Decomposing with smooth sets

TL;DR

This paper investigates the minimal number of smooth sets needed to decompose Euclidean space and explores related cardinal invariants, establishing inequalities and connections to decompositions of continuous functions.

Contribution

It introduces the invariant ${rak d}_n$ for decomposing Euclidean space into smooth sets and proves inequalities relating these invariants, linking them to function decomposition.

Findings

${rak d}_n$ can be strictly greater than ${rak d}_{n+1}$ in some models.

Established inequalities: ${rak d}_{n+1}^+ geq {rak d}_n$.

${rak d}_2$ equals the least number of differentiable functions needed to decompose any continuous function.

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 ${\frak d}_n$ 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 ${\frak d}_n > {\frak d}_{n+1} $. Moreover, the inequalities ${\frak d}_{n+1}^+ \geq ${\frak d}_n$ are established where ${\frak d}_1$ is defined to be the continuum. The cardinal invariant ${\frak d}_2$ is shown to be the same as the least $\kappa$ such that each continuous function from the reals to the reals can be decomposed into $\kappa$ differentiable functions.