On iso-dense and scattered spaces in \mathbf{ZF}
aa r X i v : . [ m a t h . GN ] J a n On iso-dense and scattered spaces in ZF Kyriakos Keremedis, Eleftherios Tachtsis and Eliza WajchDepartment of Mathematics, University of the AegeanKarlovassi, Samos 83200, [email protected] of Statistics and Actuarial-Financial Mathematics,University of the Aegean, Karlovassi 83200, Samos, [email protected] of MathematicsFaculty of Exact and Natural SciencesSiedlce University of Natural Sciences and Humanitiesul. 3 Maja 54, 08-110 Siedlce, [email protected] 11, 2021
Abstract
A topological space is iso-dense if it has a dense set of isolatedpoints. A topological space is scattered if each of its non-empty sub-spaces has an isolated point. In ZF , in the absence of the axiom ofchoice, basic properties of iso-dense spaces are investigated. A new per-mutation model is constructed in which a discrete weakly Dedekind-finite space can have the Cantor set as a remainder. A metrizationtheorem for a class of quasi-metric spaces is deduced. The statement“every compact scattered metrizable space is separable” and severalother statements about metric iso-dense spaces are shown to be equiv-alent to the countable axiom of choice for families of finite sets. Resultsconcerning the problem of whether it is provable in ZF that everynon-discrete compact metrizable space contains an infinite compactscattered subspace are also included. athematics Subject Classification (2010) : 03E25, 03E35, 54E35, 54G12,54D35 Keywords : Weak forms of the Axiom of Choice, scattered space, iso-dense space, (quasi-)metric space
In this article, the intended context for reasoning and statements of theo-rems is ZF without any form of the axiom of choice AC . However, we alsorefer to permutation models of ZFA (cf. [10] and [9]). In this article, weare concerned mainly with iso-dense and scattered spaces in ZF , defined asfollows: 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 notationand recall some known definitions in this subsection, make a list of weakerforms of AC in Subsection 1.2, and recall several known results for futurereferences in Subsection 1.3. The content of the article is described in briefin 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. Thefirst infinite ordinal number is denoted by ω . Then N = ω \ { } . If X is aset, the power set of X is denoted by P ( X ) , and the set of all finite subsetsof 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 onlyif 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 h X, d i 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 − defined by: d − ( x, y ) = d ( y, x ) for x, y ∈ X. 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 − = d ⋆ .The d - ball with centre x ∈ X and radius r ∈ (0 , + ∞ ) is the set B d ( x, r ) = { y ∈ X : d ( x, y ) < r } . The collection τ ( d ) = (cid:26) V ⊆ X : ( ∀ x ∈ V )( ∃ n ∈ ω ) B d (cid:18) x, n (cid:19) ⊆ V (cid:27) is the topology in X induced by d . For a set A ⊆ X , let δ d ( A ) = 0 if A = ∅ ,and let δ d ( A ) = sup { d ( x, y ) : x, y ∈ A } if A = ∅ . Then δ d ( A ) is the diameter of A in h X, d i .A quasi-metric d on X is called strong if τ ( d ) ⊆ τ ( d − ) .In the sequel, topological or (quasi-)metric spaces (called spaces in ab-breviation) are denoted by boldface letters, and the underlying sets of thespaces are denoted by lightface letters.For a topological space X = h X, τ i and for Y ⊆ X , let τ | Y = { V ∩ Y : V ∈ τ } and let Y = h Y, τ | Y i . Then Y is the topological subspace of X suchthat Y is the underlying set of Y . If this is not misleading, we may denotethe topological subspace Y of X by Y .A topological space X = h X, τ i is called (quasi-)metrizable if there existsa (quasi-)metric d on X such that τ = τ ( d ) .For a (quasi-)metric space X = h X, d i and for Y ⊆ X , let d Y = d ↾ Y × Y and Y = h Y, d Y 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 = h X, d i , ifnot stated otherwise, we also denote by X the topological space h X, τ ( d ) i .For every n ∈ N , R n denotes also h R n , d e i and h R n , τ ( d e ) i where d e is theEuclidean metric on R n .For a topological space X = h X, τ i and a set A ⊆ X , we denote by cl X ( A ) or by cl τ ( A ) the closure of A in X .For any topological space X = h X, τ i , let Iso τ ( X ) = { x ∈ X : x is an isolated point of X } .
3f this is not misleading, as in Definition 1.1, we use
Iso( X ) to denote Iso τ ( X ) .By transitive recursion, we define a decreasing sequence ( X ( α ) ) α ∈ ON of closedsubsets of X as follows: X (0) = X,X ( α +1) = X ( α ) \ Iso( X ( α ) ) ,X ( α ) = \ γ ∈ α X ( γ ) if α is a limit ordinal . For α ∈ ON , the set X ( α ) is called the α -th Cantor-Bendixson derivative of X . The least ordinal α such that X ( α +1) = X ( α ) is denoted by | X | CB and iscalled 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-infi-nite 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 neigh-borhoods;(ii) second-countable if X has a countable base;(iii) compact if every open cover of X has a finite subcover;4iv) locally compact if every point of X has a compact neighborhood;(v) limit point compact if every infinite subset of X has an accumulationpoint in X (cf. [14] and [15]). Definition 1.5.
Let X = h X, d i be a (quasi-)metric space.(i) Given a real number ε > , a subset D of X is called ε - dense or an ε - net in X if, for every x ∈ X , B d ( x, ε ) ∩ D = ∅ (equivalently, if X = S x ∈ D B d − ( x, ε ) ).(ii) X is called precompact (respectively, totally bounded ) if, for every realnumber ε > , there exists a finite ε -net in h X, d − i (respectively, in h X, d ⋆ i ).(iii) d is called precompact (respectively, totally bounded ) if X is precompact(respectively, totally bounded ). Remark . Definition 1.5 (ii) is based on the notions of precompact andtotally bounded quasi-uniformities defined, e.g., in [5] and [26]. Namely,given a quasi-metric d on a set X , the collection U ( d ) = (cid:26) U ⊆ X × X : ∃ n ∈ ω (cid:26) h x, y i ∈ X × X : d ( x, y ) < n (cid:27) ⊆ U (cid:27) 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., totallybounded) in the sense of Definition 1.5. Clearly d is totally bounded ifand only if, for every n ∈ ω , there exists a finite set D ⊆ X such that X = S x ∈ D (cid:0) B d ( x, n ) ∩ B d − ( x, n ) (cid:1) . The notions of a totally bounded andprecompact metric are equivalent.We recall that a (Hausdorff) compactification of a space X = h X, τ i isan ordered pair h Y , γ i where Y is a (Hausdorff) compact space and γ : X → Y is a homeomorphic embedding such that γ ( X ) is dense in Y . Acompactification h Y , γ i of X and the space Y are usually denoted by γ X .The underlying set of γ X is denoted by γX . The subspace γX \ X of γ X iscalled the remainder of γ X . A space K is said to be a remainder of X if thereexists a Hausdorff compactification γ X of X such that K is homeomorphic5o γX \ X . For compactifications α X and γ X of X , we write γ X ≤ α X ifthere exists a continuous mapping f : α X → γ X such that f ◦ α = γ . If α X and γ X are Hausdorff compactifications of X such that α X ≤ γ X and γ X ≤ α X , then we write α X ≈ γ X and say that the compactifications α X and γ X are equivalent . If n ∈ N , then a compactification γ X of X is said tobe an n -point compactification of X if γX \ X is an n -element set. Definition 1.7.
