2023-11-29T16:48:18Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/717862023-07-13T07:29:39Zcom_20.500.14352_14col_20.500.14352_15
00925njm 22002777a 4500
dc
Krzysztof, Chris
author
Natkaniec, T.
author
Rodríguez-Vidanes, D.L.
author
2022-11-14
A function f : R → R is: almost continuous in the sense of Stallings, f ∈ AC, if each open set G ⊂ R2 containing the graph of f contains also the graph of a continuous function g : R → R; Sierpiński-Zygmund, f ∈ SZ (or, more generally, f ∈ SZ(Bor)), provided its restriction f M is discontinuous (not Borel, respectively) for any M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński-Zygmund almost continuous function f : R → R (i.e., an f ∈ SZ ∩ AC) cannot be constructed in ZFC; however, an f ∈ SZ ∩ AC exists under the additional set-theoretical assumption cov(M) = c, that is, that R cannot be covered by less than c-many meager sets. The primary purpose of this paper is to show that the existence of an f ∈ SZ∩AC is also consistent with ZFC plus the negation of cov(M) = c. More precisely, we show that it is consistent with ZFC+cov(M) < c (follows from the assumption that non(N ) < cov(N ) = c) that there is an f ∈ SZ(Bor)∩AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either cov(M) = c or non(N ) < cov(N ) = c, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński-Zygmund functions. Several open problems are also stated.
[1] G. Albkwre and K. C. Ciesielski, Maximal lineability of several subclasses of Darboux-like maps on R, submitted; draft of July 2021, available at https://math.wvu.edu/˜kciesiel/publications.html.
[2] R.M. Aron, L. Bernal-González, D.M. Pellegrino, and J.B. Seoane-Sepúlveda, Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL. (2016).
[3] R.M. Aron, V. I. Gurari˘ı, and J.B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133, No. 3 (2005), 795–803.
[4] M. Balcerzak, K. Ciesielski, and T. Natkaniec, Sierpiński-Zygmund functions that are Dar�boux, almost continuous or have a perfect road, Archive for Math. Logic 37, No. 1 (1997), 29–35.
[5] K. Banaszewski and T. Natkaniec, Sierpiński-Zygmund functions that have the Cantor in�termediate value property, Real Anal. Exchange 24, No. 2 (1998–99), 827–835.
[6] A. Bartoszewicz, M. Bienias, Sz. Gł¸ab, and T. Natkaniec, Algebraic structures in the sets of
surjective functions, J. Math. Anal. Appl. 441, No. 2 (2016), 574–585.
[7] T. Bartoszyński and H. Judah, Set theory: on the structure of the real line, Wellesley, MA:
A. K. Peters Ltd. (1995).
[8] T. Bartoszyński, H. Judah, and S. Shelah, The Cichoń Diagram, J. Symbolic Logic 58, No. 2 (1993), 401–423.
[9] L. Bernal-González, D.M. Pellegrino, and J.B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51, No. 1 (2014), 71–130.
[10] J. Ceder, Some examples on continuous restrictions, Real Anal. Exchange 7 (1981–82), 155–
162.
[11] J. Cichoń, A. Kharazishvili, and B. W¸eglorz, Subsets of the real line. Łódź, Wydawnictwo Uniwersytetu Łódzkiego (1995).
[12] K.C. Ciesielski and Cheng-Han Pan, Examples of Sierpinski-Zygmund maps in the class of Darboux-like functions, Banach J. Math. Anal. 14, No. 2 (2020), 433–449.
[13] K.C.Ciesielski, J.L. Gámez-Merino, L. Mazza, and J.B. Seoane-Sepúlveda, Cardinal coeffi�cients related to surjectivity, Darboux, and Sierpiński-Zygmund maps, Proc. Amer. Math. Soc. 145, No. 3 (2017), 1041–1052.
[14] K.C. Ciesielski, J.L. Gámez-Merino, D. Pellegrino, and J.B. Seoane-Sepúlveda, Lineability, spaceability, and additivity cardinals for Darboux-like functions, Linear Algebra Appl. 440 (2014), 307–317.
