The Banach–Mazur game and σ-porosity

Miroslav Zelený

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 3, page 197-210
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Zelený, Miroslav. "The Banach–Mazur game and σ-porosity." Fundamenta Mathematicae 150.3 (1996): 197-210. <http://eudml.org/doc/212171>.

@article{Zelený1996,
abstract = {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.},
author = {Zelený, Miroslav},
journal = {Fundamenta Mathematicae},
keywords = {proper sequence; -porosity; metric space; uniformly porous sets; Banach-Mazur game},
language = {eng},
number = {3},
pages = {197-210},
title = {The Banach–Mazur game and σ-porosity},
url = {http://eudml.org/doc/212171},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Zelený, Miroslav
TI - The Banach–Mazur game and σ-porosity
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 3
SP - 197
EP - 210
AB - 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.
LA - eng
KW - proper sequence; -porosity; metric space; uniformly porous sets; Banach-Mazur game
UR - http://eudml.org/doc/212171
ER -

References

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  1. [D] E. P. Dolzhenko, Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14 (in Russian). 
  2. [K] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  3. [O] J. C. Oxtoby, Measure and Category, Springer, 1980. Zbl0435.28011
  4. [Z1] L. Zajíček, On differentiability properties of Lipschitz functions on a Banach space with a uniformly Gateaux differentiable bump function, preprint, 1995. 
  5. [Z2] L. Zajíček, Porosity and σ-porosity, Real Anal. Exchange 13 (1987-88), 314-350. 

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