## Real Analysis Exchange

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

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

Krzysztof C. Ciesielski and Juan B. Seoane--Sepúlveda

#### 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

**Subjects**

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A04

Secondary: 03E35: Consistency and independence results

**Keywords**

Covering Property Axiom CPA smooth continuous covering differentiable extensions Whitney's extension theorem

#### 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