On iso-dense and scattered spaces in ZF Kyriakos Keremedis, Eleftherios Tachtsis and Eliza Wajch Department of Mathematics, University of the Aegean Karlovassi, Samos 83200, Greece [email protected] Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece [email protected] Institute of Mathematics Faculty of Exact and Natural Sciences Siedlce University of Natural Sciences and Humanities ul. 3 Maja 54, 08-110 Siedlce, Poland [email protected] January 11, 2021 Abstract A topological space is iso-dense if it has a dense set of isolated points. 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 of arXiv:2101.02825v1 [math.GN] 8 Jan 2021 choice, 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 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 equiv- alent to the countable axiom of choice for families of finite sets. Results concerning the problem of whether it is provable in ZF that every non-discrete compact metrizable space contains an infinite compact scattered subspace are also included. 1 Mathematics Subject Classification (2010): 03E25, 03E35, 54E35, 54G12, 54D35 Keywords: Weak forms of the Axiom of Choice, scattered space, iso- dense space, (quasi-)metric space 1 Preliminaries 1.1 Set-theoretic framework and preliminary definitions 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 also refer to permutation models of ZFA (cf. [10] and [9]). In this article, we are concerned mainly with iso-dense and scattered spaces in ZF, defined as follows: Definition 1.1. A topological space X is called: (i) iso-dense if the set Iso(X) of all isolated points of X is dense in X; (ii) scattered or dispersed if, for every non-empty subspace Y of X, ∅= 6 Iso(Y ). Before we pass to the main body of the article, let us establish notation and recall some known definitions in this subsection, make a list of weaker forms of AC in Subsection 1.2, and recall several known results for future references in Subsection 1.3. The content of the article is described in brief in Subsection 1.4. Our main new results are included in Sections 2-4. We denote by ON the class of all (von Neumann) ordinal numbers. The first infinite ordinal number is denoted by ω. Then N = ω \{0}. If X is a set, the power set of X is denoted by P(X), and the set of all finite subsets of X is denoted by [X]<ω. A quasi-metric on a set X is a function d : X × X → [0, +∞) such that, for all x, y, z ∈ X, d(x, y) ≤ d(x, z)+ d(z,y) and d(x, y)=0 if and only if x = y (cf. [5], [12], [26], [31]). If a quasi-metric d on X is such that d(x, y)= d(y, x) for all x, y ∈ X, then d is a metric. A (quasi-)metric space is an ordered pair hX,di where X is a set and d is a (quasi-)metric on X. Let d be a quasi-metric on X. The conjugate of d is the quasi-metric d−1 defined by: d−1(x, y)= d(y, x) for x, y ∈ X. 2 The metric d⋆ associated with d is defined by: d⋆(x, y) = max{d(x, y),d(y, x)} for x, y ∈ X. Clearly, d is a metric if and only if d = d−1 = d⋆. The d-ball with centre x ∈ X and radius r ∈ (0, +∞) is the set Bd(x, r)= {y ∈ X : d(x, y) <r}. The collection 1 τ(d)= V ⊆ X :(∀x ∈ V )(∃n ∈ ω)B x, ⊆ V d 2n is the topology in X induced by d. For a set A ⊆ X, let δd(A)=0 if A = ∅, and let δd(A) = sup{d(x, y): x, y ∈ A} if A =6 ∅. Then δd(A) is the diameter of A in hX,di. A quasi-metric d on X is called strong if τ(d) ⊆ τ(d−1). In the sequel, topological or (quasi-)metric spaces (called spaces in ab- breviation) are denoted by boldface letters, and the underlying sets of the spaces are denoted by lightface letters. For a topological space X = hX, τi and for Y ⊆ X, let τ|Y = {V ∩ Y : V ∈ τ} and let Y = hY, τ|Y i. Then Y is the topological subspace of X such that Y is the underlying set of Y. If this is not misleading, we may denote the topological subspace Y of X by Y . A topological space X = hX, τi is called (quasi-)metrizable if there exists a (quasi-)metric d on X such that τ = τ(d). For a (quasi-)metric space X = hX,di and for Y ⊆ X, let dY = d ↾ Y ×Y and Y = hY,dY i. Then Y is the (quasi-)metric subspace of X such that Y is the underlying set of Y. Given a (quasi-)metric space X = hX,di, if not stated otherwise, we also denote by X the topological space hX, τ(d)i. n n n For every n ∈ N, R denotes also hR ,dei and hR , τ(de)i where de is the Euclidean metric on Rn. For a topological space X = hX, τi and a set A ⊆ X, we denote by clX(A) or by clτ (A) the closure of A in X. For any topological space X = hX, τi, let Isoτ (X)= {x ∈ X : x is an isolated point of X}. 3 If this is not misleading, as in Definition 1.1, we use Iso(X) to denote Isoτ (X). (α) By transitive recursion, we define a decreasing sequence (X )α∈ON of closed subsets of X as follows: X(0) = X, X(α+1) = X(α) \ Iso(X(α)), X(α) = X(γ) if α is a limit ordinal. γ\∈α For α ∈ ON, the set X(α) is called the α-th Cantor-Bendixson derivative of (α+1) (α) X. The least ordinal α such that X = X is denoted by |X|CB and is called the Cantor-Bendixson rank of X. Definition 1.2. A set X is called: (i) Dedekind-finite if there is no injection f : ω → X; Dedekind-infinite if X is not Dedekind-finite; (ii) quasi Dedekind-finite if [X]<ω is Dedekind-finite; quasi Dedekind-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; 4 (iv) locally compact if every point of X has a compact neighborhood; (v) limit point compact if every infinite subset of X has an accumulation point in X (cf. [14] and [15]). Definition 1.5. Let X = hX,di be a (quasi-)metric space. (i) Given a real number ε > 0, a subset D of X is called ε-dense or an ε-net in X if, for every x ∈ X, Bd(x, ε) ∩ D =6 ∅ (equivalently, if X = Bd−1 (x, ε)). ∈ xSD (ii) X is called precompact (respectively, totally bounded) if, for every real number ε > 0, there exists a finite ε-net in hX,d−1i (respectively, in hX,d⋆i). (iii) d is called precompact (respectively, totally bounded) if X is precompact (respectively, totally bounded). Remark 1.6. Definition 1.5 (ii) is based on the notions of precompact and totally bounded quasi-uniformities defined, e.g., in [5] and [26]. Namely, given a quasi-metric d on a set X, the collection 1 U(d)= U ⊆ X × X : ∃ ∈ hx, yi ∈ X × X : d(x, y) < ⊆ U n ω 2n is a quasi-uniformity on X called the quasi-uniformity induced by d (cf. [26, p. 504]). The quasi-uniformity U(d) is precompact (resp., totally bounded) in the sense of [5] and [26] if and only if d is precompact (resp., totally bounded) in the sense of Definition 1.5. Clearly d is totally bounded if and only if, for every n ∈ ω, there exists a finite set D ⊆ X such that 1 1 −1 X = Bd(x, 2n ) ∩ Bd (x, 2n ) . The notions of a totally bounded and x∈D precompactS metric are equivalent. We recall that a (Hausdorff) compactification of a space X = hX, τi is an ordered pair hY,γi where Y is a (Hausdorff) compact space and γ : X → Y is a homeomorphic embedding such that γ(X) is dense in Y.