## Real Analysis Exchange

### Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom

#### Abstract

The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every $\nu\in\omega\cup\{\infty\}$ there exists a family $\mathcal{F}_\nu\subset C^\nu(\mathbb{R})$ of cardinality $\omega_1<\mathfrak{c}$ such that for every $g\in D^\nu(\mathbb{R})$ the set $g\setminus \bigcup \mathcal{F}_\nu$ has cardinality $\leq\omega_1$. Moreover, we show that this result remains true for partial functions $g$ (i.e., $g\in D^\nu(X)$ for some $X\subset\mathbb{R}$) if, and only if, $\nu \in\{0,1\}$. The proof of this result is based on the following theorem of independent interest (which, for $\nu\neq 0$, seems to have been previously unnoticed): for every $X\subset\mathbb{R}$ with no isolated points, every $\nu$-times differentiable function $g\colon X\to\mathbb{R}$ admits a $\nu$-times differentiable extension $\bar g\colon B\to\mathbb{R}$, where $B \supset X$ is a Borel subset of $\mathbb{R}$. The presented arguments rely heavily on a Whitney’s Extension Theorem for the functions defined on perfect subsets of $\mathbb{R}$, for which a short, but fully detailed, proof is included. Some open questions are also posed.

#### Article information

Source
Real Anal. Exchange, Volume 43, Number 2 (2018), 359-386.

Dates
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1530064967

Digital Object Identifier
doi:10.14321/realanalexch.43.2.0359

Mathematical Reviews number (MathSciNet)
MR3942584

Zentralblatt MATH identifier
06924895

#### Citation

Ciesielski, Krzysztof C.; Seoane--Sepúlveda, Juan B. Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom. Real Anal. Exchange 43 (2018), no. 2, 359--386. doi:10.14321/realanalexch.43.2.0359. https://projecteuclid.org/euclid.rae/1530064967