Cellularity of infinite Hausdorff spaces in ZF

Keremedis, Kyriakos; Tachtsis, Eleftherios

doi:10.1016/j.topol.2020.107104

Cited by 9 publications

(18 citation statements)

References 7 publications

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“…The equivalence of (a) and (c) was independently proved recently by Keremedis and Tachtsis, see lemma 1 in [13]. They prove in theorem 3 a further equivalence in terms of the topological cellularity of the space 2 A .…”

Section: Adding a Cohen Subset To A Dedekind-finite Setmentioning

confidence: 71%

How to have more things by forgetting how to count them

Karagila

Schlicht

2020

Proc. R. Soc. A.

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Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.

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Section: Adding a Cohen Subset To A Dedekind-finite Setmentioning

confidence: 71%

How to have more things by forgetting how to count them

Karagila

Schlicht

2020

Proc. R. Soc. A.

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“…Since DF = F does not imply MC ω ω in ZF (see [HR98]), we also see by the above result of [KT20] and Theorem 2.1 that the latter implication is not reversible in ZF.…”

mentioning

confidence: 61%

“…(1) We mention an interesting topological consequence of JT, unknown until now. Keremedis and Tachtsis [KT20] have recently established that each of the weak choice forms MC ω and DF = F (which are mutually independent) implies the statement "every infinite T 2 space has a countably infinite cellular family" (where a cellular family in a space (X, T ) is a disjoint subfamily of T \ {∅}), which in turn implies "for every infinite set X, [X] <ω is Dedekind-infinite" (where [X] <ω denotes the set of finite subsets of X).…”

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confidence: 99%

On the interrelation of a theorem of Juhász and certain weak axioms of choice

Tachtsis

2022

Fund. Math.

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“…In Section 2, we establish basic notation and mainly relatively simple preliminary results. It is known, for instance, from [17] and [13] that it is independent of ZF that every denumerable compact Hausdorff space admits a denumerable cellular family. Even the sentence that all infinite discrete spaces admit denumerable cellular families is independent of ZF because it is equivalent to IWDI (see [13]).…”

Section: Introductionmentioning

confidence: 99%

“…It is known, for instance, from [17] and [13] that it is independent of ZF that every denumerable compact Hausdorff space admits a denumerable cellular family. Even the sentence that all infinite discrete spaces admit denumerable cellular families is independent of ZF because it is equivalent to IWDI (see [13]). In Section 2, it is shown that IWDI is equivalent to the following sentence:…”

Section: Introductionmentioning

confidence: 99%

Denumerable cellular families in Hausdorff spaces and towers of Boolean algebras in $\mathbf{ZF}$

Keremedis,

Wajch

2020

Preprint

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A denumerable cellular family of a topological space X is an infinitely countable collection of pairwise disjoint non-empty open sets of X. It is proved that the following statements are equivalent in ZF:(i) For every infinite set X, [X] <ω has a denumerable subset.(ii) Every infinite 0-dimensional Hausdorff space admits a denumerable cellular family.It is also proved that (i) implies the following: (iii) Every infinite Hausdorff Baire space has a denumerable cellular family.Among other results, the following theorems are also proved in ZF:(iv) Every countable collection of non-empty subsets of R has a choice function iff, for every infinite second-countable Hausdorff space 1 X, it holds that every base of X contains a denumerable cellular family of X.(v) If every Cantor cube is pseudocompact, then every non-empty countable collection of non-empty finite sets has a choice function.(vi) If all Cantor cubes are countably paracompact, then (i) holds.Moreover, among other forms independent of ZF, a partial Kinna-Wagner selection principle for families expressible as countable unions of finite families of finite sets is introduced. It is proved that if this new selection principle and (i) hold, then every infinite Boolean algebra has a tower and every infinite Hausdorff space has a denumerable cellular family.

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Cellularity of infinite Hausdorff spaces in ZF

Cited by 9 publications

References 7 publications

How to have more things by forgetting how to count them

How to have more things by forgetting how to count them

On the interrelation of a theorem of Juhász and certain weak axioms of choice

Denumerable cellular families in Hausdorff spaces and towers of Boolean algebras in $\mathbf{ZF}$

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