Decomposing with smooth sets

Abstract

A subset of Euclidean space will be said to be $n$-smooth if it has an $n$-dimensional tangent plane at each of its points. Let ${\mathfrak{d}}_n$ denote the least number $n$-smooth sets into which $n+1$-dimensional Euclidean space can be decomposed. For each $n$ it is shown to be consistent that ${\mathfrak{d}}_n>{\mathfrak{d}}_{n+1}$. Moreover, the inequalities ${\mathfrak{d}}_{n+1}^{+}\ge${\frak d}_n$areestablishedwhere${\frak d}_1$isdefinedtobethecontinuum.Thecardinalinvariant${\frak d}_2$isshowntobethesameastheleast$κ$suchthateachcontinuousfunctionfromtherealstotherealscanbedecomposedinto$κ$ differentiable functions.

References in corpus (1)

• Decomposing Baire class 1 functions into continuous functions

Cited by in corpus (2)

• Unions of Rectifiable Curves and the Dimension of Banach Spaces
• Dominating numbers for countable structures