The implications $\textbf{IDFBI}\rightarrow \textbf{INSHC}\rightarrow \textbf{NAS}$ can be deduced from Theorems 7 and 10(ii).

To prove (iv), we assume $\textbf{BPI}$ and consider any infinite set D. Let ${\textbf{D}}=\langle D, {\mathcal {P}}(D)\rangle$. In the light of Theorem 5(i) and [29, Theorem 3.27], it follows from $\textbf{BPI}$ that there exists the Čech-Stone compactification $\beta {\textbf{D}}$ of ${\textbf{D}}$. Suppose that $\beta D\setminus D$ has an isolated point $y_0$. Then there exist disjoint open subsets U, V of $\beta {\textbf{D}}$ such that $y_0\in U$, $(\beta D{\setminus } D){\setminus }\{y_0\}\subseteq V$ and $U\cap V=\emptyset$. Then the subspace $U\cup \{y_0\}$ of $\beta {\textbf{D}}$ is the Čech-Stone compactification of the subspace $U\cap D$ of ${\textbf{D}}$. It follows from Theorem 10(ii) that $U\cap D$ is amorphous but this is impossible because $\textbf{BPI}$ implies $\textbf{NAS}$ by Theorem 6. The contradiction obtained shows that $\beta D\setminus D$ is dense-in-itself, so $\beta {\textbf{D}}$ is not scattered.

(v) It was shown in [28] that the conjunction $\textbf{BPI}\wedge \lnot \textbf{IDFBI}$ has a $\textbf{ZF}$-model. This, together, with (iv), implies that there is a model of $\textbf{ZF}$ in which the conjunction $\textbf{INSHC}\wedge \lnot \textbf{IDFBI}$ is true. Hence $\textbf{INSHC}\nrightarrow \textbf{IDFBI}$.

To prove $\textbf{INSHC}\nrightarrow \textbf{BPI}$, let us use the Feferman’s forcing model ${\mathcal {M}}2$ in [13]. It is known that $\textbf{DC}\wedge \lnot \textbf{BPI}$ is true in ${\mathcal {M}}2$ (see [13, page 148]). To complete the proof, it suffices to notice that it follows from (iii) that $\textbf{INSHC}$ is also true in ${\mathcal {M}}2$. $\square$

Remark 5

(a) We do not know if the conjunction $\textbf{NAS}\wedge \lnot \textbf{INSHC}$ has a $\textbf{ZF}$-model.

• (b) It is worth noticing that it follows from Theorem 10(ii) that it holds in $\textbf{ZF}$ that $\textbf{NAS}$ is equivalent to the following statements:

  • (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 11(iii), it holds in $\textbf{ZF}$ that $\textbf{INSHC}$ follows from every form of [13] which implies $\textbf{IWDI}$. In particular, the implication $\textbf{CMC}\rightarrow \textbf{INSHC}$ holds in $\textbf{ZF}$ (see [13, page 339]).

• (d) It is known that $\textbf{BPI}$ implies $\textbf{CAC}_{fin}$ (see, for example, [13, pages 325 and 354]). Since $\textbf{BPI}$ implies $\textbf{INSHC}$ by Theorem 11(ii), it is worth noticing that the conjunction $\textbf{INSHC}\wedge \lnot \textbf{CAC}_{fin}$ has a $\textbf{ZF}$-model. To see this, let us notice that, in the Second Fraenkel Model ${\mathcal {N}}2$ in [13], the conjunction $\textbf{IWDI}\wedge \lnot \textbf{CAC}_{fin}$ is true (see [13, page 179]). Since $\textbf{IWDI}\wedge \lnot \textbf{CAC}_{fin}$ is a conjunction of two injectively boundable statements and has a permutation model, it also has a $\textbf{ZF}$-model by Pincus’ transfer theorems. This, together with Theorem 11(iii), implies that $\textbf{INSHC}\wedge \lnot \textbf{CAC}_{fin}$ has a $\textbf{ZF}$-model. This is also an alternative proof that $\textbf{INSHC}\wedge \lnot \textbf{BPI}$ has a $\textbf{ZF}$-model.

It has been shown in the proof of Theorem 11 that $\textbf{INSHC}\wedge \lnot \textbf{IDFBI}$ has a $\textbf{ZF}$-model. Thus, by Theorem 11(iii), $\textbf{INSHC}\wedge \lnot \textbf{IWDI}$ has a $\textbf{ZF}$-model. Let us recall the following open problems posed by us in [28]:

Problem 1

(i) Is there a $\textbf{ZF}$-model for $\textbf{IDFBI}\wedge \lnot \textbf{IWDI}$? (See [28, Problem (3) of Section 6].)

• (ii) Is there a model of $\textbf{ZF}$ in which a weakly Dedekind-finite set can be dyadically filterbase infinite? (See [28, Problem (6) of Section 6].)

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 ${\mathbb {N}}(\infty )$. 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 $\textbf{ZF}$ that there exists a dyadically filterbase infinite set which is weakly Dedekind-finite.

Proof

Let $\mathbf {\Phi }$ be the following statement: “There exists a dyadically filterbase infinite set which is weakly Dedekind-finite”.

Since $\mathbf {\Phi }$ is a boundable statement, by the Jech–Sochor First Embedding Theorem (see [15, Theorem 6.1]), it suffices to prove that $\mathbf {\Phi }$ 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 ${\mathcal {N}}$ in which $\mathbf {\Phi }$ is true.

In what follows, for an arbitrary non-empty set S and every permutation $\psi$ of S, we denote by ${{\,\mathrm{\textrm{supp}}\,}}(\psi )$ the support of $\psi$, that is, ${{\,\mathrm{\textrm{supp}}\,}}(\psi )=\{x\in S: \psi (x)\ne x\}$.

We start with a model M of $\textbf{ZFA}+\textbf{AC}$ with a denumerable set A of atoms such that A has a denumerable partition

$$\begin{aligned} {\mathcal {A}}=\{A_{i}:i\in \omega \} \end{aligned}$$

into infinite sets. In M, we let

$$\begin{aligned} {\mathcal {B}}=\{{\mathcal {B}}_{i}^{n}:n\in {\mathbb {N}},i\in \{1,2,\ldots , 2^{n}\}\} \end{aligned}$$

be a family with the following two properties:

• (a) For $n=1$, $\{{\mathcal {B}}_{1}^{1},{\mathcal {B}}_{2}^{1}\}$ is a partition of ${\mathcal {A}}$ into two infinite sets.

• (b) For every $n\in {\mathbb {N}}$ and for every $i\in \{1,2,\ldots ,2^{n}\}$, $\{{\mathcal {B}}_{2i-1}^{n+1},{\mathcal {B}}_{2i}^{n+1}\}$ is a partition of ${\mathcal {B}}_{i}^{n}$ into two infinite sets.

We may thus view ${\mathcal {B}}$ as an infinite binary tree, having ${\mathcal {A}}$ as its root.

Let ${\mathcal {G}}$ be the group of all permutations $\phi$ of A which satisfy the following two properties:

• (c) $\phi$ moves only finitely many elements of A.

• (d) $(\forall i\in \omega )(\exists j\in \omega )(\exists F\in [A_{j}]^{<\omega })(\phi [{{\,\mathrm{\textrm{supp}}\,}}(\phi \upharpoonright A_{i})]=F)$.

