Concerning Sets of the First Baire Category with Respect to Different Metrics

Maria Moszyńska¹ ; Grzegorz Sójka²

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland
² Department of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00-601 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) no. 1, pp. 47-54.

Résumé

We prove that if $\varrho_H$ and $\delta$ are the Hausdorff metric and the radial metric on the space $\mathcal{S}^n$ of star bodies in $\mathbb R $, with 0 in the kernel and with radial function positive and continuous, then a family $\mathcal{A} \subset \mathcal{S}^n$ that is meager with respect to $\varrho_H$ need not be meager with respect to $\delta$. Further, we show that both the family of fractal star bodies and its complement are dense in $\mathcal{S}^n$ with respect to $\delta$.

Détail

DOI : 10.4064/ba58-1-6

Keywords: prove varrho delta hausdorff metric radial metric space mathcal star bodies mathbb kernel radial function positive continuous family mathcal subset mathcal meager respect varrho meager respect delta further family fractal star bodies its complement dense mathcal respect delta