The failure of the axiom of choice implies unrest in the theory of Lindelöf metric spaces

Keremedis, Kyriakos

doi:10.1002/malq.200310017

Cited by 6 publications

(12 citation statements)

References 16 publications

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“…In this paper we continue the work done in [12]. The interested reader may consult the reference cited above for background results.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 92%

“…Then G = I ∪ G 1 is a countable dense subset of X and X is second countable. It follows (see the introduction in [12]) that X embeds in the dense-in-itself compact metric space [0, 1] ω as required. Since every dense subset of X must include its isolated points, the conclusion follows.…”

Section: Corollary 14mentioning

confidence: 94%

“…Since CAC(R ) iff M(2, hL) (Theorem 1) and CAC(R ) iff M(2,hS) (see e. g. [12]), it follows that M(2, hL) iff M(2,hS). The next theorem is the main result of this paper and it extends the latter equivalence to arbitrary Lindelöf metric spaces.…”

Section: Lemma 12mentioning

confidence: 96%

“…Furthermore, it is not hard to see that it also implies CAC fin . Indeed, if A = {A i : i ∈ ω} is a family of finite non-empty subsets, then the one point compactification X of the discrete space Y = A is compact and metrizable (see [6,12] and the proof of ((vi)⇒(vii)) of the forthcoming Theorem 15). Clearly, Z = [0, 1] × X is, in ZF, a dense-in-itself compact metric space.…”

Section: Remark 31mentioning

confidence: 98%

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Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces

Keremedis

2004

Mathematical Logic Qtrly

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Key words Axiom of choice, weak axioms of choice, metric space, Lindelöf metric space, Loeb metric space. MSC (2000) 03E25, 54D30, 54E35, 54E45We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: (i) Every Lindelöf metric space is separable and (ii) Every Lindelöf metric space is second countable (Forms 340 and 341, respectively,in [10]) are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. Notation and terminologyThe following statements "Form x" have been considered in [10], and all known implications, within the framework of Zermelo-Fraenkel set theory ZF, between these forms are given in [10,

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“…In this paper we continue the work done in [12]. The interested reader may consult the reference cited above for background results.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 92%

Section: Corollary 14mentioning

confidence: 94%

Section: Lemma 12mentioning

confidence: 96%

Section: Remark 31mentioning

confidence: 98%

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Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces

Keremedis

2004

Mathematical Logic Qtrly

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“…If X = ∅, then (X, d) is trivially Lindelöf, so let x ∈ X. Since CAC implies M(L,S) (see, e. g., [13]) it follows that for every i ∈ N ,…”

Section: Resultsmentioning

confidence: 99%

Countable sums and products of metrizable spaces in ZF

Keremedis¹,

Tachtsis

2004

Mathematical Logic Qtrly

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Key words Axiom of choice, weak axioms of choice, compact and Lindelöf metrizable spaces, sums and products of metrizable spaces. MSC (2000)03E25, 54A35, 54D30, 54D65, 54D70, 54E35, 54E45We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.In this paper we continue with the study of compact and Lindelöf metric spaces which we started in [12] and [13]. Besides the notation and terminology used in the above-mentioned papers we shall need to establish some more here. Let (X, T ) be a topological space. X is said to be metrizable iff there is a metric d on X such that the topology T d on X induced by d coincides with T .Form 9: Every infinite set has a countably infinite subset. Form 418: The disjoint union of countably many metrizable spaces is metrizable.i ∈ N} is a family of metric spaces each having the property P , then the (Tychonoff) product X = i∈N X i has the property Q.CPM le (P, Q): If {(X i , T i ) : i ∈ N} is a family of metrizable topological spaces each having the property P , then the (Tychonoff) product X = i∈N X i is metrizable and has the property Q.CSM(P, Q): If {(X i , d i ) : i ∈ N} is a disjoint family of metric spaces each having the property P , then the sum (disjoint topological union, see [19]) X = i∈N X i has the property Q.CSM le (P, Q): If {(X i , T i ) : i ∈ N} is a disjoint family of metrizable topological spaces each having the property P , then the sum X = i∈N X i is metrizable and has the property Q.CAC(ML) (resp. CAC(MC)): If {(X i , d i ) : i ∈ N} is a family of Lindelöf (resp. compact) metric spaces, then {X i : i ∈ N} has a choice function.Notice that CAC(ML) (resp. CAC(MC)) is equivalent to the proposition: Products of countably many Lindelöf (resp. compact) metric spaces are non-empty.CAC(M le L) (resp. CAC(M le C)) : If {(X i , T i ) : i ∈ N} is a family of Lindelöf (resp. compact) metrizable topological spaces, then {X i : i ∈ N} has a choice function.

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