For every $n\in {\mathbb {N}}$ and for every $i\in \{1,2,\ldots ,2^{n}\}$, we let

$$\begin{aligned} {\mathcal {Q}}_{i}^{n}=\left\{ \bigcup \{\phi (Z):Z\in {\mathcal {B}}_{i}^{n}\}:\phi \in {\mathcal {G}}\right\} . \end{aligned}$$

We also let

$$\begin{aligned} {\mathcal {Q}}=\bigcup \{{\mathcal {Q}}_{i}^{n}:n\in {\mathbb {N}},i\in \{1,2,\ldots ,2^{n}\}\}. \end{aligned}$$

For every $E\in [{\mathcal {Q}}]^{<\omega }$, we let

$$\begin{aligned} {\mathcal {G}}_{E}=\{\phi \in {\mathcal {G}}:(\forall Q\in E)(\phi (Q)=Q)\}. \end{aligned}$$

Then ${\mathcal {G}}_E$ is a subgroup of ${\mathcal {G}}$. Furthermore, since for all $E,E^{\prime }\in [{\mathcal {Q}}]^{<\omega }$, ${\mathcal {G}}_{E\cup E^{\prime }}\subseteq {\mathcal {G}}_E\cap {\mathcal {G}}_{E^{\prime }}$, the collection $\{{\mathcal {G}}_E: E\in [{\mathcal {Q}}]^{<\omega }\}$ is a base for a filter in the set of all subgroups of ${\mathcal {G}}$. Let ${\mathcal {F}}$ be the filter of subgroups of ${\mathcal {G}}$ generated by $\{{\mathcal {G}}_E: E\in [{\mathcal {Q}}]^{<\omega }\}$. To check that ${\mathcal {F}}$ is a normal filter on ${\mathcal {G}}$, we need to show that ${\mathcal {F}}$ has the following two properties:

$$\begin{aligned} (\forall a\in A)(\{\pi \in {\mathcal {G}}:\pi (a)=a\}\in {\mathcal {F}}) \end{aligned}$$

(1)

and

$$\begin{aligned} (\forall \pi \in {\mathcal {G}})(\forall H\in {\mathcal {F}})(\pi H\pi ^{-1}\in {\mathcal {F}}). \end{aligned}$$

(2)

To argue for (1), let $a\in A$. Since ${\mathcal {A}}$ is a partition of A in M, there exists a unique $i\in \omega$ such that $a\in A_{i}$. Since the set $\{{\mathcal {B}}_{1}^{1},{\mathcal {B}}_{2}^{1}\}$ is a partition of ${\mathcal {A}}$, either $A_{i}\in {\mathcal {B}}_{1}^{1}$ or $A_{i}\in {\mathcal {B}}_{2}^{1}$. Suppose that $A_{i}\in {\mathcal {B}}_{1}^{1}$ (the argument is similar if $A_{i}\in {\mathcal {B}}_{2}^{1}$). Pick an $A_{j}\in {\mathcal {B}}_{2}^{1}$ and an $a'\in A_{j}$. Let $\phi \in {\mathcal {G}}$ be the transposition $(a,a')$ (i.e. $\phi$ interchanges a and $a'$ and fixes all other atoms). Then

$$\begin{aligned} \bigcup \{\phi (Z):Z\in {\mathcal {B}}_{2}^{1}\}=\left( \bigcup ({\mathcal {B}}_{2}^{1}\setminus \{A_{j}\})\right) \cup ((A_j\setminus \{a^{\prime }\})\cup \{a\}). \end{aligned}$$

Let $E=\{\bigcup {\mathcal {B}}_{1}^{1},\bigcup \{\phi (Z):Z\in {\mathcal {B}}_{2}^{1}\}\}$. Then, $E\in [{\mathcal {Q}}_{1}^{1}\cup {\mathcal {Q}}_{2}^{1}]^{<\omega }\subset [{\mathcal {Q}}]^{<\omega }$, so $E\in [{\mathcal {Q}}]^{<\omega }$. Furthermore, ${\mathcal {G}}_{E}\subseteq \{\pi \in {\mathcal {G}}:\pi (a)=a\}$. Indeed, let $\pi \in {\mathcal {G}}_E$. Towards a contradiction, assume that $\pi (a)=b$ for some $b\in A\setminus \{a\}$. Since $a\in \bigcup {\mathcal {B}}_{1}^{1}$ and $\pi$ fixes $\bigcup {\mathcal {B}}_{1}^{1}$, it follows that $b=\pi (a)\in \pi (\bigcup {\mathcal {B}}_{1}^{1})=\bigcup {\mathcal {B}}_{1}^{1}$. But then, since $\pi \in {\mathcal {G}}_{E}$, we have the following:

$$\begin{aligned}{} & {} a\in \bigcup \{\phi (Z):Z\in {\mathcal {B}}_{2}^{1}\}\}\rightarrow \pi (a)\in \pi \left( \bigcup \{\phi (Z):Z\in {\mathcal {B}}_{2}^{1}\}\}\right) \\{} & {} \rightarrow b\in \bigcup \{\phi (Z):Z\in {\mathcal {B}}_{2}^{1}\}\}=\left( \bigcup ({\mathcal {B}}_{2}^{1}\setminus \{A_{j}\})\right) \cup ((A_j\setminus \{a^{\prime }\})\cup \{a\}), \end{aligned}$$

which is a contradiction. Therefore, (1) holds.

To argue for (2), let $\pi \in {\mathcal {G}}$ and $H\in {\mathcal {F}}$. There exists $E\in [{\mathcal {Q}}]^{<\omega }$ such that ${\mathcal {G}}_{E}\subseteq H$. By the definition of ${\mathcal {Q}}$, we have $\pi [E]\in [{\mathcal {Q}}]^{<\omega }$. We assert that ${\mathcal {G}}_{\pi [E]}\subseteq \pi H\pi ^{-1}$. Let $\rho \in {\mathcal {G}}_{\pi [E]}$. For every $T\in E$, we have the following:

$$\begin{aligned} \rho (\pi T)=\pi T\rightarrow \pi ^{-1}\rho \pi (T)=T. \end{aligned}$$

Hence, since ${\mathcal {G}}_{E}\subseteq H$, we have:

$$\begin{aligned} \pi ^{-1}\rho \pi \in {\mathcal {G}}_{E}\rightarrow \rho \in \pi {\mathcal {G}}_{E}\pi ^{-1}\subseteq \pi H\pi ^{-1}. \end{aligned}$$

Therefore, $\rho \in \pi H\pi ^{-1}$. Since $\rho$ is an arbitrary element of ${\mathcal {G}}_{\pi [E]}$, we conclude that ${\mathcal {G}}_{\pi [E]}\subseteq \pi H\pi ^{-1}$. Thus, $\pi H\pi ^{-1}\in {\mathcal {F}}$, so (2) holds. This completes the proof that ${\mathcal {F}}$ is a normal filter on ${\mathcal {G}}$.

Let ${\mathcal {N}}$ be the permutation model determined by M, ${\mathcal {G}}$ and ${\mathcal {F}}$. We say that an element $x\in {\mathcal {N}}$ has support $E\in [{\mathcal {Q}}]^{<\omega }$ if, for all $\phi \in {\mathcal {G}}_{E}$, $\phi (x)=x$.

In ${\mathcal {N}}$, the set $({\mathcal {P}}(A))^{{\mathcal {N}}}=({\mathcal {P}}(A))^M\cap {\mathcal {N}}$ is the power set of A. To prove that A is dyadically filterbase infinite in ${\mathcal {N}}$, let us show that, in ${\mathcal {N}}$, the discrete space $\langle A, ({\mathcal {P}}(A))^{{\mathcal {N}}}\rangle$ satisfies condition (c) of Theorem 7. To this aim, for every $n\in {\mathbb {N}}$ and for every $i\in \{1,2,\ldots ,2^{n}\}$, we let

$$\begin{aligned} {\mathcal {V}}_{i}^{n}=\left\{ \bigcap {\mathcal {R}}:{\mathcal {R}}\in [{\mathcal {Q}}_{i}^{n}]^{<\omega }\setminus \{\emptyset \}\right\} \cup \left\{ \bigcup {\mathcal {C}}: {\mathcal {C}}\in [{\mathcal {Q}}_{i}^{n}]^{<\omega }\setminus \{\emptyset \}\right\} , \end{aligned}$$

and we also let

$$\begin{aligned} {\mathcal {V}}=\{{\mathcal {V}}_{i}^{n}:n\in {\mathbb {N}},i\in \{1,2,\ldots ,2^{n}\}\}. \end{aligned}$$

We notice that any permutation of A in ${\mathcal {G}}$ fixes ${\mathcal {V}}$ pointwise. Hence, ${\mathcal {V}}\in {\mathcal {N}}$ and, moreover, ${\mathcal {V}}$ is well-orderable in the model ${\mathcal {N}}$ (see [15, page 47]). Since ${\mathcal {V}}$ is denumerable in M and well-orderable in ${\mathcal {N}}$, it follows that ${\mathcal {V}}$ is also denumerable in ${\mathcal {N}}$. Furthermore, in view of the properties of the family ${\mathcal {B}}$ and of the elements of ${\mathcal {G}}$, and the construction of ${\mathcal {V}}$, it is easy to see that, if we put $X=A$ and ${\textbf{X}}=\langle A, ({\mathcal {P}}(A))^{{\mathcal {N}}}\rangle$, then ${\mathcal {V}}$ 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 ${\mathcal {N}}$.

To complete the proof, it remains to show that A is weakly Dedekind-finite in ${\mathcal {N}}$. By way of contradiction, we assume that A is weakly Dedekind-infinite in ${\mathcal {N}}$. Thus, it holds in ${\mathcal {N}}$ that there exists a denumerable disjoint family ${\mathcal {U}}=\{U_{n}:n\in \omega \}$ of $({\mathcal {P}}(A))^{{\mathcal {N}}}$. Let $E\in [{\mathcal {Q}}]^{<\omega }$ be a support of $U_{n}$ for all $n\in \omega$. By the definitions of ${\mathcal {G}}$, ${\mathcal {Q}}$ and the supports, as well as the fact that ${\mathcal {U}}$ is infinite and disjoint, it follows that there exist distinct $k,m\in \omega$ and atoms $x\in U_{k}$, $y\in U_{m}$ such that the transposition $\psi =(x,y)$ of A is an element of ${\mathcal {G}}_{E}$. Since E is a support of $U_{k}$ and $\psi \in {\mathcal {G}}_{E}$, $\psi (U_{k})=U_{k}$, and so $y=\psi (x)\in \psi (U_{k})=U_{k}$. This is impossible because $y\in U_{m}$ and $U_{k}\cap U_{m}=\emptyset$. The contradiction obtained shows that A is weakly Dedekind-finite in ${\mathcal {N}}$. $\square$

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], $\textbf{NAS}$ is false in the model from [28, the proof of Theorem 5.14], whereas it is unknown to us whether or not $\textbf{NAS}$ is true in the model ${\mathcal {N}}$ of the proof of Theorem 12. With regards to the latter model, let us notice that, for every $i\in \omega$, no $E \in [{\mathcal {Q}}]^{<\omega }$ can support $A_i$ and, in consequence, $A_i\notin {\mathcal {N}}$. This is the chief reason why one should not be tempted to mimic the proof of Theorem 5.15 in [28] to show that $\textbf{NAS}$ fails in ${\mathcal {N}}$. We also do not know if $\textbf{INSHC}$ holds in ${\mathcal {N}}$.

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 13

$(\textbf{ZF})$ Let d be a quasi-metric on a set X such that either $d^{-1}$ is precompact or the space ${\textbf{X}}=\langle X, \tau (d)\rangle$ is limit point compact. Then the set ${{\,\mathrm{\textrm{Iso}}\,}}_{\tau (d^{-1})}(X)$ is a cuf set.

Proof

Let $D={{\,\mathrm{\textrm{Iso}}\,}}_{\tau (d^{-1})}(X)$. For every $x\in D$, let

$$\begin{aligned} n_x=\min \left\{ n\in {\mathbb {N}}: B_{d^{-1}}\left( x, \frac{1}{n}\right) =\{x\}\right\} . \end{aligned}$$

For every $n\in {\mathbb {N}}$, let $A_n=\{x\in D: n=n_x\}$. Suppose that $n_0\in {\mathbb {N}}$ is such that $A_{n_0}$ is infinite. Let us show that there exist $x_0\in X$ and $x_1\in A_{n_0}$ such that $x_0\ne x_1$ and $x_1\in B_d\left( x_0,\frac{1}{n_0}\right)$. If $x_0, x_1$ are such points, we notice that $d^{-1}(x_1, x_0)=d(x_0, x_1)<\frac{1}{n_0}$, so $x_0\in B_{d^{-1}}\left( x_1,\frac{1}{n_0}\right)$ which is impossible by the definition of $A_{n_0}$.

If ${\textbf{X}}$ is limit point compact, there exists an accumulation point of $A_{n_0}$ in ${\textbf{X}}$. In this case, for a fixed accumulation point $x_0$ of $A_{n_0}$, we can fix $x_1\in A_{n_0}$ such that $x_1\ne x_0$ and $x_1\in B_d\left( x_0, \frac{1}{n_0}\right)$. Assuming that $d^{-1}$ is precompact, we can fix a finite set $F\subseteq X$ such that $X=\bigcup \nolimits _{x\in F}B_d\left( x, \frac{1}{n_0}\right)$ and, since $A_{n_0}$ is infinite, we can fix $x_0\in F$ and $x_1\in A_{n_0}$ such that $x_1\ne x_0$ and $x_1\in B_d\left( x_0, \frac{1}{n_0}\right)$. Hence, the assumption that $A_{n_0}$ is infinite leads to a contradiction. Therefore, $D=\bigcup \nolimits _{n\in {\mathbb {N}}}A_n$ is a cuf set. $\square$

Corollary 1

$(\textbf{ZF})$ Let ${\textbf{X}}=\langle X, d\rangle$ be a metric space which is either limit point compact or totally bounded. Then ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is a cuf set. Furthermore, if ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is infinite, then X is quasi Dedekind-infinite.

Proof

That ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is a cuf set follows from Theorem 13. The second assertion is straightforward. $\square$

Remark 6

Let ${\textbf{X}}$ be a compact metrizable space. Then, using Proposition 4(ii), we may deduce that ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is a cuf set. Namely, suppose that ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is infinite. Let $Y=\text {cl}_{{\textbf{X}}}({{\,\mathrm{\textrm{Iso}}\,}}(X))$. Then ${\textbf{Y}}$ is a metrizable compactification of the discrete space ${{\,\mathrm{\textrm{Iso}}\,}}(X)$, so ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is a cuf set by Proposition 4(ii).

Theorem 3(b) improves the well-known result of $\textbf{ZFC}$ 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 $\textbf{AC}$) choice principle $\textbf{CAC}$ suffices for the proof. An open problem posed in [32] is whether it can be proved in $\textbf{ZF}$ 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

$(\textbf{ZF})$ Let d be a strong quasi-metric on a set X such that $\langle X, \tau (d)\rangle$ is a $T_3$-space. Then the following conditions are satisfied:

• (i) if $\langle X, \tau (d^{-1})\rangle$ has a dense cuf set, then $\langle X, \tau (d)\rangle$ is metrizable;

• (ii) if $\langle X, \tau ( d^{-1})\rangle$ is iso-dense and either $\langle X, \tau (d)\rangle$ is limit point compact or $d^{-1}$ is precompact, then the space $\langle X, \tau (d)\rangle$ is metrizable.

Proof

(i) Assume that $A=\bigcup \nolimits _{n\in \omega }A_n$ is a dense set in $\langle X, \tau (d^{-1})\rangle$ such that, for every $n\in \omega$, $A_n$ is a non-empty finite set. For $m,n\in \omega$, we define

$$\begin{aligned} {\mathcal {B}}_{m,n}=\left\{ B_d\left( x,\frac{1}{m+1}\right) : x\in A_n\right\} . \end{aligned}$$

Since d is strong, in much the same way, as in the proof of Theorem 4.6 in [32], one can show that ${\mathcal {B}}=\bigcup \nolimits _{n,m\in \omega }{\mathcal {B}}_{n,m}$ is a base of $\langle X, \tau (d)\rangle$. Since ${\mathcal {B}}$ is a cuf set, the space $\langle X, \tau (d)\rangle$ is metrizable by Theorem 2(ii).

(ii) Now, we assume that $\langle X, \tau ( d^{-1})\rangle$ is iso-dense and either $\langle X, \tau (d)\rangle$ is limit point compact or $d^{-1}$ is precompact. Let $E={{\,\mathrm{\textrm{Iso}}\,}}_{\tau (d^{-1})}(X)$. Then E is dense in $\langle X, \tau (d^{-1})\rangle$. By Theorem 13, the set E is a cuf set. Hence, to conclude the proof, it suffices to apply (i). $\square$

Corollary 2

$(\textbf{ZF})$ Let $\langle X, d\rangle$ be a compact Hausdorff quasi-metric space such that $\langle X, \tau (d^{-1})\rangle$ is iso-dense. Then $\langle X, \tau (d)\rangle$ is metrizable.

Proof

This follows immediately from Proposition 2 and Theorem 14. $\square$

Remark 7

In Corollary 2, we cannot omit the assumption that $\langle X, \tau (d)\rangle$ is Hausdorff. Indeed, there is a quasi-metric d on $\omega$ such that $\tau (d)$ is the cofinite topology on $\omega$ and $\tau (d^{-1})$ is the discrete topology on $\omega$ (see [32]). Then d is a strong quasi-metric such that $\langle \omega , \tau (d)\rangle$ is a compact $T_1$-space which is not metrizable.

Theorem 15

$(\textbf{ZF})$ Let $\langle X, d\rangle$ be an iso-dense metric space such that either d is totally bounded or $\langle X, \tau (d)\rangle$ is limit point compact. Then $\langle X, \tau (d)\rangle$ has a cuf base and can be embedded in a metrizable Tychonoff cube.

Proof

It follows from the proof of Theorem 14 that $\langle X, \tau (d)\rangle$ has a cuf base. Since $\langle X, \tau (d)\rangle$ is a $T_3$-space, to conclude the proof, it suffices to apply Theorem 2(ii). $\square$

4 $\textbf{CAC}_{fin}$ via Iso-dense Metrizable Spaces

It is known that it holds in $\textbf{ZFC}$ 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 $\textbf{ZF}$ is different from the one in $\textbf{ZFC}$. To begin, let us recall the following lemma proved in [27]:

Lemma 1

$(\textbf{ZF})$. Let ${\textbf{X}}$ be a non-empty metrizable space and let ${\mathcal {B}}$ be a base of ${\textbf{X}}$. Then ${\textbf{X}}$ embeds in $[0, 1]^{{\mathcal {B}}\times {\mathcal {B}}}$.

If ${\textbf{X}}=\langle X, d\rangle$ is a metric space and ${\textbf{Y}}$ is a topological space, then we say that ${\textbf{X}}$ embeds in ${\textbf{Y}}$ if the space $\langle X, \tau (d)\rangle$ embeds in ${\textbf{Y}}$.

The following theorem is a characterization of $\textbf{CAC}_{fin}$ in terms of iso-dense (limit point) compact metrizable spaces and in terms of iso-dense totally bounded metric spaces.

Theorem 16

$(\textbf{ZF})$ The following conditions are all equivalent:

• (i) $\textbf{CAC}_{fin}$;

• (ii) for every iso-dense metric space ${\textbf{X}}$, if ${\textbf{X}}$ is either limit point compact or totally bounded, then ${\textbf{X}}$ is separable;

• (iii) for every iso-dense metric space ${\textbf{X}}$, if ${\textbf{X}}$ is either limit point compact or totally bounded, then ${\textbf{X}}$ embeds in the Hilbert cube $[0,1]^{{\mathbb {N}}}$;

• (iv) for every iso-dense metric space ${\textbf{X}}$, if ${\textbf{X}}$ is either limit point compact or totally bounded, then $|{{\,\mathrm{\textrm{Iso}}\,}}(X)|\le |{\mathbb {R}}|$;

• (v) for every iso-dense metric space ${\textbf{X}}$, if ${\textbf{X}}$ is either limit point compact or totally bounded, then the set ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is countable.

In (ii)-(v), the term “iso-dense” can be replaced by “scattered”.

Proof

Let ${\textbf{X}}=\langle X, d\rangle$ be an iso-dense (respectively, scattered) metric space such that ${\textbf{X}}$ is either limit point compact or totally bounded. By Corollary 1, the set ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is a cuf set. Hence, it follows from $\textbf{CAC}_{fin}$ that ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ 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 $\textbf{ZF}$ that (ii) implies (iii).

Now, to show that (iii) implies (iv), suppose that $\langle X, \tau (d)\rangle$ is homeomorphic to a subspace of $[0, 1]^{{\mathbb {N}}}$. Then ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is equipotent to a subset of $[0, 1]^{{\mathbb {N}}}$. Since it holds in $\textbf{ZF}$ that $[0, 1]^{{\mathbb {N}}}$ and ${\mathbb {R}}$ are equipotent, we deduce that ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is equipotent to a subset of ${\mathbb {R}}$. Hence, (iii) implies (iv).

It is obvious that, in $\textbf{ZF}$, every cuf subset of ${\mathbb {R}}$ is countable as a countable union of finite well-ordered sets. Hence, if ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is equipotent to a subset of ${\mathbb {R}}$, then ${{\,\mathrm{\textrm{Iso}}\,}}(X)$ is countable as a set equipotent to a cuf set contained in ${\mathbb {R}}$. This shows that (iv) implies (v).

Finally, suppose that $\textbf{CAC}_{fin}$ fails. Then there exists an uncountable discrete cuf space ${\textbf{D}}$. It follows from Proposition 4(i) that the Alexandroff compactification ${\textbf{D}}(\infty )$ of ${\textbf{D}}$ is metrizable. Since ${\textbf{D}}(\infty )$ is an iso-dense compact mertizable space whose set of all isolated points is uncountable, (v) fails if $\textbf{CAC}_{fin}$ fails. Hence (v) implies (i). $\square$

Theorem 17

$(\textbf{ZF})$

• (i) For every totally bounded metric space ${\textbf{X}}$, the Cantor-Bendixson rank of ${\textbf{X}}$ 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 ${\textbf{X}}=\langle X, d\rangle$ be an infinite totally bounded metric space. Let $\alpha =|{\textbf{X}}|_{\text {CB}}$. Then

$$\begin{aligned} X=\bigcup \nolimits _{\gamma \in \alpha }{{\,\mathrm{\textrm{Iso}}\,}}(X^{(\gamma )})\cup X^{(\alpha )}. \end{aligned}$$

For every $\gamma \in \alpha$ and every $x\in {{\,\mathrm{\textrm{Iso}}\,}}(X^{(\gamma )})$, let

$$\begin{aligned} n(x,\gamma )=\min \left\{ n\in {\mathbb {N}}: B_d\left( x,\frac{1}{n}\right) \cap X^{(\gamma )}=\{x\}\right\} . \end{aligned}$$

For every $\gamma \in \alpha$ and every $n\in {\mathbb {N}}$, let

$$\begin{aligned} A_{\gamma , n}=\{x\in {{\,\mathrm{\textrm{Iso}}\,}}(X^{(\gamma )}): n(x,\gamma )=n\}. \end{aligned}$$

We have already shown in the proof of Theorem 13 that, for every $\gamma \in \alpha$ and every $n\in {\mathbb {N}}$, the set $A_{\gamma , n}$ is finite and ${{\,\mathrm{\textrm{Iso}}\,}}(X^{(\gamma )})=\bigcup \nolimits _{n\in {\mathbb {N}}}A_{\gamma , n}$.

