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Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps


Authors: K. C. Ciesielski, J. L. Gámez-Merino, L. Mazza and J. B. Seoane-Sepúlveda
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 15A03, 26A15, 26B05, 54A25
Published electronically: September 15, 2016
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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}$.


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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
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
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.
Email: jseoane@ucm.es

DOI: https://doi.org/10.1090/proc/13294
Keywords: Additivity, lineability, cardinal invariant, Darboux
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
Article copyright: © Copyright 2016 American Mathematical Society