Ideal approach to convergence in functional spaces

Bardyla, Serhii; Šupina, Jaroslav; Zdomskyy, Lyubomyr

doi:10.1090/tran/9008

Ideal approach to convergence in functional spaces
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by Serhii Bardyla, Jaroslav Šupina and Lyubomyr Zdomskyy

Trans. Amer. Math. Soc. 376 (2023), 8495-8528

DOI: https://doi.org/10.1090/tran/9008

Published electronically: September 12, 2023

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Abstract:

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [Topology Appl. 14 (1982), pp. 151–161], which was later reposed by A. Miller, T. Orenshtein, and B. Tsaban. Namely, we show that under $\mathfrak {p}=\mathfrak {c}$ there is a $\delta$-set that is not a $\gamma$-set. Thus we constructed a subset $A$ of reals such that the space $\mathrm {C}_p(A)$ of all real-valued continuous functions on $A$ is not Fréchet–Urysohn, but possesses the Pytkeev property. Moreover, under $\mathbf {CH}$ we construct a $\pi$-set that is not a $\delta$-set solving a problem by M. Sakai. In fact, we construct various examples of $\delta$-sets that are not $\gamma$-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Fréchet–Urysohn property for many different Borel ideals in the realm of functional spaces.

References

A. V. Arhangel′skiĭ, Some topological spaces that arise in functional analysis, Uspehi Mat. Nauk 31 (1976), no. 5(191), 17–32 (Russian). MR 458366
A. V. Arkhangel′skiĭ, Topological function spaces, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR 1144519, DOI 10.1007/978-94-011-2598-7
Karen Bakke Haga, David Schrittesser, and Asger Törnquist, Maximal almost disjoint families, determinacy, and forcing, J. Math. Log. 22 (2022), no. 1, Paper No. 2150026, 42. MR 4439581, DOI 10.1142/S0219061321500264
Tomek Bartoszyński and Boaz Tsaban, Hereditary topological diagonalizations and the Menger-Hurewicz conjectures, Proc. Amer. Math. Soc. 134 (2006), no. 2, 605–615. MR 2176030, DOI 10.1090/S0002-9939-05-07997-9
PawełBarbarski, RafałFilipów, Nikodem Mrożek, and Piotr Szuca, When does the Katětov order imply that one ideal extends the other?, Colloq. Math. 130 (2013), no. 1, 91–102. MR 3034318, DOI 10.4064/cm130-1-9
Andreas Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489. MR 2768685, DOI 10.1007/978-1-4020-5764-9_{7}
Piotr Borodulin-Nadzieja and Barnabás Farkas, Cardinal coefficients associated to certain orders on ideals, Arch. Math. Logic 51 (2012), no. 1-2, 187–202. MR 2864404, DOI 10.1007/s00153-011-0260-9
Jörg Brendle and Jana Flašková, Generic existence of ultrafilters on the natural numbers, Fund. Math. 236 (2017), no. 3, 201–245. MR 3600759, DOI 10.4064/fm730-5-2016
Jörg Brendle, Barnabás Farkas, and Jonathan Verner, Towers in filters, cardinal invariants, and Luzin type families, J. Symb. Log. 83 (2018), no. 3, 1013–1062. MR 3868039, DOI 10.1017/jsl.2017.52
Lev Bukovský, The structure of the real line, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 71, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2778559, DOI 10.1007/978-3-0348-0006-8
Lev Bukovský, Selection principle $\rm S_1$ and combinatorics of open covers, Topology Appl. 258 (2019), 239–250. MR 3924516, DOI 10.1016/j.topol.2019.02.054
Lev Bukovský, Pratulananda Das, and Jaroslav Šupina, Ideal quasi-normal convergence and related notions, Colloq. Math. 146 (2017), no. 2, 265–281. MR 3622377, DOI 10.4064/cm6520-3-2016
Pratulananda Das, Certain types of open covers and selection principles using ideals, Houston J. Math. 39 (2013), no. 2, 637–650. MR 3080458
Gabriel Debs and Jean Saint Raymond, Filter descriptive classes of Borel functions, Fund. Math. 204 (2009), no. 3, 189–213. MR 2520152, DOI 10.4064/fm204-3-1