Let X = h X, τ i is a non-compact locally compact Hausdorffspace and let K ( X ) be the collection of all compact subsets of X . For anelement ∞ / ∈ X , we define X ( ∞ ) = X ∪ {∞} , τ ( ∞ ) = τ ∪ { X ( ∞ ) \ K : K ∈ K ( X ) } and X ( ∞ ) = h X ( ∞ ) , τ ( ∞ ) i . Then X ( ∞ ) is called the Alexandroff compact-ification of X .For every non-compact locally compact Hausdorff space X , X ( ∞ ) is theunique (up to ≈ ) one-point Hausdorff compactification of X . Therefore, ev-ery one-point Hausdorff compactification of X is called the Alexandroff com-pactification of X . Chandler’s book [3] is a good introduction to Hausdorffcompactifications in ZFC . Basic facts about Hausdorff compactifications in ZF can be found in [23]. If X is a space which has the Čech-Stone compact-ification, then, as usual, β X stands for the Čech-Stone compactification of X . Given a collection { X j : j ∈ J } of sets, for every i ∈ J , we denote by π i the projection π i : Q j ∈ J X j → X i defined by π i ( x ) = x ( i ) for each x ∈ Q j ∈ J X j .If τ j is a topology in X j , then X = Q j ∈ J X j denotes the Tychonoff productof the topological spaces X j = h X j , τ j i with j ∈ J . If X j = X for every j ∈ J , then X J = Q j ∈ J X j . As in [4], for an infinite set J and the unit interval [0 , of R , the cube [0 , J is called the Tychonoff cube . If J is denumerable,then the Tychonoff cube [0 , J is called the Hilbert cube . We denote by the discrete space with the underlying set { , } . If J is an infinite set,the space J is called the Cantor cube .We recall that if Q j ∈ J X j = ∅ , then it is said that the family { X j : j ∈ J } has a choice function, and every element of Q j ∈ J X j is called a choice function of the family { X j : j ∈ J } . A multiple choice function of { X j : j ∈ J } is every6unction f ∈ Q j ∈ J ([ X j ] <ω \ {∅} ) . If J is infinite, a function f is called partial ( multiple ) choice function of { X j : j ∈ J } if there exists an infinite subset I of J such that f is a (multiple) choice function of { X j : j ∈ I } . Given anon-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 choicefunction of A .Let { X j : j ∈ J } be a disjoint family of sets, that is, X i ∩ X j = ∅ for eachpair i, j of distinct elements of J . If τ j is a topology in X j for every j ∈ J ,then L j ∈ J X j denotes the direct sum of the spaces X j = h X j , τ j i with j ∈ J . Definition 1.8. (Cf. [1], [27] and [19].)(i) A space X is said to be Loeb (respectively, weakly Loeb ) if the family ofall non-empty closed subsets of X has a choice function (respectively,a multiple choice function).(ii) If X is a (weakly) Loeb space, then every (multiple) choice function ofthe family of all non-empty closed subsets of X is called a ( weak ) Loebfunction of X .Other topological notions used in this article but not defined here arestandard. They can be found, for instance, in [4] and [30]. AC In this subsection, for readers’ convenience, we define and denote the weakerforms 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 recordedin [9]. Definition 1.9. 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 hasa choice function. 7.
CAC fin ( [9, Form 10]): Every denumerable family of non-empty finitesets has a choice function.6.
WOAC fin ([9, Form 122]): Every well-orderable non-empty family ofnon-empty finite sets has a choice function.7. MC (the Axiom of Multiple Choice, [9, Form 67]): Every non-emptyfamily of non-empty sets has a multiple choice function.8. CMC (the Countable Axiom of Multiple Choice, [9, Form 126]): Everydenumerable family of non-empty sets has a multiple choice function.9.
WoAm ([9, Form 133]): Every infinite set is either well-orderable orhas 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 )( h x, y i ∈ S ) , there exists a ∈ A ω such that: ( ∀ n ∈ ω )( h a ( n ) , a ( n + 1) i ∈ S ) . BPI (the Boolean Prime Ideal Principle, [9, Form 14]): Every Booleanalgebra 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 ω is a remainder ofthe discrete space h D, P ( D ) i . (Cf. [22].)15. INSHC : Every infinite discrete space has a non-scattered Hausdorffcompactification.The form
IDFBI has been introduced and investigated in [22] recently.More comments about
IDFBI are included in Remark 1.22. New facts con-cerning
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 isessentially weaker than
IDFBI is shown in Section 2.8 .3 Some known results
In this subsection, we quote several known results that we refer to in thesequel. Some of the quoted results have been obtained recently, so they canbe unknown to possible readers of this article.
Proposition 1.10. (Cf. [16].) ( ZF ) A topological space X is scattered ifand only if there exists α ∈ ON such that X ( α ) = ∅ . If X is scattered then | X | CB = min { α ∈ ON : X ( α ) = ∅} . Moreover, if X is a non-empty scattered compact space, then | X | CB is asuccessor ordinal. Theorem 1.11. (Cf., e.g., [2].) ( ZF ) Every non-empty dense-in-itself com-pact 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 aquasi-metric on X such that h X, 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 T -space is metrizable.(ii) (Cf. [22].) Every T -space which has a cuf base can be embedded in ametrizable Tychonoff cube and that, it is metrizable.(iii) (Cf. [22].) A T -space X is embeddable in a compact metrizable Ty-chonoff cube if and only if X is embeddable in the Hilbert cube [0 , ω . Several essential applications of Theorem 1.13(ii), especially to the theoryof Hausdorff compactifications in ZF , have been shown in [22] recently. Weshow some other applications of Theorem 1.13(ii) in the forthcoming Sections3 and 4. Theorem 1.14. (Cf. [25].) ( ZF ) ( a ) For every compact Hausdorff, quasi metric space X = h X, d i the fol-lowing are equivalent:(i) X is Loeb; ii) h X, d − i is separable;(iii) X and h X, d − i are both separable;(iv) X is second-countable.In particular, every compact, Hausdorff, quasi-metrizable Loeb space ismetrizable.( b ) CAC implies that every compact, Hausdorff quasi-metrizable space ismetrizable.
Proposition 1.15. (Cf. [22].) ( ZF ) Every weakly Loeb regular space whichadmits a cuf base has a dense cuf set.
Theorem 1.16. (Cf. [27].) ( ZF ) Let κ be an infinite cardinal number ofvon Neumann, { X i : i ∈ κ } be a family of compact spaces, { f i : i ∈ κ } be acollection of functions such that for every i ∈ κ, f i is a Loeb function of X i .Then the Tychonoff product X = Q i ∈ κ X i is compact. Theorem 1.17. (Cf., e.g., [8, Theorem 4.37] and [9, Forms: 14, 14 A, 14J].) ( ZF ) The following statements are equivalent to
BPI :(i) For every non-empty set X , every filter in P ( X ) can be enlarged to anultrafilter 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 impli-cation 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 re-mainder of X ;(b) the Cantor cube ω is a remainder of X ;(c) there exists a family V = {V ni : n ∈ N , i ∈ { , . . . , n }} such that, forevery n ∈ N , the following conditions are satisfied: i) for every i ∈ { , . . . , n } , V ni is a non-empty family of open setsof X such that V ni is stable under finite unions and finite inter-sections, and, for every U ∈ V ni , the set cl X U is non-compact;(ii) for every i ∈ { , . . . , n } and for any U, V ∈ V ni , cl X ( U ) \ V iscompact;(iii) for every pair i, j of distinct elements of { , . . . , n } , for any W ∈V ni and G ∈ V nj , there exist U ∈ V n +12 i − , V ∈ V n +12 i with cl X ( U ∪ V ) \ W compact and cl X (( U ∪ V ) ∩ G ) compact;(iv) if, for every i ∈ { , . . . , n } , V i ∈ V ni , then X \ n S i =1 V i is compact. Theorem 1.20. (Cf. [22].) ( ZF ) For a set D , let D = h D, P ( D ) i . Thenthe following statements hold:(i) If D is a cuf space, then every non-empty second-countable compactHausdorff space is a remainder of a metrizable compactification of D .In particular, all non-empty second countable compact Hausdorff spacesare remainders of metrizable compactifications of N .(ii) If D is weakly Dedekind-infinite, then every non-empty second-count-able 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 remain-ders of metrizable compactifications of N ” is equivalent to M ( C, S ) and,thus, it implies CAC fin .Remark . In ZFC , an archetype of Theorem 1.19 is included in [7, Theo-rem 2.1]; however, in [ ] , Theorem 2.1 of [7] has been shown to be unprovablein ZF . In [22], an infinite set D is called dyadically filterbase infinite if ω is a remainder of the discrete space h D, P ( D ) i . An equivalent purely set-theoretic definition of a dyadically filterbase infinite set is given in [22] and itcan be easily obtained from condition ( c ) of Theorem 1.19 applied to discretespaces. Clearly, IDFBI is equivalent to the sentence “Every infinite set isdyadically filterbase infinite”.
Theorem 1.23. (Cf. [23].) ( ZF ) 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 thediscrete space D = h D, 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 = h D, 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 thesequel, 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, weuse 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 orto 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. Thenotions of boundable and injectively boundable statements can be found in[28], [10, Problem 1 on page 94] and [9, Note 3, page 284]. Every boundablestatement is equivalent to an injectively boundable one (see [28] or [9, Note 3,page 285]). We recommend [10, Chapter 4] as an introduction to permutationmodels. 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 scat-tered spaces; similarly, the class of all iso-dense compact metrizable spacesis 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 inevery model of ZF + BPI , is independent of ZF , does not imply BPI and isstrictly weaker than
IDFBI (see Theorem 2.3). We construct a new permu-tation model to prove that a dyadically filterbase infinite set can be weakly12edekind-finite (see the proof to Theorem 2.6. This solves an open problemposed by us in [22].In Section 3, we prove in ZF that if h X, d i is a quasi-metric T -spacesuch that d is strong and either h X, τ ( d ) i is limit point compact or d − isprecompact, then the space h X, τ ( d ) i is metrizable (see Theorem 3.4). Thisresult and its direct consequence that if h X, d i is compact Hausdorff quasi-metric space such that h X, τ ( d − ) i is iso-dense, then h X, τ ( d ) i is metrizable(see Corollary 3.5) are new applications of Theorem 1.13(ii) and adjuncts toTheorem 1.14. By applying Theorem 1.13, we show in ZF that if h X, d i isan iso-dense metric space such that either d is totally bounded or h X, τ ( d ) i is limit point compact, then h X, τ ( d ) i has a cuf base and can be embeddedin a metrizable Tychonoff cube (see Theorem 3.7).Section 4 concerns equivalents of CAC fin in terms of scattered or iso-dense spaces (see Theorems 4.2 and 4.4). Among our new equivalents of
CAC fin 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 iscountable; (c) every compact metrizable scattered space is countable; (d)every totally bounded, complete scattered metric space is compact. We showthat, in ZF , every compact metrizable cuf space is scattered (see Theorem4.5). We prove that WOAC fin is equivalent to the sentence: for every well-orderable non-empty set S and every family {h X s , d s i : s ∈ S } of compactscattered metric spaces, the product Q s ∈ S h X s , τ ( d s ) i is compact (see Theorem4.7). Several other relevant results are included in Section 4, too.Section 5 concerns the problem of whether it is provable in ZF thatevery non-empty dense-in-itself compact metrizable space contains an infinitecompact scattered subspace. Among other results of Section 4, we showthe following; (a) each of IDI , WoAm and
BPI implies that every infinitecompact first-countable Hausdorff space contains a copy of N ( ∞ ) ; (b) everyinfinite first-countable compact Hausdorff separable space contains a copyof N ( ∞ ) ; (c) every infinite first-countable compact Hausdorff space havingan infinite cuf subset contains a copy of D ( ∞ ) for some infinite discrete cufspace (see Theorem 5.2). We prove that the sentence “every infinite first-countable Hausdorff compact space contains an infinite metrizable compactscattered subspace” implies neither CAC fin nor
IQDI , nor
CMC in ZFA (see Theorem 5.4).Section 6 contains a shortlist of new open problems strictly relevant to13he topic of this paper.
INSHC
Since every isolated point of an open subspace of a topological space X is anisolated 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 itis a Hausdorff compactification of a discrete space. Every iso-dense locallycompact Hausdorff space which satisfies condition (c) of Theorem 1.19 hasan iso-dense non-scattered Hausdorff compactification. In particular, for ev-ery dyadically filterbase infinite set D , the discrete space h D, P ( D ) i has anon-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) metrizablescattered spaces is essentially smaller than the class of all iso-dense (compact)metrizable spaces, and the class of all (compact Hausdorff) scattered spaces isessentially smaller than the class of all (compact Hausdorff) iso-dense spaces.It was proved in [16] that it holds in ZF that every compact Hausdorffscattered space is zero-dimensional, so completely regular. Let us show thata 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 compactHausdorff 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 . ByTheorem 1.23(i), it holds in M that there exists a Hausdorff compactification γ D of a discrete space D such that γD \ D is homeomorphic to Z . Then, in M , γ D is a non-scattered, iso-dense compact Hausdorff space which is notcompletely regular.Let us shed more light on the forms INSHC and
IDFBI . Theorem 2.3. ( ZF ) i) Every compact Hausdorff space is a subspace of a compact Hausdorffiso-dense space.(ii) Every compact second-countable Hausdorff space is a subspace of a com-pact second-countable iso-dense space.(iii) DC → IWDI → IDFBI → INSHC → NAS ;(iv)
BPI → INSHC ;(v)
INSHC IDFBI and
INSHC BPI .Proof.
To show that (i) holds, it suffices to apply Theorem 1.23. It followsdirectly from Theorem 1.20(i) that (ii) holds.(iii) It is known that DC implies IWDI (see, e.eg., [9, pages 326 and339]). It has been noticed in [22] that, by Theorem 1.20(ii),
IWDI implies
IDFBI . The implications
IDFBI → INSHC → NAS can be deduced fromTheorems 1.19 and 1.23(ii).To prove (iv), we assume
BPI and consider any infinite set D . Let D = h D, P ( D ) i . In the light of Theorem 1.17(i) and [23, Theorem 3.27], itfollows from BPI that there exists the Čech-Stone compactification β D of D .Suppose that βD \ D has an isolated point y . Then there exist disjoint opensubsets U, V of β D such that y ∈ U , ( βD \ D ) \ { y } ⊆ V and U ∩ V = ∅ .Then the subspace U ∪ { y } of β D is the Čech-Stone compactification ofthe subspace U ∩ D of D . It follows from Theorem 1.23(ii) that U ∩ D isamorphous but this is impossible because BPI implies
NAS by Theorem1.18. The contradiction obtained shows that βD \ D is dense-in-itself, so β D 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 whichthe conjunction INSHC ∧ ¬
IDFBI is true. Hence
INSHC IDFBI .To prove
INSHC BPI , let us use the Feferman’s forcing model M in [9]. It is known that DC ∧ ¬ BPI is true in M (see [9, page 148]). Tocomplete the proof, it suffices to notice that it follows from (iii) that INSHC is also true in M . Remark . ( 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 holdsin ZF that NAS is equivalent to the following statements:15i) Every infinite discrete space has a two-point Hausdorff compactifica-tion.(ii) For every natural number n , every infinite discrete space has an n -pointHausdorff compactification.( c ) In view of Theorem 2.3(iii), it holds in ZF that INSHC followsfrom 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
CAC fin (see, e.g., [9, pages 325 and354]). Since
BPI implies
INSHC by Theorem 2.3(ii), it is worth noticingthat the conjunction
INSHC ∧ ¬
CAC fin has a ZF -model. To see this,let us notice that, in the Second Fraenkel Model N in [9], the conjunction IWDI ∧ ¬
CAC fin is true (see [9, page 179]). Since
IWDI ∧ ¬
CAC fin is aconjunction of two injectively boundable statements and it has a permutationmodel, it also has a ZF -model by Pincus’ transfer theorems. This, togetherwith Theorem 2.3(iii), implies that INSHC ∧ ¬
CAC fin 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 . (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 bedyadically filterbase infinite? (See [22, Problem (6) of Section 6].)In [22, the proof to Theorem 5.14], a permutation model has been con-structed in which there exists a weakly Dedekind-finite discrete space whichhas a remainder homeomorphic to N ( ∞ ) . Now, we are in a position to solveProblem 2.5(ii) (that is, Problem (6) from Section 6 in [22]) by the followingtheorem: Theorem 2.6.
It is relatively consistent with ZF that there exists a dyadi-cally filterbase infinite set which is weakly Dedekind-finite.Proof. Let Φ be the following statement: “There exists a dyadically filterbaseinfinite set which is weakly Dedekind-finite”.Since Φ is a boundable statement, by the Jech–Sochor First EmbeddingTheorem (see [10, Theorem 6.1]), it suffices to prove that Φ has a permutation16odel. To this end, let us modify the model constructed in [22, the proof toTheorem 5.14] to get a new permutation model N in which Φ is true.In what follows, for an arbitrary non-empty set S and every permutation ψ of S , we denote by supp( ψ ) the support of ψ , that is, supp( ψ ) = { x ∈ S : ψ ( x ) = x } .We start with a model M of ZFA + AC with a denumerable set A ofatoms such that A has a denumerable partition A = { A i : i ∈ ω } into infinitesets. In M , we let B = {B ni : n ∈ N , i ∈ { , , . . . , n }} be a family with the following two properties:( a ) For n = 1 , {B , B } is a partition of A = { A i : i ∈ ω } into two infinitesets.( b ) For every n ∈ N and for every i ∈ { , , . . . , n } , {B n +12 i − , B n +12 i } is apartition of B ni 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 followingtwo properties:( c ) φ moves only finitely many elements of A .( d ) ( ∀ i ∈ ω )( ∃ j ∈ ω )( ∃ F ∈ [ A j ] <ω )( φ [supp( φ ↾ A i )] = F ) .For every n ∈ N and for every i ∈ { , , . . . , n } , we let Q ni = { [ { φ ( Z ) : Z ∈ B ni } : φ ∈ G} . We also let Q = [ {Q ni : n ∈ N , i ∈ { , , . . . , n }} . For every E ∈ [ Q ] <ω , we let G E = { φ ∈ G : ∀ Q ∈ E ( φ ( Q ) = Q ) } . Then G E is a subgroup of G . Furthermore, since for all E, E ′ ∈ [ Q ] <ω , G E ∪ E ′ ⊆ G E ∩ G E ′ , the collection {G E : E ∈ [ Q ] <ω } is a base for a filter in theset of all subgroups of G . Let F be the filter of subgroups of G generated by17 G E : E ∈ [ Q ] <ω } . To check that F is a normal filter on G , we need to showthat F has the following two properties:(1) ∀ a ∈ A ( { π ∈ G : π ( a ) = a } ∈ F ) and(2) ( ∀ π ∈ G )( ∀ H ∈ F )( πHπ − ∈ F ) . To argue for (1), let a ∈ A . Since A is a partition of A in M , there existsa unique i ∈ ω such that a ∈ A i . Since the set {B , B } is a partition of A ,either A i ∈ B or A i ∈ B . Suppose that A i ∈ B (the argument is similar if A i ∈ B ). Pick an A j ∈ B and an a ′ ∈ A j . Let φ ∈ G be the transposition ( a, a ′ ) (i.e. φ interchanges a and a ′ and fixes all other atoms). Then [ { φ ( Z ) : Z ∈ B } = ( [ ( B \ { A j } )) ∪ (( A j \ { a ′ } ) ∪ { a } ) . Let E = { S B , S { φ ( Z ) : Z ∈ B }} . Then, E ∈ [ Q ∪ Q ] <ω ⊂ [ Q ] <ω ,so E ∈ [ Q ] <ω . Furthermore, G E ⊆ { π ∈ G : π ( a ) = a } . Indeed, let π ∈ G E .Towards a contradiction, assume that π ( a ) = b for some b ∈ A \ { a } . Since a ∈ S B and π fixes S B , it follows that b = π ( a ) ∈ π ( S B ) = S B . Butthen, since π ∈ G E , we have the following: a ∈ [ { φ ( Z ) : Z ∈ B }} → π ( a ) ∈ π ( [ { φ ( Z ) : Z ∈ B }} ) → b ∈ [ { φ ( Z ) : Z ∈ B }} = ( [ ( B \ { A j } )) ∪ (( A j \ { a ′ } ) ∪ { a } ) , which is a contradiction. Therefore, (1) holds.To argue for (2), let π ∈ G and H ∈ F . There exists E ∈ [ Q ] <ω suchthat G E ⊆ H . By the definition of Q , we have π [ E ] ∈ [ Q ] <ω . We assert that G π [ E ] ⊆ πHπ − . Let ρ ∈ G π [ E ] . For every T ∈ E we have the following: ρ ( πT ) = πT → π − ρπ ( T ) = T ; Hence, since G E ⊆ H , we have: π − ρπ ∈ G E → ρ ∈ π G E π − ⊆ πHπ − . ρ ∈ πHπ − . Since ρ is an arbitrary element of G π [ E ] , we concludethat G π [ E ] ⊆ πHπ − . Thus, πHπ − ∈ F , so (2) holds. This completes theproof that F is a normal filter on G .Let N be the permutation model determined by M , G and F . We saythat an element x ∈ N has support E ∈ [ Q ] <ω if, for all φ ∈ G E , 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 discretespace