(i) Suppose that $\alpha$ is uncountable. For every $n\in {\mathbb {N}}$, let

$$\begin{aligned} B_n=\{\gamma \in \alpha : A_{\gamma , n}\ne \emptyset \}. \end{aligned}$$

Since $\alpha$ is supposed to be uncountable, there exists $n_0\in {\mathbb {N}}$ such that $B_{n_0}$ is infinite. We fix such an $n_0$ and put

$$\begin{aligned} {\mathcal {U}}=\left\{ B_d\left( x, \frac{1}{3n_0}\right) : x\in X\right\} . \end{aligned}$$

By the total boundedness of d, the open cover ${\mathcal {U}}$ of ${\textbf{X}}$ has a finite subcover. Hence, there exists a non-empty finite subset F of X such that $X=\bigcup \nolimits _{x\in F}B_d\left( x,\frac{1}{3n_0}\right)$. Since $B_{n_0}$ is infinite, there exist $\gamma _1,\gamma _2\in B_{n_0}$ and elements $x_0\in F$, $x_1\in A_{\gamma _1, n_0}\cap B_d\left( x_0,\frac{1}{3n_0}\right)$ and $x_2\in A_{\gamma _2, n_0}\cap B_d\left( x_0, \frac{1}{3n_0}\right)$, such that $x_1\ne x_2$. We may assume that $\gamma _1\le \gamma _2$. Then $X^{(\gamma _2)}\subseteq X^{(\gamma _1)}$ and it follows from the definition of $A_{\gamma _1, n_0}$ that $d(x_1, x_2)\ge \frac{1}{n_0}$. On the other hand, since $x_1, x_2\in B_d(x_0,\frac{1}{3n_0})$, we have $d(x_1, x_2)\le \frac{2}{3n_0}<\frac{1}{n_0}$. The contradiction obtained proves that $\alpha$ is countable.

(ii) Now, suppose that the space ${\textbf{X}}$ is also scattered. Then it follows from Proposition 1 that $X^{(\alpha )}=\emptyset$. Hence, $X=\bigcup \{A_{\gamma , n}: \gamma \in \alpha \text { and } n\in {\mathbb {N}}\}$. Since $\alpha$ is countable, the set $\alpha \times {\mathbb {N}}$ is countable. This implies that the family $\{A_{\gamma , n}: \gamma \in \alpha \text { and }n\in {\mathbb {N}}\}$ is also countable. We have already shown that, for every $\gamma \in \alpha$ and every $n\in {\mathbb {N}}$, the set $A_{\gamma , n}$ is finite. Hence, X is a cuf set.

It follows immediately from (ii) and Theorem 15 that (iii) holds. $\square$

Theorem 18

$(\textbf{ZF})$ The following conditions are all equivalent:

• (i) $\textbf{CAC}_{fin}$;

• (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 $\textbf{CAC}_{fin}$ implies that all cuf sets are countable, it follows from Theorem 17 that (i) implies (ii) and (iii). It is provable in $\textbf{ZF}$ that every totally bounded, complete countable metric space is compact. Hence, in the light of Theorem 17, (i) implies (iv).

Assume that $\textbf{CAC}_{fin}$ is false. Then there exists a family $\{A_n: n\in \omega \}$ of non-empty pairwise disjoint finite sets such that the set $D=\bigcup \nolimits _{n\in \omega }A_n$ is Dedekind-finite (see [13, Form 10 M]). Let ${\textbf{D}}=\langle D, {\mathcal {P}}(D)\rangle$. By Proposition 4(i), the space ${\textbf{D}}(\infty )$ is metrizable. Let d be any metric which induces the topology of ${\textbf{D}}(\infty )$. Since ${\textbf{D}}(\infty )$ is compact, the metric d is totally bounded. Moreover, ${\textbf{D}}(\infty )$ is scattered but uncountable. For $\rho =d\upharpoonright D\times D$, the metric space $\langle D, \rho \rangle$ is also totally bounded. Since D is Dedekind-finite and $\langle D, \rho \rangle$ is discrete, the metric $\rho$ is complete. Clearly, $\langle D, \rho \rangle$ is not compact. All this taken together completes the proof. $\square$

Theorem 19

$(\textbf{ZF})$ 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 ${\textbf{X}}=\langle X, d\rangle$ is a compact metric space such that the set X is a non-empty cuf set. Towards a contradiction, suppose that ${\textbf{X}}$ is dense-in-itself. We fix a partition $\{X_n: n\in \omega \}$ of ${\textbf{X}}$ into non-empty finite sets.

Let $S=\bigcup \{\{0,1\}^n: n\in {\mathbb {N}}\}$. For $n\in {\mathbb {N}}$, $s\in \{0,1\}^n$ and $t\in \{0,1\}$, let $s\smallfrown t\in \{0,1\}^{n+1}$ be defined by: $s\smallfrown t(i)=s(i)$ for every $i\in n$, and $s\smallfrown t(n)=t$. Using ideas from [3], let us define by induction (with respect to n) a family $\{B_s: s\in S\}$ such that, for every $s\in S$, the following conditions are satisfied:

• (1) $B_s$ is a non-empty open subset of ${\textbf{X}}$;

• (2) for every $t\in \{0,1\}$, $B_{s\smallfrown t}\subseteq B_s$;

• (3) ${{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(B_{s\smallfrown 0})\cap {{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(B_{s\smallfrown 1})=\emptyset$.

To start the induction, for $n=1=\{0\}$ and every $s\in \{0,1\}^1$, we define:

$$\begin{aligned} B_s=\bigcup \left\{ B_d\left( x, \frac{d(X_0, X_1)}{3}\right) : x\in X_{s(0)}\right\} . \end{aligned}$$

Now, suppose that $n\in {\mathbb {N}}$ is such that, for every $s\in \bigcup \nolimits _{i=1}^n\{0,1\}^i$, we have defined a non-empty open subset $B_s$ of ${\textbf{X}}$. For an arbitrary $s\in \{0,1\}^{n+1}$, we consider the set $B_{s\upharpoonright n}$. We put $n_s=\min \{m\in \omega : X_m\cap B_{s\upharpoonright n}\ne \emptyset \}$. Since ${\textbf{X}}$ is dense-in-itself, we have $\emptyset \ne \{m\in \omega : X_m\cap (B_{s\upharpoonright n}{\setminus } X_{n_s})\ne \emptyset \}$, so we can define $k_s=\min \{m\in \omega : X_m\cap (B_{s\upharpoonright n}{\setminus } X_{n_s})\ne \emptyset \}$. Now, we put $Y_{s,0}=X_{n_s}\cap B_{s\upharpoonright n}$ and $Y_{s,1}=X_{k_s}\cap (B_{s\upharpoonright n}\setminus X_{n_s})$. We define

$$\begin{aligned} B_s=\bigcup \left\{ B_d\left( y,\frac{d(Y_{s,0}, Y_{s,1})}{3}\right) : y\in Y_{s,s(n)}\right\} . \end{aligned}$$

In this way, we have inductively defined the required family $\{B_s: s\in S\}$.

We notice that it follows from (2) that, for every $f\in \{0,1\}^{\omega }$ and $n\in {\mathbb {N}}$, $\emptyset \ne {{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(B_{f\upharpoonright (n+1)})\subseteq {{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(B_{f\upharpoonright n})$. Thus, by the compactness of ${\textbf{X}}$, for every $f\in \{0,1\}^{\omega }$, the set

$$\begin{aligned} M_f=\bigcap \left\{ {{\,\mathrm{\textrm{cl}}\,}}_{{\textbf{X}}}(B_{f\upharpoonright n}): n\in {\mathbb {N}}\right\} \end{aligned}$$

is non-empty. For every $f\in \{0, 1\}^{\omega }$, let $m_f=\min \{n\in \omega : X_n\cap M_f\ne \emptyset \}$. We define a mapping $F:\{0, 1\}^{\omega }\rightarrow \bigcup \{{\mathcal {P}}(X_n): n\in \omega \}$ by putting:

$$\begin{aligned} F(f)=X_{m_f}\cap M_f\text { for every } f\in \{0,1\}^{\omega }. \end{aligned}$$

It follows from (3) that F is an injection. In consequence, the set $\{0, 1\}^{\omega }$ is equipotent to a subset of the cuf set $\bigcup \{{\mathcal {P}}(X_n): n\in \omega \}$. But this is impossible because $\{0, 1\}^{\omega }$, being equipotent to ${\mathbb {R}}$, 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 ${\textbf{X}}$ be any compact metrizable cuf space. We have proved that every non-empty compact subspace of ${\textbf{X}}$ has an isolated point. Hence ${\textbf{X}}$ cannot contain non-empty dense-in-itself subspaces. This implies that ${\textbf{X}}$ is scattered. $\square$

Now, we can give the following modification of Theorem 1:

Theorem 20

$(\textbf{ZF})$ Let ${\textbf{X}}$ be a compact Hausdorff, non-scattered space which has a cuf base. If ${\textbf{X}}$ is weakly Loeb, then $|{\mathbb {R}}|\le |[X]^{<\omega }|$. If ${\textbf{X}}$ is Loeb, then $|{\mathbb {R}}|\le |X|$.

Proof

Without loss of generality, we may assume that ${\textbf{X}}$ is dense-in-itself because we can replace ${\textbf{X}}$ with its non-empty dense-in-itself compact subspace. By Theorem 1.13(ii), ${\textbf{X}}$ is metrizable. It is known that every compact metrizable Loeb space is second-countable (see, e.g., [23]). Hence, if ${\textbf{X}}$ is Loeb, then $|{\mathbb {R}}|\le |X|$ by Theorem 1. Suppose that ${\textbf{X}}$ is weakly Loeb. Let d be any metric on X which induces the topology of ${\textbf{X}}$. It follows from Proposition 3 that ${\textbf{X}}$ has a dense cuf set. Since ${\textbf{X}}$ is non-empty and dense-in-itself, every dense subset of ${\textbf{X}}$ is infinite. Therefore, we can fix a disjoint family $\{X_n: n\in \omega \}$ of non-empty finite subsets of ${\textbf{X}}$ such that the set $\bigcup \{X_n: n\in \omega \}$ is dense in ${\textbf{X}}$. Mimicking the proof of Theorem 19, we can define an injection $F:\{0,1\}^{\omega }\rightarrow {\mathcal {P}}(X)$ such that, for every $f\in \{0,1\}^{\omega }$, the set $M_f=F(f)$ is a non-empty closed subset of ${\textbf{X}}$ and, for every pair f, g of distinct functions from $\{0,1\}^{\omega }$, $M_f\cap M_g=\emptyset$. Let $\psi$ be a weak Loeb function for ${\textbf{X}}$. Then $\psi \circ F$ is an injection from $\{0, 1\}^{\omega }$ into $[X]^{<\omega }$. Hence $|{\mathbb {R}}|=|\{0, 1\}^{\omega }|\le |[X]^{<\omega }|$. $\square$

Taking the opportunity, let us give a proof of the following theorem:

Theorem 21

$(\textbf{ZF})$

• (a) For every non-empty set I and every family $\{\langle X_i, \tau _i\rangle : i\in I\}$ of denumerable metrizable compact spaces, the family $\{X_i: i\in I\}$ has a multiple choice function.

• (b) For a non-zero ordinal $\alpha$, let $\{{\textbf{X}}_\gamma : \gamma \in \alpha \}$ be a family of pairwise disjoint non-empty countable, compact metrizable spaces. Then the direct sum ${\textbf{X}}=\bigoplus \nolimits _{\gamma \in \alpha }{\textbf{X}}_{\gamma }$ is weakly Loeb.

• (c) The following conditions are all equivalent:

  • (i) $\textbf{WOAC}_{fin}$;

  • (ii) for every well-orderable set S and every family $\{\langle X_s, d_s\rangle : s\in S\}$ of scattered totally bounded metric spaces, the union $\bigcup \nolimits _{s\in S}X_s$ is well-orderable;

  • (iii) for every well-orderable non-empty set S and every family $\{\langle X_s, d_s\rangle : s\in S\}$ of compact scattered metric spaces, the product $\prod \nolimits _{s\in S}\langle X_s, \tau (d_s)\rangle$ is compact.

Proof

(a) Let ${\textbf{X}}_i=\langle X_i, \tau _i\rangle$ be a denumerable metrizable compact space for every $i\in I$ with $I\ne \emptyset$. For every $i\in I$, let $\alpha _i=|{\textbf{X}}_i|_{{{\,\mathrm{\textrm{CB}}\,}}}$. We fix $i\in I$ and observe that, since ${\textbf{X}}_i$ is scattered by Theorem 19, it follows from Proposition 1 that $X_i^{(\alpha _i)}=\emptyset$ and $\alpha _i$ is a successor ordinal. Let $\beta _i\in ON$ be such that $\alpha _i=\beta _i+1$. Then $X_i^{(\beta _i)}={{\,\mathrm{\textrm{Iso}}\,}}(X_i^{(\beta _i)})$. If the set ${{\,\mathrm{\textrm{Iso}}\,}}(X_i^{(\beta _i)})$ were infinite, it would have an accumulation point in $X_i^{(\beta _i)}$ by the compactness of ${\textbf{X}}_i$. Hence ${{\,\mathrm{\textrm{Iso}}\,}}(X_i^{(\beta _i)})$ is a non-empty finite set. In consequence, by assigning to any $i\in I$ the set ${{\,\mathrm{\textrm{Iso}}\,}}(X_i^{(\beta _i)})$, we obtain a multiple choice function of $\{X_i: i\in I\}$.

(b) Let us consider the family ${\mathcal {F}}$ of all non-empty closed sets of ${\textbf{X}}$. By the proof of (a), there exists a family $\{f_{\gamma }: \gamma \in \alpha \}$ such that, for every $\gamma \in \alpha$, $f_{\gamma }$ is a weak Loeb function of ${\textbf{X}}_{\gamma }$. For every $F\in {\mathcal {F}}$, let $\gamma (F)=\min \{\gamma \in \alpha : F\cap X_{\gamma }\ne \emptyset \}$ and let $f(F)=f_{\gamma (F)}(F\cap X_{\gamma })$. Then f is a weak Loeb function of ${\textbf{X}}$.

(c) (i)$\rightarrow$(ii) Let us assume $\textbf{WOAC}_{fin}$. Suppose that S is a well-orderable non-empty set and, for every $s\in S$, ${\textbf{X}}_s=\langle X_s, d_s\rangle$ is a non-empty scattered totally bounded metric space. To prove that $X=\bigcup \nolimits _{s\in S}X_s$ is well-orderable, without loss of generality, we may assume that $S=\alpha$ for some non-zero ordinal $\alpha$, and $X_i\cap X_j=\emptyset$ for every pair i, j of distinct elements of $\alpha$. In much the same way, as in the proof of Theorem 17, we can define a family $\{A_{i, n}: i\in \alpha , n\in {\mathbb {N}}\}$ of non-empty finite sets such that, for every $i\in \alpha$, $X_i=\bigcup \nolimits _{n\in {\mathbb {N}}}A_{i,n}$. Now, we can easily define a family $\{M_i: i\in \alpha \}$ of subsets of ${\mathbb {N}}$ and a family $\{F_{i,n}: i\in \alpha , n\in M_i\}$ of pairwise disjoint non-empty finite sets such that, for every $i\in \alpha$, $X_i=\bigcup \nolimits _{n\in M_i}F_{i,n}$. The set $J=\{\langle i, n\rangle : i\in \alpha , n\in M_i\}$ is well-orderable, so we can fix an ordinal number $\gamma$ and a bijection $h:\gamma \rightarrow J$. For every $j\in \gamma$, let $n(j)\in \omega$ be equipotent to $F_{h(j)}$, and let

$$\begin{aligned} B_j=\{ f\in F_{h(j)}^{n(j)}: f\text { is a bijection}\}. \end{aligned}$$

By $\textbf{WOAC}_{fin}$, there exists $\psi \in \prod \nolimits _{j\in \gamma }B_{j}$. Now, we can define a well-ordering $\le$ on $X=\bigcup \nolimits _{j\in \gamma }F_{h(j)}$ as follows: for $i,j\in \gamma$, $x\in F_{h(i)}$, $y\in F_{h(j)}$, we put:

$$\begin{aligned} x\le y\text { if either } i\in j, \text { or } i=j \text { and } \psi (i)^{-1}(x)\subseteq \psi (i)^{-1}(y). \end{aligned}$$

Hence, (i) implies (ii)

(ii)$\rightarrow$(iii) Let S be a non-empty well-orderable set and, for every $s\in S$, let $\langle X_s, d_s\rangle$ be a compact scattered metric space with $X_s\ne \emptyset$. Clearly, we may assume that S is a well-ordered cardinal. We notice that if $X=\bigcup \nolimits _{s\in S}X_s$ is well-orderable, then we can define a family $\{f_s: s\in S\}$ such that, for every $s\in S$, $f_s$ is a Loeb function of ${\textbf{X}}_s=\langle X_s, \tau (d_s)\rangle$ and, therefore, by Theorem 4, the product $\prod \nolimits _{s\in S}{\textbf{X}}_s$ is compact. Hence, (ii) implies (iii).

(iii)$\rightarrow$(i) Now, let S be a well-orderable set and let $\{A_s: s\in S\}$ be a family of non-empty finite sets. Take an element $\infty \notin \bigcup \nolimits _{s\in S}A_s$ and put $Y_s=A_s\cup \{\infty \}$ for $s\in S$. Let $\rho _s$ be the discrete metric on $Y_s$ and let ${\textbf{Y}}_s=\langle Y_s, {\mathcal {P}}(Y_s)\rangle$ for every $s\in S$. Assuming (iii), we obtain that the space ${\textbf{Y}}=\prod \nolimits _{s\in S}{\textbf{Y}}_s$ is compact. In much the same way, as in the proof of Kelley’s theorem that the Tychonoff Product Theorem implies $\textbf{AC}$ (see [16]), one can show that the compactness of ${\textbf{Y}}$ implies that $\prod \nolimits _{s\in S}A_s\ne \emptyset$. Hence, (iii) implies (i). $\square$

5 Alexandroff Compactifications of Infinite Discrete Cuf Spaces as Subspaces

This section is closely 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” (recall that, by [26], the latter proposition is not provable in $\textbf{ZF}$).

First, note that ${\mathbb {N}}(\infty )$ or, more generally, for every infinite discrete cuf space ${\textbf{D}}$, ${\textbf{D}}(\infty )$ 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 ${\textbf{D}}(\infty )$ for some infinite discrete cuf space ${\textbf{D}}$. Let us start with the following simple proposition:

Proposition 7

$(\textbf{ZF})$ Let ${\textbf{X}}$ be a non-discrete first-countable Loeb $T_3$-space. Then ${\textbf{X}}$ contains a copy of ${\mathbb {N}}(\infty )$. In particular, every non-discrete metrizable Loeb space contains a copy of ${\mathbb {N}}(\infty )$.

Proof

Let $x_0$ be an accumulation point of ${\textbf{X}}$ and let f be a Loeb function of ${\textbf{X}}$. Since ${\textbf{X}}$ is a first-countable $T_3$-space and $x_0$ is an accumulation point of ${\textbf{X}}$, there exists a base $\{U_n: n\in {\mathbb {N}}\}$ of open neighborhoods of $x_0$ such that $\text {cl}_{{\textbf{X}}}(U_{n+1})\subset U_n$ for every $n\in {\mathbb {N}}$. Let $x_n=f(\text {cl}_{{\textbf{X}}}(U_n){\setminus } U_{n+1})$ for every $n\in {\mathbb {N}}$. Then the subspace $\{x_{2n}: n\in \omega \}$ of ${\textbf{X}}$ is a copy of ${\mathbb {N}}(\infty )$. $\square$

Theorem 22

$(\textbf{ZF})$

• (i) $\textbf{IQDI}$ implies that every infinite compact first-countable Hausdorff space contains a copy of ${\textbf{D}}(\infty )$ for some infinite discrete cuf space ${\textbf{D}}$.

• (ii) Each of $\textbf{IDI}$, $\textbf{WoAm}$ and $\textbf{BPI}$ implies that every infinite compact first-countable Hausdorff space contains a copy of ${\mathbb {N}}(\infty )$.

• (iii) Every infinite first-countable compact Hausdorff separable space contains a copy of ${\mathbb {N}}(\infty )$. Every infinite first-countable compact Hausdorff space having an infinite cuf subset contains a copy of ${\textbf{D}}(\infty )$ for some infinite discrete cuf space ${\textbf{D}}$.

Proof

Let ${\textbf{X}}=\langle X, \tau \rangle$ be an infinite compact first-countable Hausdorff space. For a point $x_0\in X$, let $\{U_n: n\in \omega \}$ be a base of neighborhoods of $x_0$ such that $\text {cl}_{{\textbf{X}}}(U_{n+1})\subseteq U_n$ for every $n\in \omega$.

(i) Assuming $\textbf{IQDI}$, we can fix a disjoint family $\{F_n: n\in \omega \}$ of non-empty finite subsets of X. Since ${\textbf{X}}$ is compact, the set $F=\bigcup \nolimits _{n\in \omega }F_n$ has an accumulation point. Let $x_0$ be an accumulation point of F. We may assume that $x_0\notin F$. Let $n_0=m_0=\min \{n\in \omega : U_0\cap F_{n}\ne \emptyset \}$, $M_0=\omega$ and $Y_0=F_{n_0}\cap U_0$. Since ${\textbf{X}}$ is a $T_1$-space, $Y_0$ is a finite set and $x_0\notin Y_0$, the set $M_1=\{n\in \omega : U_n\cap Y_0=\emptyset \}$ is non-empty. Let $m_1=\min M_1$, $n_1=\min \{n\in \omega : U_{m_1}\cap F_n\ne \emptyset \}$ and $Y_1= U_{m_1}\cap F_{n_1}$. Suppose that $k\in \omega \setminus \{0\}$ is such that we have already defined $n_k, m_k\in \omega$ such that $Y_k=U_{m_k}\cap F_{n_k}\ne \emptyset$. We put $M_{k+1}=\{n\in \omega : U_n\cap Y_{k}=\emptyset \}$, $m_{k+1}=\min M_{k+1}$, $n_{k+1}=\min \{n\in \omega : U_{m_{k+1}}\cap F_n\ne \emptyset \}$ and $Y_{k+1}=U_{m_{k+1}}\cap F_{n_{k+1}}$. This terminates our inductive definition. Let $D=\bigcup \nolimits _{k\in \omega }Y_k$ and $Y=D\cup \{x_0\}$. Then ${\textbf{D}}$ is a discrete cuf subspace of ${\textbf{X}}$, the subspace ${\textbf{Y}}$ of ${\textbf{X}}$ is compact and $x_0$ is the unique accumulation point of ${\textbf{Y}}$. Hence, ${\textbf{Y}}$ is a copy of ${\textbf{D}}(\infty )$.

