Decomposing Euclidean space
with a small number of smooth sets
Author:
Juris Steprans
Journal:
Trans. Amer. Math. Soc. 351 (1999), 1461-1480
MSC (1991):
Primary 04A30; Secondary 28A15
DOI:
https://doi.org/10.1090/S0002-9947-99-02197-2
MathSciNet review:
1473455
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let the cardinal invariant denote the least number of continuously smooth
-dimensional surfaces into which
-dimensional Euclidean space can be decomposed. It will be shown to be consistent that
is greater than
. These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.
- 1. Uri Abraham, Matatyahu Rubin, and Saharon Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ℵ₁-dense real order types, Ann. Pure Appl. Logic 29 (1985), no. 2, 123–206. MR 801036, https://doi.org/10.1016/0168-0072(84)90024-1
- 2. V. Aversa, M. Laczkovich, and D. Preiss, Extension of differentiable functions, Comment. Math. Univ. Carolin. 26 (1985), no. 3, 597–609. MR 817830
- 3. James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, https://doi.org/10.1016/0003-4843(79)90010-X
- 4. A. S. Besicovitch. On tangents to general sets of points. Fund. Math., 22:49-53, 1934.
- 5. J. Cichoń and M. Morayne, Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), no. 1, 83–89. MR 915721, https://doi.org/10.1090/S0002-9939-1988-0915721-6
- 6. J. Cichoń, M. Morayne, J. Pawlikowski, and S. Solecki, Decomposing Baire functions, J. Symbolic Logic 56 (1991), no. 4, 1273–1283. MR 1136456, https://doi.org/10.2307/2275474
- 7. S. Saks. Theory of the Integral. Hafner, New York, 1937.
- 8. Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
- 9. Saharon Shelah and Juris Steprāns, Decomposing Baire class 1 functions into continuous functions, Fund. Math. 145 (1994), no. 2, 171–180. MR 1297403
- 10. S. Solecki. Decomposing Borel sets and functions and the structure of Baire class 1 functions. J. Amer. Math. Soc. 11:521-550, 1998. CMP 98:13
- 11. Juris Steprāns, A very discontinuous Borel function, J. Symbolic Logic 58 (1993), no. 4, 1268–1283. MR 1253921, https://doi.org/10.2307/2275142
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Additional Information
Juris Steprans
Affiliation:
Department of Mathematics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3
Email:
steprans@mathstat.yorku.ca
DOI:
https://doi.org/10.1090/S0002-9947-99-02197-2
Keywords:
Cardinal invariant,
Sacks real,
tangent plane,
covering number
Received by editor(s):
March 9, 1995
Received by editor(s) in revised form:
May 5, 1997
Additional Notes:
Research for this paper was partially supported by NSERC of Canada. The author would also like to acknowledge that this paper has significantly benefitted from several remarks of A. Miller
Article copyright:
© Copyright 1999
American Mathematical Society