Natasha Dobrinen, High dimensional Ellentuck spaces and initial chains in the Tukey structure of non-p-points, J. Symb. Log. 81 (2016), no. 1, 237–263. MR 3471138, DOI 10.1017/jsl.2015.10
R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc. 112 (1964), 483–507. MR 175081, DOI 10.1090/S0002-9947-1964-0175081-6
Ilijas Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, xvi+177. MR 1711328, DOI 10.1090/memo/0702
Barnabás Farkas and Lajos Soukup, More on cardinal invariants of analytic $P$-ideals, Comment. Math. Univ. Carolin. 50 (2009), no. 2, 281–295. MR 2537837
RafałFilipów and Piotr Szuca, Three kinds of convergence and the associated $\scr I$-Baire classes, J. Math. Anal. Appl. 391 (2012), no. 1, 1–9. MR 2899832, DOI 10.1016/j.jmaa.2012.02.041
S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107–115. MR 180954, DOI 10.4064/fm-57-1-107-115
Fred Galvin and Arnold W. Miller, $\gamma$-sets and other singular sets of real numbers, Topology Appl. 17 (1984), no. 2, 145–155. MR 738943, DOI 10.1016/0166-8641(84)90038-5
J. Gerlits and Zs. Nagy, Some properties of $C(X)$. I, Topology Appl. 14 (1982), no. 2, 151–161. MR 667661, DOI 10.1016/0166-8641(82)90065-7
Gerhard Grimeisen, Ein Approximationssatz für Bairesche Funktionen, Math. Ann. 146 (1962), 189–194 (German). MR 148810, DOI 10.1007/BF01470961
Michael Hrušák, Combinatorics of filters and ideals, Set theory and its applications, Contemp. Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 29–69. MR 2777744, DOI 10.1090/conm/533/10503
Michael Hrušák, Katětov order on Borel ideals, Arch. Math. Logic 56 (2017), no. 7-8, 831–847. MR 3696069, DOI 10.1007/s00153-017-0543-x
Michael Hrušák, David Meza-Alcántara, and Hiroaki Minami, Pair-splitting, pair-reaping and cardinal invariants of $F_\sigma$-ideals, J. Symbolic Logic 75 (2010), no. 2, 661–677. MR 2648159, DOI 10.2178/jsl/1268917498
W. Hurewicz, Über Folgen stetiger Funktionen, Fund. Math. 9 (1927), 193–204.
S. Jackson, A. S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Log. 2 (2002), no. 1, 1–80. MR 1900547, DOI 10.1142/S0219061302000138
Winfried Just and Adam Krawczyk, On certain Boolean algebras ${\scr P}(\omega )/I$, Trans. Amer. Math. Soc. 285 (1984), no. 1, 411–429. MR 748847, DOI 10.1090/S0002-9947-1984-0748847-1
Winfried Just, Arnold W. Miller, Marion Scheepers, and Paul J. Szeptycki, The combinatorics of open covers. II, Topology Appl. 73 (1996), no. 3, 241–266. MR 1419798, DOI 10.1016/S0166-8641(96)00075-2
Miroslav Katětov, Products of filters, Comment. Math. Univ. Carolinae 9 (1968), 173–189. MR 250257
M. Katětov, On descriptive classification of functions, General Topology and its Relations to Modern Analysis and Algebra, Proceedings of the Third Prague Topological Symposium, Prague, 1972, 235–242.
Ljubiša D. R. Kočinac and Marion Scheepers, Combinatorics of open covers. VII. Groupability, Fund. Math. 179 (2003), no. 2, 131–155. MR 2029229, DOI 10.4064/fm179-2-2
Adam Kwela and Ireneusz Recław, Ranks of $\scr {F}$-limits of filter sequences, J. Math. Anal. Appl. 398 (2013), no. 2, 872–878. MR 2990109, DOI 10.1016/j.jmaa.2012.09.052
Adam Kwela and Marcin Staniszewski, Ideal equal Baire classes, J. Math. Anal. Appl. 451 (2017), no. 2, 1133–1153. MR 3624783, DOI 10.1016/j.jmaa.2016.11.062
V. I. Malykhin and G. Tironi, Weakly Fréchet-Urysohn and Pytkeev spaces, Proceedings of the French-Japanese Conference “Hyperspace Topologies and Applications” (La Bussière, 1997), 2000, pp. 181–190. MR 1780904, DOI 10.1016/S0166-8641(99)00027-9
A. Miller, On $\gamma$-sets, Plenary lecture, Second Workshop on Coverings, Selections, and Games in Topology, Lecce, Italy, 19–22 Dec 2005, Lecture notes, http://u.cs.biu.ac.il/~tsaban/SPMC05/Miller.pdf.