(ii) If $\textbf{IDI}$ holds or X is well-orderable, then we can fix a disjoint family $\{F_n: n\in \omega \}$ of singletons of X and in much the same way, as in the proof of (i), we can find a copy of ${\mathbb {N}}(\infty )$ in ${\textbf{X}}$. By Proposition 5 of [27], $\textbf{WoAm}$ implies that every first-countable limit point compact $T_1$-space is well-orderable. Hence, $\textbf{WoAm}$ implies that ${\textbf{X}}$ contains a copy of ${\mathbb {N}}(\infty )$.

Now, let us assume $\textbf{BPI}$. Let $x_0$ be an accumulation point of ${\textbf{X}}$. Without loss of generality, we may aassume that $\text {cl}_{{\textbf{X}}}(U_{n+1})\ne U_n$ for every $n\in \omega$. Let $G_n=\{x_0\}\cup ( \text {cl}_{{\textbf{X}}}(U_n){\setminus } U_{n+1})$ for every $n\in \omega$. Then the subspaces ${\textbf{G}}_n$ of ${\textbf{X}}$ are compact. By Theorem 5(ii), the product ${\textbf{G}}=\prod \nolimits _{n\in \omega }{\textbf{G}}_n$ is compact. Therefore, since the family ${\mathcal {G}}=\{\pi _n^{-1}(\text {cl}_{{\textbf{X}}}(U_n){\setminus } U_{n+1}): n\in \omega \}$ has the finite intersection property and consists of closed subsets of the compact space ${\textbf{G}}$, there exists $f\in \bigcap \nolimits _{n\in \omega }\pi _n^{-1}(\text {cl}_{{\textbf{X}}}(U_n){\setminus } U_{n+1})$. Then the subspace $\{x_0\}\cup \{f(2n): n\in \omega \}$ of ${\textbf{X}}$ is a copy of ${\mathbb {N}}(\infty )$. This completes the proof of (ii).

(iii) This can be deduced from the proofs of (i) and (ii). $\square$

Corollary 3

$(\textbf{ZF})$

• (i) Each of $\textbf{IQDI}$, $\textbf{WoAm}$ and $\textbf{BPI}$ 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 [13]: the Basic Fraenkel Model ${\mathcal {N}}1$, the Second Fraenkel Model ${\mathcal {N}}2$ and the Mostowski Linearly Ordered Model ${\mathcal {N}}3$: $\textbf{WoAm}$ is true in ${\mathcal {N}}1$, $\textbf{IWDI}$ (and hence the stronger $\textbf{IQDI}$, which is implied by $\textbf{CMC}$) is false in both ${\mathcal {N}}1$ and ${\mathcal {N}}3$, $\textbf{CAC}_{fin}$ is true in ${\mathcal {N}}1$ but it is false in ${\mathcal {N}}2$, $\textbf{MC}$ is true in ${\mathcal {N}}2$, and $\textbf{BPI}$ (and hence $\textbf{CAC}_{fin}$) is true in ${\mathcal {N}}3$ (see [13] for the above facts). All this, taken together with Theorem 22 and its proof, yields the following theorem:

Theorem 23

• (i) In ${\mathcal {N}}1$, every first-countable limit point compact $T_1$-space is well-orderable.

• (ii) The statement “Every infinite first-countable Hausdorff compact space contains a copy of ${\mathbb {N}}(\infty )$” is true in ${\mathcal {N}}1$ and in ${\mathcal {N}}3$.

• (iii) The statement “Every infinite first-countable Hausdorff compact space contains an infinite metrizable compact scattered subspace” is true in ${\mathcal {N}}2$.

• (iv) The statement “Every infinite first-countable Hausdorff compact space contains an infinite metrizable compact scattered subspace” implies neither $\textbf{CAC}_{fin}$ nor $\textbf{IQDI}$, nor $\textbf{CMC}$ in $\textbf{ZFA}$.

Remark 8

(i) In [27], it was shown that ${\textbf{M}}(C, S)\wedge \lnot \textbf{IDI}$ has a $\textbf{ZF}$-model. Using similar arguments, one can show that ${\textbf{M}}(C, S)\wedge \lnot \textbf{IQDI}$ also has a $\textbf{ZF}$-model. This, together with Corollary 3, proves that the statement “Every infinite compact metrizable space contains an infinite compact scattered subspace” does not imply $\textbf{IQDI}$ in $\textbf{ZF}$. This can be also deduced from Theorem 23.

(ii) Suppose that $\{A_n: n\in \omega \}$ is a disjoint family of non-empty finite sets which does not have any partial choice function (see, for example, [13, Model ${\mathcal {M}}7$]). Let $D=\bigcup \nolimits _{n\in \omega }A_n$ and ${\textbf{D}}=\langle D, {\mathcal {P}}(D)\rangle$. Then ${\textbf{D}}(\infty )$ is a compact metrizable scattered space which does not contain a copy of ${\mathbb {N}}(\infty )$.

Let us complete this section with the following proposition:

Proposition 8

$(\textbf{ZF})$ A metrizable space ${\textbf{X}}$ contains an infinite compact scattered subspace if and only if ${\textbf{X}}$ contains a copy of ${\textbf{D}}(\infty )$ for some infinite discrete cuf space ${\textbf{D}}$.

Proof

$(\rightarrow )$ Suppose that ${\textbf{X}}$ is a metrizable space which has an infinite compact scattered subspace ${\textbf{Y}}$. Then ${{\,\mathrm{\textrm{Iso}}\,}}(Y)$ is a dense discrete subspace of ${\textbf{Y}}$, so ${\textbf{Y}}$ is a metrizable compactification of the discrete subspace ${{\,\mathrm{\textrm{Iso}}\,}}(Y)$ of ${\textbf{Y}}$. We deduce from Proposition 4(ii) that ${{\,\mathrm{\textrm{Iso}}\,}}(Y)$ is an infinite cuf set. Thus, by Theorem 22(iii), ${\textbf{Y}}$ contains a copy of ${\textbf{D}}(\infty )$ for some infinite discrete cuf space ${\textbf{D}}$.

$(\leftarrow )$ This is trivial because the one-point Hausdorff compactification of an infinite discrete space is a scattered space. $\square$

6 A Shortlist of New Open Problems

Problem 2

Find, if possible, a $\textbf{ZF}$-model for $\textbf{NAS}\wedge \lnot \textbf{INSHC}$.

Problem 3

Find, if possible, a $\textbf{ZF}$-model for $\textbf{INSHC}\wedge \lnot \textbf{BPI}\wedge \lnot \textbf{IDFBI}$.

Problem 4

Make a list of the forms from [13] that are true in the model ${\mathcal {N}}$ constructed in the proof of Theorem 12.

Problem 5

Is it provable in $\textbf{ZF}$ that if ${\textbf{X}}$ is a compact Hausdorff non-scattered weakly Loeb space which has a cuf base, then $|{\mathbb {R}}|\le |X|$?