Open Access
2018 Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom
Krzysztof C. Ciesielski, Juan B. Seoane--Sepúlveda
Real Anal. Exchange 43(2): 359-386 (2018). DOI: 10.14321/realanalexch.43.2.0359


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.


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Krzysztof C. Ciesielski. Juan B. Seoane--Sepúlveda. "Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom." Real Anal. Exchange 43 (2) 359 - 386, 2018.


Published: 2018
First available in Project Euclid: 27 June 2018

zbMATH: 06924895
MathSciNet: MR3942584
Digital Object Identifier: 10.14321/realanalexch.43.2.0359

Primary: 26A04 , 26A24
Secondary: 03E35

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

Rights: Copyright © 2018 Michigan State University Press

Vol.43 • No. 2 • 2018
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