Compactness in Countable Tychonoff Products and Choice

Howard, Paul E.; Keremedis, Kyriakos; Rubin, Jean E.; Stanley, Adrienne

doi:10.1002/(sici)1521-3870(200001)46:1<3::aid-malq3>3.0.co;2-e

Cited by 15 publications

(23 citation statements)

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“…It is well known that the countable version of Tychonoff's compactness theorem, i. e., countable products of compact topological spaces are compact, implies the axiom of countable choice CAC, see [11]. The status of the reverse implication is still an open problem, see [1,5,8]. However, it is known, see [12], that if we restrict to the class of compact metric spaces, then Tychonoff's theorem for countable products of compact metric spaces, CPM(C,C), is a theorem of ZF + CAC.…”

Section: Introduction and Some Known 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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Section: Introduction and Some Known Resultsmentioning

confidence: 99%

Countable sums and products of metrizable spaces in ZF

Keremedis¹,

Tachtsis

2004

Mathematical Logic Qtrly

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“…(ii)→(i) Let A = (A n ) n∈N be a disjoint family of non-empty sets such that no infinite subfamily of A has a choice function and consider the pseudometric d on X = {A n : n ∈ N} given by (5). Clearly, X is complete.…”

Section: Clearly Ac Implies the Statement: (H)mentioning

confidence: 99%

On the metric reflection of a pseudometric space in ZF

Herrlich¹,

Keremedis²

2015

Commentationes Mathematicae Universitatis Carolinae

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“…) 3. CFE(TP) (Closed Filter Extendability in Tychonoff Products): If {(X i , T i ) : i ∈ I} is a family of compact topological spaces, then every family G ⊂ K(X), X = i∈I X i , with the fip extends to a closed ultrafilter F. 4 Kelley's proof of the implication "TCT → AC" can be adopted to show: Theorem 2.2 CTCT implies CAC. However, one cannot adopt his proof of the implication "AC → TCT" to establish "CAC → CTCT".…”

Section: An Open Set O Of X Is Called Restricted Open If There Is a Fmentioning

confidence: 99%

“…Several partial answers to Question 2.3 have been given in [4], where, in addition, the following characterization of CTCT has been established:…”

Section: Let X Be a Non Empty Set And E ⊂ P(x) \ {∅} A Non Empty Colmentioning

confidence: 99%

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Tychonoff products of compact spaces in ZF and closed ultrafilters

Keremedis

2010

Mathematical Logic Quarterly

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Compactness in Countable Tychonoff Products and Choice

Abstract: Abstract. We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.Mathematics Subject Classification: 03E25, 04A25, 54B10.

Cited by 15 publications

References 11 publications

Countable sums and products of metrizable spaces in ZF

Countable sums and products of metrizable spaces in ZF

On the metric reflection of a pseudometric space in ZF

Tychonoff products of compact spaces in ZF and closed ultrafilters

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