h A, ( P ( A )) N i satisfies condition ( c ) of Theorem 1.19. To this aim, forevery n ∈ N and for every i ∈ { , , . . . , n } , we let V ni = n\ R : R ∈ [ Q ni ] <ω \ {∅} o ∪ n[ C : C ∈ [ Q ni ] <ω \ {∅} o , and we also let V = {V ni : n ∈ N , i ∈ { , , . . . , n }} . We notice that any permutation of A in G fixes V pointwise. Hence, V ∈ N and, moreover, V is well-orderable in the model N (see [10, page 47]). Since V is denumerable in the ground model M , it follows that V is also denumerablein N . Furthermore, in view of the properties of the family B and of theelements of G , and the construction of V , it is easy to see that, if we put X = A and X = h A, ( P ( A )) N i , then V has properties (i)-(iv) of condition c ofTheorem 1.19. This, together with Theorem 1.19, proves that A is dyadicallyfilterbase infinite in the model N . To complete the proof, it remains to showthat A is weakly Dedefind-finite in N .By way of contradiction, we assume that A is weakly Dedekind-infinitein N . Thus, it holds in N that there exists a denumerable disjoint family U = { U n : n ∈ ω } of ( P ( A )) N . Let E ∈ [ Q ] <ω be a support of U n for all n ∈ ω . It is not hard to verify now that there exist a pair k, m of distinctelements of ω and a pair x, y of atoms, such that x ∈ U k and y ∈ U m . Thetransposition ψ = ( x, y ) from the group G is an element of G E . It followsthat ψ ( U k ) = U k , and so y = ψ ( x ) ∈ ψ ( U k ) = U k . This is impossible because y ∈ U m and U k ∩ U m = ∅ . The contradiction obtained shows that A is weaklyDedekind-finite in N .The model constructed in [22, the proof to Theorem 5.14] is different fromthat we have just introduced in the proof to Theorem 2.6. By Theorem 5.15of [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 notknow if NAS holds N . Let us notice that, for every i ∈ ω , no E ∈ [ Q ] <ω A i and, in consequence, A i / ∈ N . This is why we cannot mimicthe proof to Theorem 5.15 in [22] to show that NAS fails in N . We also donot know if INSHC holds in N . The following theorem is of significant importance because of its consequencesthat 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 − is precompact or the space X = h X, τ ( d ) i is limit point compact. Then Iso τ ( d − ) ( X ) is a cuf set.Proof. Let D = Iso τ ( d − ) ( X ) . For every x ∈ D , let n x = min (cid:26) n ∈ N : B d − (cid:18) x, n (cid:19) = { x } (cid:27) . For every n ∈ N , let A n = { x ∈ D : n = n x } . Suppose that n ∈ N is suchthat A n is infinite. Let us show that there exist x ∈ X and x ∈ A n suchthat x = x and x ∈ B d (cid:16) x , n (cid:17) . If x , x are such points, we notice that d − ( x , x ) = d ( x , x ) < n , so x ∈ B d − (cid:16) x , n (cid:17) which is impossible bythe definition of A n .If X is limit point compact, there exists an accumulation point of A n in X . In this case, for a fixed accumulation point x of A n , we can fix x ∈ A n such that x = x and x ∈ B d (cid:16) x , n (cid:17) . Assuming that d − isprecompact, we can fix a finite set F ⊆ X such that X = S x ∈ F B d (cid:16) x, n (cid:17) and, since A n is infinite, we can fix x ∈ F and x ∈ A n such that x = x and x ∈ B d (cid:16) x , n (cid:17) . Hence, the assumption that A n is infinite leads to acontradiction. Therefore, D = S n ∈ N A n is a cuf set. Corollary 3.2. ( ZF ) Let X = h X, d i be a metric space which is either limitpoint compact or totally bounded. Then Iso( X ) is a cuf set. Furthermore, if Iso( X ) is infinite, then X is quasi Dedekind-infinite. roof. That
Iso( X ) is a cuf set follows from Theorem 3.1. The second asser-tion is straightforward. Remark . Let X be a compact metrizable space. Then, using Proposition1.24(ii), we may deduce that Iso( X ) is a cuf set. Namely, suppose that Iso( X ) is infinite. Let Y = cl X (Iso( X )) . Then Y is a metrizable compactification ofthe 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 com-pact 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 h X, τ ( d ) i is a T -space. Then the following conditions are satisfied:(i) if h X, τ ( d − ) i has a dense cuf set, then h X, τ ( d ) i is metrizable;(ii) if h X, τ ( d − ) i is iso-dense and either h X, τ ( d ) i is limit point compactor d − is precompact, then the space h X, τ ( d ) i is metrizable.Proof. (i) Assume that A = S n ∈ ω A n is a dense set in h X, τ ( d − ) i such that,for every n ∈ ω , A n is a non-empty finite set. For n, m ∈ ω , we define B m,n = (cid:8) B d (cid:0) x, m +1 (cid:1) : x ∈ A n (cid:9) . Since d is strong, in much the same way, asin the proof to Theorem 4.6 in [25], one can show that B = S n,m ∈ ω B n,m is abase of h X, τ ( d ) i . Since B is a cuf set, the space h X, τ ( d ) i is metrizable byTheorem 1.13(ii).(ii) Now, we assume that h X, τ ( d − ) i is iso-dense and either h X, τ ( d ) i islimit point compact or d − is precompact. Let E = Iso τ ( d − ) ( X ) . Then E is dense in h X, τ ( d − ) i . By Theorem 3.1, the set E is a cuf set. Hence, toconclude the proof, it suffices to apply (i). Corollary 3.5. ( ZF ) Let h X, d i be a compact Hausdorff quasi-metric spacesuch that h X, τ ( d − ) i is iso-dense. Then h X, τ ( d ) i is metrizable.Proof. This follows immediately from Proposition 1.12 and Theorem 3.4.21 emark . In Corollary 3.5, we cannot omit the assumption that h X, τ ( d ) i is Hausdorff. Indeed, there is a quasi-metric d on ω such that τ ( d ) is thecofinite topology on ω and τ ( d − ) is the discrete topology on ω (see [25]).Then d is a strong quasi-metric such that h ω, τ ( d ) i is a compact T -spacewhich is not metrizable. Theorem 3.7. ( ZF ) Let h X, d i be an iso-dense metric space such that either d is totally bounded or h X, τ ( d ) i is limit point compact. Then h X, τ ( d ) i hasa cuf base and can be embedded in a metrizable Tychonoff cube.Proof. It follows from the proof to Theorem 3.4 that h X, τ ( d ) i has a cufbase. Since h X, τ ( d ) i is a T -space, to conclude the proof, it suffices to applyTheorem 1.13(ii). CAC f in via iso-dense metrizable spaces
It is known that it holds in
ZFC that every iso-dense compact metrizablespace is separable and every scattered compact metrizable space is countable.In this section, we show that the situation of compact iso-dense metrizablespaces 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 abase of X . Then X embeds in [0 , B×B . If X = h X, d i is a metric space and Y is a topological space, then we saythat X embeds in Y if the space h X, τ ( d ) i embeds in Y .The following theorem is a characterization of CAC fin in terms of iso-dense (limit point) compact metrizable spaces and in terms of iso-dense to-tally bounded metric spaces.
Theorem 4.2. ( ZF ) The following conditions are all equivalent:(i)
CAC fin ;(ii) for every iso-dense metric space X , if X is either limit point compactor totally bounded, then X is separable;(iii) for every iso-dense metric space X , if X is either limit point compactor totally bounded, then X embeds in the Hilbert cube [0 , N ; iv) for every iso-dense metric space X , if X is either limit point compactor totally bounded, then | Iso( X ) | ≤ | R | ;(v) for every iso-dense metric space X , if X is either limit point compactor totally bounded, then the set Iso( X ) is countable.In (ii)-(v), the term “iso-dense” can be replaced with “scattered”.Proof. Let X = h X, d i be an iso-dense (respectively, scattered) metric spacesuch that X is either limit point compact or totally bounded. By Corollary3.2, the set Iso( X ) is a cuf set. Hence, it follows from CAC fin that
Iso( X ) iscountable. In consequence, (i) implies (ii). Since every separable metrizablespace 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 h X, τ ( d ) i is homeomor-phic to a subspace of [0 , N . Then Iso( X ) is equipotent to a subset of [0 , N .Since it holds in ZF that [0 , 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 countableunion of finite well-ordered sets. Hence, if Iso( X ) is equipotent to a subsetof R , then Iso( X ) is countable as a set equipotent to a cuf set contained in R . This shows that (iv) implies (v).Finally, suppose that CAC fin fails. Then there exists an uncountable dis-crete cuf space D . It follows from Proposition 1.24(i) that the Alexandroffcompactification D ( ∞ ) of D is metrizable. Since D ( ∞ ) is an iso-dense com-pact mertizable space whose set of all isolated points D ( ∞ ) is uncountable,(v) fails if CAC fin 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 = h X, d i be an infinite totally bounded metric space. Let α = | X | CB . Then X = [ γ ∈ α Iso( X ( γ ) ) ∪ X ( α ) . γ ∈ α and every x ∈ Iso( X ( γ ) ) , let n ( x, γ ) = min (cid:26) n ∈ N : B d (cid:18) x, n (cid:19) ∩ X ( γ ) = { x } (cid:27) . For every γ ∈ α and every n ∈ N , let A γ,n = { x ∈ Iso( X ( γ ) ) : n ( x, γ ) = n } . We have already shown in the proof to Theorem 3.1 that, for every γ ∈ α and every n ∈ N , the set A γ,n is finite and Iso( X ( γ ) ) = S n ∈ N A γ,n .(i) Suppose that α is uncountable. For every n ∈ N , let B n = { γ ∈ α : A γ,n = ∅} . Since α is supposed to be uncountable, there exists n ∈ N such that B n isinfinite. We fix such an n and put U = (cid:26) B d (cid:18) x, n (cid:19) : x ∈ X (cid:27) . By the total boundedness of d , the open cover U of X has a finite sub-cover. Hence, there exists a non-empty finite subset F of X such that X = S x ∈ F B d (cid:16) x, n (cid:17) . Since B n is infinite, there exist γ , γ ∈ B n andelements x ∈ F , x ∈ A γ ,n ∩ B d (cid:16) x , n (cid:17) and x ∈ A γ ,n ∩ B d (cid:16) x , n (cid:17) ,such that x = x . We may assume that γ ≤ γ . Then X ( γ ) ⊆ X ( γ ) and itfollows from the definition of A γ ,n that d ( x , x ) ≥ n . On the other hand,since x , x ∈ B d ( x , n ) , we have d ( x , x ) ≤ n < n . The contradictionobtained proves that α is countable.(ii) Now, suppose that the space X is also scattered. Then it follows fromProposition 1.10 that X ( α ) = ∅ . Hence X = S { A γ,n : γ ∈ α and n ∈ N } .Since α is countable, the set α × N is countable. This implies that the family { A γ,n : γ ∈ α and n ∈ N } is also countable. We have already shown that,for every γ ∈ α and every n ∈ N , the set A γ,n is finite. Hence X is a cuf set.It follows immediately from (ii) and Theorem 3.7 that (iii) holds. Theorem 4.4. ( ZF ) The following conditions are all equivalent:(i)
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
CAC fin implies that all cuf sets are countable, it follows fromTheorem 4.3 that (i) implies (ii) and (iii). It is provable in ZF that everytotally bounded, complete countable metric space is compact. Hence, in thelight of Theorem 4.3, (i) implies (iv).Assume that CAC fin is false. Then there exists a family { A n : n ∈ ω } of non-empty pairwise disjoint finite sets such that the set D = S n ∈ ω A n isDedekind-finite (see [9, Form [10M] ]). Let D = h D, P ( D ) i . By Proposition1.24(i), the space D ( ∞ ) is metrizable. Let d be any metric which induces thetopology of D ( ∞ ) . Since D ( ∞ ) is compact, the metric d is totally bounded.Moreover, D ( ∞ ) is scattered but uncountable. For ρ = d ↾ D × D , the metricspace h D, ρ i is also totally bounded. Since D is Dedekind-finite and h D, ρ i is discrete, the metric ρ is complete. Clearly, h D, ρ i is not compact. All thistaken together completes the proof. Theorem 4.5. ( ZF ) Every compact metrizable cuf space is scattered. Inparticular, every compact metrizable countable space is scattered.Proof.
Our first step is to prove that every non-empty compact metrizablecuf space has an isolated point. To this aim, suppose that X = h X, d i is acompact metric space such that the set X is a non-empty cuf set. Towardsa contradiction, suppose that X is dense-in-itself. We fix a partition { X n : n ∈ ω } of X into non-empty finite sets.Let S = S {{ , } n : n ∈ N } . For n ∈ N , s ∈ { , } n and t ∈ { , } ,let s a t ∈ { , } 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 { B s : s ∈ S } such that, for every s ∈ S , the following conditionsare satisfied:(1) B s is a non-empty open subset of X ;(2) for every t ∈ { , } , B s a t ⊆ B s ;(3) cl X ( B s a ) ∩ cl X ( B s a ) = ∅ .To start the induction, for n = 1 = { } and every s ∈ { , } , we define: B s = [ (cid:26) B d (cid:18) x, d ( X , X )3 (cid:19) : x ∈ X s (0) (cid:27) . n ∈ N is such that, for every s ∈ n S i =1 { , } i , we havedefined a non-empty open subset B s of X . For an arbitrary s ∈ { , } n +1 , weconsider the set B s ↾ n . We put n s = min { m ∈ ω : X m ∩ B s ↾ n = ∅} . Since X isdense-in-itself, we have ∅ 6 = { m ∈ ω : X m ∩ ( B s ↾ n \ X n s ) = ∅} , so we can define k s = min { m ∈ ω : X m ∩ ( B s ↾ n \ X n s ) = ∅} . Now, we put Y s, = X n s ∩ B s ↾ n and Y s, = X k s ∩ ( B s ↾ n \ X n s ) . We define B s = [ (cid:26) B d (cid:18) y, d ( Y s, , Y s, )3 (cid:19) : y ∈ Y s,s ( n ) (cid:27) . In this way, we have inductively defined the required family { B s : s ∈ S } .We notice that it follows from (2) that, for every f ∈ { , } ω and n ∈ N , ∅ 6 = cl X ( B f ↾ ( n +1) ) ⊆ cl X ( B f ↾ n ) . Thus, by the compactness of X , for every f ∈ { , } ω , the set M f = \ { cl X ( B f ↾ n ) : n ∈ N } is non-empty. For every f ∈ { , } ω , let m f = min { n ∈ ω : X n ∩ M f = ∅} .We define a mapping F : { , } ω → S {P ( X n ) : n ∈ ω } by putting: F ( f ) = X m f ∩ M f for every f ∈ { , } ω . It follows from (3) that F is an injection. In consequence, the set { , } ω is equipotent to a subset of the cuf set S {P ( X n ) : n ∈ ω } . But this isimpossible because { , } ω , being equipotent to R , is not a cuf set. Thecontradiction obtained shows that every non-empty compact metrizable cufspace 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 isolatedpoint. Hence X cannot contain non-empty dense-in-itself subspaces. Thisimplies 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 spacewhich has a cuf base. If X is weakly Loeb, then | R | ≤ | [ X ] <ω | . If X is Loeb,then | R | ≤ | X | . roof. Without loss of generality, we may assume that X is dense-in-itselfbecause we can replace X with its non-empty dense-in-itself compact sub-space. By Theorem 1.13(ii), X is metrizable. It is known that every compactmetrizable Loeb space is second-countable (see, e.g., [18]). Hence, if X isLoeb, 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 fromProposition 1.15 that X has a dense cuf set. Since X is non-empty anddense-in-itself, every dense subset of X is infinite. Therefore, we can fix adisjoint family { X n : n ∈ ω } of non-empty finite subsets of X such that theset S { X n : n ∈ ω } is dense in X . Mimicking the proof to Theorem 4.5, wecan define an injection F : { , } ω → P ( X ) such that, for every f ∈ { , } ω ,the set M f = F ( f ) is a non-empty closed subset of X and, for every pair f, g of distinct functions from { , } ω , M f ∩ M g = ∅ . Let ψ be a weak Loebfunction for X . Then ψ ◦ F is an injection from { , } ω into [ X ] <ω . Hence | R | = |{ , } 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 {h X i , τ i i : i ∈ I } of de-numerable metrizable compact spaces, the family { X i : i ∈ I } has amultiple choice function.( b ) For a non-zero von Neumann ordinal α , let { X γ : γ ∈ α } be a family ofpairwise disjoint non-empty countable, compact metrizable spaces. Thenthe direct sum X = L γ ∈ α X γ is weakly Loeb.( c ) The following conditions are all equivalent:(i) WOAC fin ;(ii) for every well-orderable set S and every family {h X s , d s i : s ∈ S } of scattered totally bounded metric spaces, the union S s ∈ S X s is well-orderable.(iii) for every well-orderable non-empty set S and every family {h X s , d s i : s ∈ S } of compact scattered metric spaces, the product Q s ∈ S h X s , τ ( d s ) i is compact.Proof. ( a ) Let X i = h X i , τ i i be a denumerable metrizable compact space forevery i ∈ I with I = ∅ . For every i ∈ I , let α i = | X i | CB . We fix i ∈ I and27bserve that, since X i is scattered by Theorem 4.5, it follows from Proposition1.10 that X ( α i ) i = ∅ and α i is a successor ordinal. Let β i ∈ ON be such that α i = β i + 1 . Then X ( β i ) i = Iso( X ( β i ) i ) . If the set Iso( X ( β i ) i ) were infinite, itwould have an accumulation point in X ( β i ) i by the compactness of X i . Hence Iso( X ( β i ) i ) is a non-empty finite set. In consequence, by assigning to any i ∈ I the set Iso( X ( β i ) i ) , we obtain a multiple choice function of { X i : i ∈ I } .( b ) Let us consider the family F of all non-empty closed sets of X . Bythe proof of ( a ), there exists a family { f γ : γ ∈ α } such that, for every γ ∈ α , f γ is a weak Loeb function of X γ . For every F ∈ F , let γ ( F ) = min { γ ∈ α : F ∩ X γ = ∅} and let f ( F ) = f γ ( F ) ( F ∩ X γ ) . Then f is a weak Loeb functionof X .( c ) (i) → (ii) Let us assume WOAC fin . Suppose that S is a well-orderablenon-empty set and, for every s ∈ S , X s = h X s , d s i is a non-empty scatteredtotally bounded metric space. To prove that X = S s ∈ S X s is well-orderable,without loss of generality, we may assume that S = α for some non-zero vonNeumann ordinal α , and X i ∩ X j = ∅ for every pair i, j of distinct elementsof α . In much the same way, as in the proof to Theorem 4.3, we can definea family { A i,n : i ∈ α, n ∈ N } of non-empty finite sets such that, for every i ∈ α , X i = S n ∈ N A i,n . Now, we can easily define a family { M i : i ∈ α } of subsets of N and a family { F i,n : i ∈ α, n ∈ M i } of pairwise disjointnon-empty finite sets such that, for every i ∈ α , X i = S n ∈ M i F i,n . The set J = {h i, n i : i ∈ α, n ∈ M i } is well-orderable, so we can fix a von Neumannordinal number γ and a bijection h : γ → J . For every j ∈ γ , let n ( j ) ∈ ω be equipotent to F h ( j ) , and let B j = { f ∈ F n ( j ) h ( j ) : f is a bijection } . By WOAC fin , there exists ψ ∈ Q j ∈ γ B j . Now, we can define a well-ordering ≤ on X = S j ∈ γ F h ( j ) as follows: for i, j ∈ γ , x ∈ F h ( i ) , y ∈ F h ( j ) , we put x ≤ y ifeither i ∈ j or i = j and ψ ( i ) − ( x ) ⊆ ψ ( i ) − ( y ) . Hence (i) implies (ii)(ii) → (iii) Let S be a non-empty well-orderable set and, for every s ∈ S ,let h X s , d s i be a compact scattered metric space. Clearly, we may assumethat S is a von Neumann cardinal. We notice that if X = S s ∈ S X s is well-orderable, then we can define a family { f s : s ∈ S } such that, for every s ∈ S , f s is a Loeb function of X s = h X s , τ ( d s ) i and, therefore, by Theorem 1.16,the product Q s ∈ S X s is compact. Hence (ii) implies (iii).28iv) → (i) Now, let S be a well-orderable set and let { A s : s ∈ S } bea family of non-empty finite sets. Take an element ∞ / ∈ S s ∈ S A s and put Y s = A s ∪ {∞} for s ∈ S . Let ρ s be the discrete metric on Y s and let Y s = h Y s , P ( Y s ) i for every s ∈ S . Assuming (iii), we obtain that the space Y = Q s ∈ S Y s is compact. In much the same way, as in the proof of Kelley’stheorem that the Tychonoff Product Theorem implies AC (see [11]), one canshow that the compactnes of Y implies that Q s ∈ S A s = ∅ . Hence (iii) implies(i). 