Small uncountable cardinals in large-scale topology

Банах, Тарас

doi:10.1016/j.topol.2022.108277

Cited by 1 publication

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that the inequality ń ≥ d has been observed before, and can be considered folklore. It is (arguably) implicit in [17], and was observed explicitly by Hrušak in [13] and later by Banakh in [1]. Anticipating the main theorem in Section 3, note that this inequality gives us an easy way of excluding an initial segment of the uncountable cardinals from sp(closed): simply make d big.…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 83%

“…Throughout the proof, if X ⊆ Y and P is a partition of Y, then P X = {K ∩ X : K ∈ P} denotes the restriction of P to X. We prove that (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (1).…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 96%

“…Lemma 2.2. For any uncountable cardinal κ, the following are equivalent: (1) There is a partition of 2 into κ closed sets.…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 99%

“…The main results are due to Stern [25], Miller [17], Newelski [19], and Spinas [24], who studied this cardinal implicitly without giving it a name, and Hrušak [12,13], who denotes it a T . The name "Sierpi ński cardinal" and the notation ń were suggested by Banakh in [1].…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 99%

“…Forcing an (almost) arbitrary spectrum. Every κ ∈ sp(Borel) is an example what Blass calls a "Δ 1 1 characteristic" in [2]. Blass proves [2, Theorems 8 and 9] that there can be many cardinals between ℵ 1 and c that are not Δ 1 1 characteristics, and therefore are not in sp(Borel).…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 99%

See 4 more Smart Citations

Partitioning the Real Line Into Borel Sets

Brian

2023

J. symb. log.

View full text Add to dashboard Cite

For which infinite cardinals $\kappa $ is there a partition of the real line ${\mathbb R}$ into precisely $\kappa $ Borel sets? Work of Lusin, Souslin, and Hausdorff shows that ${\mathbb R}$ can be partitioned into $\aleph _1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of ${\mathbb R}$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega $ with $0,1 \in A$ , there is a forcing extension in which ${A = \{ n :\, \text {there is a partition of } {{\mathbb R}} \text { into }\aleph _n\text { Borel sets}\}}$ . We also look at the corresponding question for partitions of ${\mathbb R}$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa $ such that there is a partition of ${\mathbb R}$ into precisely $\kappa $ closed sets can be fairly arbitrary.

show abstract

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 83%

“…Throughout the proof, if X ⊆ Y and P is a partition of Y, then P X = {K ∩ X : K ∈ P} denotes the restriction of P to X. We prove that (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (1).…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 96%

“…Lemma 2.2. For any uncountable cardinal κ, the following are equivalent: (1) There is a partition of 2 into κ closed sets.…”

Section: §1 Introduction By Work Of Lusin and Sierpi ńSki Inmentioning

confidence: 99%