A New Proof of the Tychonoff Theorem

Loeb, Peter A.

doi:10.2307/2314411

Cited by 33 publications

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“…(Cf. [2], [29] and [21].) A space X is said to be Loeb if the family of all non-empty closed subsets of X has a choice function.…”

Section: The Collectionmentioning

confidence: 99%

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

Keremedis,

Tachtsis,

Wajch

2021

Preprint

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The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of ZF, statements: "Every countable product of compact metrizable spaces is separable (respectively, compact)" and "Every countable product of compact metrizable spaces is metrizable". Statements related to the above-mentioned ones are also studied. Permutation models (among them new ones) are shown in which a countable sum (also a countable product) of metrizable spaces need not be metrizable, countable unions 1 of countable sets are countable and there is a countable family of nonempty sets of size at most 2 ℵ 0 which does not have a choice function. A new permutation model is constructed in which every uncountable compact metrizable space is of size at least 2 ℵ 0 but a denumerable family of denumerable sets need not have a multiple choice function.

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“…(Cf. [2], [29] and [21].) A space X is said to be Loeb if the family of all non-empty closed subsets of X has a choice function.…”

Section: The Collectionmentioning

confidence: 99%

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

Keremedis,

Tachtsis,

Wajch

2021

Preprint

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“…In addition, if each X i is finite then Z is a closed subspace of 2 Y . Theorem 2.12 [2], [12] (ZF) Let (X i ) i∈ℵ be a family of compact T 1 spaces. Then, the product X = i∈ℵ X i is compact and Loeb iff there exists a family ( f i ) i∈ℵ such that for all i ∈ ℵ, f i is a Loeb function for X i .…”

Section: Theorem 29mentioning

confidence: 99%

The Boolean prime ideal theorem and products of cofinite topologies

Keremedis

2013

Math. Log. Quart.

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We show: The Boolean Prime Ideal theorem sans-serifBPI is equivalent to each one of the statements: AC fin (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”. CAC fin (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”. BPI(ω) (: every filter of ℘(ω) extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space boldY having a base of size ≤|R| and for every set X of size ≤|R| the product YX is compact”.

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“…Proposition 2.1 is a good motivation for the introduction of the R‐Loeb spaces. For the history, Loeb in 9 proved that if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(\mathbf {X} _{i})_{i\in \aleph }$\end{document} is a family of compact spaces and ( f i ) i ∈ ℵ is a family of functions such that for all i ∈ ℵ, f i is a Loeb function of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {X}_{i}$\end{document}, then the Tychonoff product \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {X}={\prod _{i\in \aleph }} \mathbf {X}_{i}$\end{document} is compact. Subsequently, Brunner in 2 named these spaces Loeb space, in honour of Loeb.…”

Section: Introductionmentioning

confidence: 99%

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Keremedis

2012

Mathematical Logic Qtrly

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Given a set X, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} denotes the statement: “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set” and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$\end{document} denotes the family of all closed subsets of the topological space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {2}^{X}$\end{document} whose definition depends on a finite subset of X. We study the interrelations between the statements \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document} and “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document}has a choice set”. We show: \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$\end{document} iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$\end{document}. \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}_{\mathrm{fin}}$\end{document} (\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}$\end{document} restricted to families of finite sets) iff for every set X, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set. \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}_{\mathrm{fin}}$\end{document} does not imply “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set(\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {X})$\end{document} is the family of all closed subset...

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A New Proof of the Tychonoff Theorem

Cited by 33 publications

References 0 publications

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

The Boolean prime ideal theorem and products of cofinite topologies

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

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