Real Analysis Exchange

Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom

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

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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

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

First available in Project Euclid: 27 June 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A04
Secondary: 03E35: Consistency and independence results

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


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.

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