Souslin’s hypothesis and convergence in category
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- by Arnold W. Miller PDF
- Proc. Amer. Math. Soc. 124 (1996), 1529-1532 Request permission
Abstract:
A sequence of functions $f_n\colon X\to \mathbb R$ from a Baire space $X$ to the reals $\mathbb R$ is said to converge in category iff every subsequence has a subsequence which converges on all but a meager set. We show that if there exists a Souslin tree, then there exists a nonatomic Baire space $X$ such that every sequence which converges in category converges everywhere on a comeager set. This answers a question of Wagner and Wilczynski who proved the converse.References
- Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski, $\scr I$-density continuous functions, Mem. Amer. Math. Soc. 107 (1994), no. 515, xiv+133. MR 1188595, DOI 10.1090/memo/0515
- Ju. I. Gribanov, Remark on convergence almost everywhere and in measure, Comment. Math. Univ. Carolinae 7 (1966), 297–300 (Russian). MR 201600
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- W. Poreda, E. Wagner-Bojakowska, and W. Wilczyński, A category analogue of the density topology, Fund. Math. 125 (1985), no. 2, 167–173. MR 813753, DOI 10.4064/fm-125-2-167-173
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Juris Steprāns, Trees and continuous mappings into the real line, Topology Appl. 12 (1981), no. 2, 181–185. MR 612014, DOI 10.1016/0166-8641(81)90019-5
- R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201–245. MR 294139, DOI 10.2307/1970860
- S. Todorčević, Trees and linearly ordered sets, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 235–293. MR 776625
- E. Wagner and W. Wilczyński, Convergence of sequences of measurable functions, Acta Math. Acad. Sci. Hungar. 36 (1980), no. 1-2, 125–128. MR 605180, DOI 10.1007/BF01897101
Additional Information
- Arnold W. Miller
- Affiliation: University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
- Email: miller@math.wisc.edu
- Received by editor(s): November 2, 1994
- Additional Notes: I want to thank Krzysztof Ciesielski for many helpful conversations
The results presented in this paper were obtained during the Joint US–Polish Workshop in Real Analysis, Łódź, Poland, July 1994. The workshop was partially supported by the NSF grant INT–9401673 - Communicated by: Franklin D. Tall
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1529-1532
- MSC (1991): Primary 28A20; Secondary 03E65, 54E52
- DOI: https://doi.org/10.1090/S0002-9939-96-03409-0
- MathSciNet review: 1328364