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Splitting chains, tunnels and twisted sums

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Abstract

We study splitting chains in \(\mathcal{P}(\omega)\), that is, families of subsets of ω which are linearly ordered by ⊆* and which are splitting. We prove that their existence is independent of axioms of ZFC. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of C(ω*) = ℓ∞/c0 and c0(c). Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.

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Authors and Affiliations

  1. Departamento de Matemáticas and IMUEx, Universidad de Extremadura, 06071, Badajoz, Spain

    Félix Cabello-Sánchez

  2. Departamento de Matemáticas, Universidad de Murcia, 30100, Murcia, Spain

    Antonio Avilés

  3. Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland

    Piotr Borodulin-Nadzieja

  4. Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic

    David Chodounský

  5. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Morelia, Michoacan, 58089, Mexico

    Osvaldo Guzmán

Authors
  1. Félix Cabello-Sánchez
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  2. Antonio Avilés
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  3. Piotr Borodulin-Nadzieja
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  4. David Chodounský
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  5. Osvaldo Guzmán
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Corresponding author

Correspondence to Piotr Borodulin-Nadzieja.

Additional information

FCS was supported in part by DGICYT project MTM2016-76958-C2-1-P (Spain) and Junta de Extremadura program IB-16056.

AA was supported by projects MTM2017-86182-P (AEI, Government of Spain and ERDF, EU) and 20797/PI/18 by Fundación Séneca, ACyT Región de Murcia.

PBN was supported by Polish National Science Center grant no. 2018/29/B/ST1/00223. He is also indebted to Universidad de Murcia for financing his stay in Murcia, during which AA introduced him to the concept of splitting chains.

DCh was supported by the GACR project 17-33849L and RVO: 67985840.

OG was partially supported by NSERC grant number 455916, PAPIIT grants IN 100317 and IN 104220, and a CONACyT grant A1-S-16164.

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Cabello-Sánchez, F., Avilés, A., Borodulin-Nadzieja, P. et al. Splitting chains, tunnels and twisted sums. Isr. J. Math. 241, 955–989 (2021). https://doi.org/10.1007/s11856-021-2121-5

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  • DOI: https://doi.org/10.1007/s11856-021-2121-5

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