Arnold W. Miller, Boaz Tsaban, and Lyubomyr Zdomskyy, Selective covering properties of product spaces, II: $\gamma$ spaces, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2865–2889. MR 3449260, DOI 10.1090/tran/6581
Tal Orenshtein and Boaz Tsaban, Linear $\sigma$-additivity and some applications, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3621–3637. MR 2775821, DOI 10.1090/S0002-9947-2011-05228-1
Tal Orenshtein and Boaz Tsaban, Pointwise convergence of partial functions: the Gerlits-Nagy problem, Adv. Math. 232 (2013), 311–326. MR 2989985, DOI 10.1016/j.aim.2012.09.017
Alexander V. Osipov, Classification of selectors for sequences of dense sets of $C_p(X)$, Topology Appl. 242 (2018), 20–32. MR 3802170, DOI 10.1016/j.topol.2018.04.010
E. G. Pytkeev, Tightness of spaces of continuous functions, Uspekhi Mat. Nauk 37 (1982), no. 1(223), 157–158 (Russian). MR 643782
E. G. Pytkeev, Sequentiality of spaces of continuous functions, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 197–198 (Russian). MR 676634
E. G. Pytkeev, On maximally resolvable spaces, Proc. Steklov Inst. Math. 154 (1984), 225–230.
Dilip Raghavan and Juris Steprāns, The almost disjointness invariant for products of ideals, Topology Appl. 323 (2023), Paper No. 108295, 11. MR 4518092, DOI 10.1016/j.topol.2022.108295
Ireneusz Recław, Every Lusin set is undetermined in the point-open game, Fund. Math. 144 (1994), no. 1, 43–54. MR 1271477, DOI 10.4064/fm-144-1-43-54
Miroslav Repický, Spaces not distinguishing ideal convergences of real-valued functions, Real Anal. Exchange 46 (2021), no. 2, 367–394. MR 4336563
Miroslav Repický, Spaces not distinguishing ideal convergences of real-valued functions, II, Real Anal. Exchange 46 (2021), no. 2, 395–421. MR 4336564
Masami Sakai, The Pytkeev property and the Reznichenko property in function spaces, Note Mat. 22 (2003/04), no. 2, 43–52. MR 2112730
Masami Sakai, Special subsets of reals characterizing local properties of function spaces, Selection principles and covering properties in topology, Quad. Mat., vol. 18, Dept. Math., Seconda Univ. Napoli, Caserta, 2006, pp. 195–225. MR 2395755
Marion Scheepers, Combinatorics of open covers. I. Ramsey theory, Topology Appl. 69 (1996), no. 1, 31–62. MR 1378387, DOI 10.1016/0166-8641(95)00067-4
Petr Simon and Boaz Tsaban, On the Pytkeev property in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1125–1135. MR 2361889, DOI 10.1090/S0002-9939-07-09070-3
Viera Šottová and Jaroslav Šupina, Principle $\textrm {S}_1(\mathcal {P},\mathcal {R})$: ideals and functions, Topology Appl. 258 (2019), 282–304. MR 3924519, DOI 10.1016/j.topol.2019.02.060
Jaroslav Šupina, Ideal QN-spaces, J. Math. Anal. Appl. 435 (2016), no. 1, 477–491. MR 3423409, DOI 10.1016/j.jmaa.2015.10.041
Jaroslav Šupina, Pseudointersection numbers, ideal slaloms, topological spaces, and cardinal inequalities, Arch. Math. Logic 62 (2023), no. 1-2, 87–112. MR 4535928, DOI 10.1007/s00153-022-00832-8
Piotr Szewczak and Boaz Tsaban, Products of general Menger spaces, Topology Appl. 255 (2019), 41–55. MR 3902779, DOI 10.1016/j.topol.2019.01.005
Boaz Tsaban, Menger’s and Hurewicz’s problems: solutions from “the book” and refinements, Set theory and its applications, Contemp. Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 211–226. MR 2777750, DOI 10.1090/conm/533/10509
Boaz Tsaban and Tomasz Weiss, Products of special sets of real numbers, Real Anal. Exchange 30 (2004/05), no. 2, 819–835. MR 2177439, DOI 10.14321/realanalexch.30.2.0819
Boaz Tsaban and Lyubomyr Zdomskyy, Scales, fields, and a problem of Hurewicz, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 3, 837–866. MR 2421163, DOI 10.4171/JEMS/132
Boaz Tsaban and Lyubomyr Zdomskyy, On the Pytkeev property in spaces of continuous functions. II, Houston J. Math. 35 (2009), no. 2, 563–571. MR 2519548