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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. 145 (2017), 1041-1052
MSC (2010): Primary 15A03, 26A15, 26B05, 54A25
DOI: https://doi.org/10.1090/proc/13294
Published electronically: September 15, 2016
MathSciNet review: 3589304
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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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  • [1] N. Albuquerque, L. Bernal-González, D. Pellegrino, and J. B. Seoane-Sepúlveda, Peano curves on topological vector spaces, Linear Algebra Appl. 460 (2014), 81-96. MR 3250532, https://doi.org/10.1016/j.laa.2014.07.029
  • [2] Richard M. Aron, Luis Bernal González, Daniel M. Pellegrino, and Juan B. Seoane Sepúlveda, Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3445906
  • [3] Richard M. Aron, José A. Conejero, Alfredo Peris, and Juan B. Seoane-Sepúlveda, Uncountably generated algebras of everywhere surjective functions, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 3, 571-575. MR 2731374
  • [4] R. M. Aron, F. J. García-Pacheco, D. Pérez-García, and J. B. Seoane-Sepúlveda, On dense-lineability of sets of functions on $ \mathbb{R}$, Topology 48 (2009), no. 2-4, 149-156. MR 2596209, https://doi.org/10.1016/j.top.2009.11.013
  • [5] Richard Aron, V. I. Gurariy, and J. B. Seoane, Lineability and spaceability of sets of functions on $ \mathbb{R}$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795-803 (electronic). MR 2113929, https://doi.org/10.1090/S0002-9939-04-07533-1
  • [6] Richard M. Aron, David Pérez-García, and Juan B. Seoane-Sepúlveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (2006), no. 1, 83-90. MR 2261701, https://doi.org/10.4064/sm175-1-5
  • [7] Richard M. Aron and Juan B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on $ \mathbb{C}$, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25-31. MR 2327324
  • [8] Marek Balcerzak, Krzysztof Ciesielski, and Tomasz Natkaniec, Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road, Arch. Math. Logic 37 (1997), no. 1, 29-35. MR 1485861, https://doi.org/10.1007/s001530050080
  • [9] Artur Bartoszewicz, Marek Bienias, Szymon Glab, and Tomasz Natkaniec, Algebraic structures in the sets of surjective functions, J. Math. Anal. Appl. 441 (2016), no. 2, 574-585. MR 3491544, https://doi.org/10.1016/j.jmaa.2016.04.013
  • [10] Artur Bartoszewicz and Szymon Glab, Additivity and lineability in vector spaces, Linear Algebra Appl. 439 (2013), no. 7, 2123-2130. MR 3090459, https://doi.org/10.1016/j.laa.2013.06.007
  • [11] Artur Bartoszewicz, Szymon Glab, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Algebrability, non-linear properties, and special functions, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3391-3402. MR 3080162, https://doi.org/10.1090/S0002-9939-2013-11641-2
  • [12] F. Bastin, J. A. Conejero, C. Esser, and J. B. Seoane-Sepúlveda, Algebrability and nowhere Gevrey differentiability, Israel J. Math. 205 (2015), no. 1, 127-143. MR 3314585, https://doi.org/10.1007/s11856-014-1104-1
  • [13] Luis Bernal-González and Manuel Ordóñez Cabrera, Lineability criteria, with applications, J. Funct. Anal. 266 (2014), no. 6, 3997-4025. MR 3165251, https://doi.org/10.1016/j.jfa.2013.11.014
  • [14] Luis Bernal-González, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71-130. MR 3119823, https://doi.org/10.1090/S0273-0979-2013-01421-6
  • [15] K. Ciesielski, Set-theoretic real analysis, J. Appl. Anal. 3 (1997), no. 2, 143-190. MR 1619547, https://doi.org/10.1515/JAA.1997.143
  • [16] Krzysztof Ciesielski, José L. Gámez-Merino, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Lineability, spaceability, and additivity cardinals for Darboux-like functions, Linear Algebra Appl. 440 (2014), 307-317. MR 3134273, https://doi.org/10.1016/j.laa.2013.10.033
  • [17] Krzysztof Ciesielski and Jan Jastrzebski, Darboux-like functions within the classes of Baire one, Baire two, and additive functions, Topology Appl. 103 (2000), no. 2, 203-219. MR 1758794, https://doi.org/10.1016/S0166-8641(98)00169-2
  • [18] Krzysztof Ciesielski and Arnold W. Miller, Cardinal invariants concerning functions whose sum is almost continuous, Real Anal. Exchange 20 (1994/95), no. 2, 657-672. MR 1348089
  • [19] Krzysztof Ciesielski and Tomasz Natkaniec, On Sierpiński-Zygmund bijections and their inverses, Topology Proc. 22 (1997), no. Spring, 155-164. MR 1657922
  • [20] Krzysztof Ciesielski and Tomasz Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79 (1997), no. 1, 75-99. MR 1462608, https://doi.org/10.1016/S0166-8641(96)00128-9
  • [21] Krzysztof Ciesielski and Janusz Pawlikowski, The covering property axiom, CPA, Cambridge Tracts in Mathematics, vol. 164, Cambridge University Press, Cambridge, 2004. A combinatorial core of the iterated perfect set model. MR 2176267
  • [22] Krzysztof Ciesielski and Ireneusz Recław, Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange 21 (1995/96), no. 2, 459-472. MR 1407262
  • [23] Per H. Enflo, Vladimir I. Gurariy, and Juan B. Seoane-Sepúlveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc. 366 (2014), no. 2, 611-625. MR 3130310, https://doi.org/10.1090/S0002-9947-2013-05747-9
  • [24] José L. Gámez-Merino, Large algebraic structures inside the set of surjective functions, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 2, 297-300. MR 2848805
  • [25] José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, Víctor M. Sánchez, and Juan B. Seoane-Sepúlveda, Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3863-3876. MR 2679609, https://doi.org/10.1090/S0002-9939-2010-10420-3
  • [26] José L. Gámez, Gustavo A. Muñoz-Fernández, and Juan B. Seoane-Sepúlveda, Lineability and additivity in $ \mathbb{R}^{\mathbb{R}}$, J. Math. Anal. Appl. 369 (2010), no. 1, 265-272. MR 2643865, https://doi.org/10.1016/j.jmaa.2010.03.036
  • [27] José L. Gámez-Merino and Juan B. Seoane-Sepúlveda, An undecidable case of lineability in $ \mathbb{R}^{\mathbb{R}}$, J. Math. Anal. Appl. 401 (2013), no. 2, 959-962. MR 3018041, https://doi.org/10.1016/j.jmaa.2012.10.067
  • [28] Francis Edmund Jordan, Cardinal numbers connected with adding Darboux-like functions, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)-West Virginia University. MR 2697132
  • [29] Kenneth R. Kellum, Almost continuity and connectivity--sometimes it's as easy to prove a stronger result, Real Anal. Exchange 8 (1982/83), no. 1, 244-252. MR 694512
  • [30] T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (1991/92), no. 2, 462-520. MR 1171393
  • [31] Tomasz Natkaniec, New cardinal invariants in real analysis, Bull. Polish Acad. Sci. Math. 44 (1996), no. 2, 251-256. MR 1416428
  • [32] Krzysztof Płotka, Algebraic structures within subsets of Hamel and Sierpiński-Zygmund functions, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 3, 447-454. MR 3396995
  • [33] J. H. Roberts, Zero-dimensional sets blocking connectivity functions, Fund. Math. 57 (1965), 173-179. MR 0195065
  • [34] Juan B. Seoane, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)-Kent State University. MR 2709064

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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

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