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Transactions of the American Mathematical Society

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Decomposing Euclidean space with a small number of smooth sets

Author: Juris Steprāns
Journal: Trans. Amer. Math. Soc. 351 (1999), 1461-1480
MSC (1991): Primary 04A30; Secondary 28A15
MathSciNet review: 1473455
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Abstract: Let the cardinal invariant ${\mathfrak s}_{n}$ denote the least number of continuously smooth $n$-dimensional surfaces into which $(n+1)$-dimensional Euclidean space can be decomposed. It will be shown to be consistent that ${\mathfrak s}_{n}$ is greater than ${\mathfrak s}_{n+1}$. These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.

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Juris Steprāns
Affiliation: Department of Mathematics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3

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