标题： 使用平滑集合进行分解 显示英文标题

标题： Decomposing with smooth sets

摘要： 欧几里得空间的一个子集被称为$n$-光滑，如果在其每个点处都有一个$n$-维的切平面。 令${\frak d}_n$表示将$n+1$-维欧几里得空间分解为$n$-光滑集合的最少数量。 对于每个$n$，证明它是相容的，即${\frak d}_n > {\frak d}_{n+1} $。 此外，不等式 ${\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 适用于可微函数。 显示英文摘要

摘要： 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.