[15] K.C. Ciesielski and T. Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79, No. 1, (1997), 75–99.
[16] K.C. Ciesielski and T. Natkaniec, Different notions of Sierpiński-Zygmund functions, Rev.
Mat. Complut. 34, No. 1 (2021), 151-173
[17] K.C. Ciesielski, T. Natkaniec, D.L. Rodríguez-Vidanes, and J.B. Seoane-Sepúlveda, Additivity coefficients for all classes in the algebra of Darboux-like maps on R, Results Math. 76, No. 1 (2021), Paper No. 7 (38 pages).
[18] K.C. Ciesielski and J.B. Seoane-Sepúlveda, A century of Sierpiński-Zygmund functions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113, No. 4 (2019), 3863–3901.
[19] J. L. Gámez-Merino, Gustavo A. Muñoz-Fernández, and J.B. Seoane-Sepúlveda, Lineability and additivity in RR, J. Math. Anal. Appl. 369(2) (2010), 265–272.
[20] J. L. Gámez-Merino and J. B. Seoane-Sepúlveda, An undecidable case of lineability in RR, J. Math. Anal. Appl. 401(2) (2013), 959–962.
[21] R.G. Gibson and F. Roush, The Cantor intermediate value property, Proceedings of the 1982
Topology Conference (Annapolis, Md., 1982), (1982), 55–62.
[22] R.G. Gibson and T. Natkaniec, Darboux like functions, Real Anal. Exchange 2 (1996–97), 492–533.
[23] F. Jordan, Cardinal invariants connected with adding real functions, Real Anal. Exchange 2 (1996–97), 696–713.
[24] K. R. Kellum, Almost continuity and connectivity – sometimes it’s as easy to prove a stronger result, Real Anal. Exchange 8 (1982–83), 244–252.
[25] A. Kharazishvili, Strange functions in real analysis, 2nd ed., Pure and Applied Mathematics, vol. 272, Chapman & Hall/CRC, Boca Raton, Florida, 2006.
[26] A. Kharazishvili, Some remarks on Sierpiński-Zygmund functions in the strong sense, Geor�gian Math.J. 27, No. 4 (2020), 569–575.
[27] K. Kunen, Random and Cohen reals, in: Kunen, K., Vaughan, J.E. (eds.): Handbook of Set-Theoretic Topology. North-Holland Publishing Co., Amsterdam (1984), 887-911.
[28] K. Kuratowski, Topologie, Vol. I. 4th ed. Monografie Matematyczne, Vol. 20, Warszawa: PWN-Polish Scientific Publishers, 1958.
[29] I. Maximoff, Sur les fonctions ayant la propriété de Darboux, Prace Mat. Fiz. 43 (1936), 241–265.
[30] T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (1991–92), 462–520.
[31] T. Natkaniec and H. Rosen, An example of an additive almost continuous Sierpiński- Zyg�mund function, Real Anal. Exchange 30, No. 1 (2004–05), 261–265.
[32] T. Natkaniec and H. Rosen, Additive Sierpiński-Zygmund functions, Real Anal. Exchange
31, No. 1 (2005–06), 253–269.
[33] K. Płotka, Algebraic structures within subsets of Hamel and Sierpinski-Zygmund functions,
Bull. Belg. Math. Soc. - Simon Stevin 22, No. 3 (2015), 447–454.
[34] F. Rothberger, Eine Homöomorphiebedingung für orientierbare Mannigfaltigkeiten von drei
Dimensionen, Monatsh. Math. Phys. 41, No. 1 (1934), 353–357.
[35] W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math. 4 (1923), 316–318.
[36] S.M. Srivastava„ A Course of Borel Sets, Springer, New York (1998).
[37] J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249–263.
[38] F.D. Tall, The density topology, Pacific Math. J. 62, No. 1 (1976), 275–284
1579-1505
10.1007/s13398-022-01347-w
https://hdl.handle.net/20.500.14352/71786
https://doi.org/10.1007/s13398-022-01347-w
Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions