It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.
@article{bwmeta1.element.bwnjournal-article-fmv150i3p197bwm,
author = {Miroslav Zelen\'y},
title = {The Banach--Mazur game and $\sigma$-porosity},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {197-210},
zbl = {0877.54024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv150i3p197bwm}
}
Zelený, Miroslav. The Banach–Mazur game and σ-porosity. Fundamenta Mathematicae, Tome 149 (1996) pp. 197-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv150i3p197bwm/
[00000] [D] E. P. Dolzhenko, Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14 (in Russian).
[00001] [K] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
[00002] [O] J. C. Oxtoby, Measure and Category, Springer, 1980. | Zbl 0435.28011
[00003] [Z1] L. Zajíček, On differentiability properties of Lipschitz functions on a Banach space with a uniformly Gateaux differentiable bump function, preprint, 1995.
[00004] [Z2] L. Zajíček, Porosity and σ-porosity, Real Anal. Exchange 13 (1987-88), 314-350.