Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps
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- by K. C. Ciesielski, J. L. Gámez-Merino, L. Mazza and J. B. Seoane-Sepúlveda PDF
- Proc. Amer. Math. Soc. 145 (2017), 1041-1052 Request permission
Abstract:
We investigate the additivity $A$ and lineability $\mathcal {L}$ cardinal coefficients for the following classes of functions: $\operatorname {ES} \setminus \operatorname {SES}$ of everywhere surjective functions that are not strongly everywhere surjective, Darboux-like, Sierpiński-Zygmund, surjective, and their corresponding intersections. The classes $\operatorname {SES}$ and $\operatorname {ES}$ have been shown to be $2^{\mathfrak {c}}$-lineable. In contrast, although we prove here that $\operatorname {ES} \setminus \operatorname {SES}$ is ${\mathfrak {c}}^+$-lineable, it is still unclear whether it can be proved in ZFC that $\operatorname {ES} \setminus \operatorname {SES}$ is $2^{\mathfrak {c}}$-lineable. Moreover, we prove that if $\mathfrak {c}$ is a regular cardinal number, then $A(\operatorname {ES} \setminus \operatorname {SES})\leq \mathfrak {c}$. This shows that, for the class $\operatorname {ES} \setminus \operatorname {SES}$, there is an unusually large gap between the numbers $A$ and $\mathcal {L}$.References
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Additional Information
- K. C. Ciesielski
- Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310 – and – Department of Radiology, MIPG, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6021
- MR Author ID: 49415
- Email: KCies@math.wvu.edu
- J. L. Gámez-Merino
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, 28040 Madrid, Spain
- MR Author ID: 634110
- Email: jlgamez@mat.ucm.es
- L. Mazza
- Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
- Email: lmazza@mix.wvu.edu
- J. B. Seoane-Sepúlveda
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, 28040 Madrid, Spain – and – Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, UAM, 28049 Madrid, Spain.
- MR Author ID: 680972
- Email: jseoane@ucm.es
- Received by editor(s): March 5, 2016
- Received by editor(s) in revised form: May 16, 2016
- Published electronically: September 15, 2016
- Additional Notes: The second and fourth authors were supported by grant MTM2015-65825-P
- Communicated by: Ken Ono
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1041-1052
- MSC (2010): Primary 15A03, 26A15, 26B05, 54A25
- DOI: https://doi.org/10.1090/proc/13294
- MathSciNet review: 3589304