Small Coverings with Smooth Functions under the Covering Property Axiom

Ciesielski, Krzysztof ; Pawlikowski, Janusz

Canadian journal of mathematics, Tome 57 (2005) no. 3, pp. 471-493

Résumé

In the paper we formulate a Covering Property Axiom, $\text{CP}{{\text{A}}_{\text{prism}}}$ , which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.(a) There exists a family $\mathcal{F}$ of less than continuum many ${{C}^{1}}$ functions from $\mathbb{R}$ to $\mathbb{R}$ such that ${{\mathbb{R}}^{2}}$ is covered by functions from $\mathcal{F}$ , in the sense that for every $\left\langle x,\,y \right\rangle \,\in \,{{\mathbb{R}}^{2}}$ ${{\mathbb{R}}^{2}}$ there exists an $f\,\in \,\mathcal{F}$ such that either $f\left( x \right)\,=\,y$ or $f\left( y \right)\,=\,x$ .(b) For every Borel function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many “ ${{C}^{1}}$ ” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$ .(c) For every $n\,>\,0$ and a ${{D}^{n}}$ function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many ${{C}^{n}}$ functions whose graphs cover the graph of $f$ .We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiî.

Détail

DOI : 10.4153/CJM-2005-020-8

Mots-clés : 26A24, 03E35, continuous, smooth, covering

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