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 containsan infinite compact scattered subspace. Unfortunately, we are still unableto give a satisfactiory solution to this problem. Certainly, N ( ∞ ) or, moregenerally, for every infinite discrete cuf space D , D ( ∞ ) is a simple exampleof an infinite compact metrizable scattered space. So, let us search for condi-tions 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 simpleproposition: Proposition 5.1. ( ZF ) Let X be a non-discrete first-countable Loeb T -space. Then X contains a copy of N ( ∞ ) . In particular, every non-discretemetrizable Loeb space contains a copy of N ( ∞ ) .Proof. Let x be an accumulation point of X and let f be a Loeb functionof X . Since X is a first-countable T -space and x is an accumulation pointof X , there exists a base { U n : n ∈ N } of open neighborhoods of x suchthat cl X ( U n +1 ) ⊂ U n for every n ∈ N . Let x n = f ( cl X ( U n ) \ U n +1 ) for every n ∈ N . Then the subspace { x n : n ∈ ω } of X is a copy of N ( ∞ ) . Theorem 5.2. ( ZF ) (i) IQDI implies that every infinite compact first-countable Hausdorff spacecontains a copy of D ( ∞ ) for some infinite discrete cuf space D . 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 containsa copy of N ( ∞ ) . Every infinite first-countable compact Hausdorff spacehaving an infinite cuf subset contains a copy of D ( ∞ ) for some infinitediscrete cuf space.Proof. Let X = h X, τ i be an infinite compact first-countable Hausdorff space.For a point x ∈ X , let { U n : n ∈ ω } be a base of neighborhoods of x suchthat cl X ( U n +1 ) ⊆ U n for every n ∈ ω .(i) Assuming IQDI , we can fix a disjoint family { F n : n ∈ ω } of non-empty finite subsets of X . Since X is compact, the set F = S n ∈ ω F n hasan accumulation point. Let x be an accumulation point of F . We mayassume that x / ∈ F . Let n = m = min { n ∈ ω : U ∩ F n = ∅} , M = ω and Y = F n ∩ U . Since X is a T -space, Y is a finite set and x / ∈ Y ,the set M = { n ∈ ω : U n ∩ Y = ∅} is non-empty. Let m = min M , n = min { n ∈ ω : U m ∩ F n = ∅} and Y = U m ∩ F n . Suppose that k ∈ ω \ { } is such that we have already defined n k , m k ∈ ω such that Y k = U m k ∩ F n k = ∅ . We put M k +1 = { n ∈ ω : U n ∩ Y k = ∅} , m k +1 = min M k +1 , n k +1 = min { n ∈ ω : U m k +1 ∩ F n = ∅} and Y k +1 = U m k +1 ∩ F n k +1 . Thisterminates our inductive definition. Let D = S k ∈ ω Y k and Y = D ∪ { x } .Then D is a discrete cuf subspace of X , the subspace Y of X is compact and x 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 { F n : n ∈ ω } of singletons of X and in much the same way, as in theproof 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 T -space iswell-orderable. Hence, WoAm implies that X contains a copy of N ( ∞ ) .Now, let us assume BPI . Let x be an accumulation point of X . Withoutloss of generality, we may aassume that cl X ( U n +1 ) = U n for every n ∈ ω . Let G n = { x } ∪ ( cl X ( U n ) \ U n +1 ) for every n ∈ ω . Then the subspaces G n of X are compact. By Theorem 1.17(ii), the product G = Q n ∈ ω G n is compact.Therefore, since the family G = { π − n ( cl X ( U n ) \ U n +1 ) : n ∈ ω } has the finiteintersection property and consists of closed subsets of the compact space G ,there exists f ∈ T n ∈ ω π − n ( cl X ( U n ) \ U n +1 ) . Then the subspace { x } ∪ { f (2 n ) : n ∈ ω } of X is a copy of N ( ∞ ) . This completes the proof to (ii).30iii) 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 compactHausdorff first-countable space contains an infinite metrizable compactscattered subspace.(ii) Every infinite compact Hausdorff first-countable space which has an in-finite cuf subset contains an infinite compact metrizable scattered sub-space.
Let us recall a few known facts about the following permutation modelsin [9]: the Basic Fraenkel Model N , the Second Fraenkel Model N and theMostowski Linearly Ordered Model N .It is known that WoAm is true in N , IWDI (and hence the stronger
IQDI , which is implied by
CMC ) is false in both N and N , CAC fin istrue in N but it is false in N , MC is true in N , and BPI (and hence
CAC fin ) is true in N . All this, taken together with Theorem 5.2 and itsproof, implies the following theorem: Theorem 5.4. (i) It is true in N that every first-countable limit pointcompact T -space is well-orderable.(ii) The sentence “every infinite first-countable Hausdorff compact spacecontains a copy of N ( ∞ ) ” is true in N and in N .(iii) The sentence “every infinite first-countable Hausdorff compact spacecontains an infinite metrizable compact scattered subspace” is true in N .(iv) The sentence “every infinite first-countable Hausdorff compact spacecontains an infinite metrizable compact scattered subspace” implies nei-ther CAC fin nor
IQDI , nor
CMC in ZFA .Remark . 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 “everyinfinite compact metrizable space contains an infinite compact scattered sub-space” does not imply IQDI in ZF . This can be also deduced from Theorem5.4.(ii) Suppose that { A n : n ∈ ω } is a disjoint family of non-empty finitesets which does not have any partial choice function. Let D = S n ∈ ω A n and31 = h D, P ( D ) i . Then D ( ∞ ) is a compact metrizable scattered space whichdoes 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 compactscattered subspace if and only if X contains a copy of D ( ∞ ) for some infinitediscrete cuf space D .Proof. ( → ) Suppose that X is a metrizable space which has an infinite com-pact 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 discretecuf space D . ( ← ) This is trivial because the one point Hausdorff compactification ofan infinite discrete space is a scattered space.
Problem . Find, if possible, a ZF -model for NAS ∧ ¬
INSHC . Problem . Find, if possible, a ZF -model for INSHC ∧ ¬
BPI ∧ ¬
IDFBI . Problem . Make a list of the forms from [9] that are true in the model N constructed in the proof to Theorem 2.6. Problem . Is it provable in ZF that every non-empty dense-in-itself com-pact metrizable space contains an infinite compact scattered subspace? Problem . 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