A remark on a theorem of Solecki

Holický, Petr, Zajíček, Luděk, and Zelený, Miroslav. "A remark on a theorem of Solecki." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 43-54. <http://eudml.org/doc/249555>.

@article{Holický2005,
abstract = {S. Solecki proved that if $\mathcal \{F\}$ is a system of closed subsets of a complete separable metric space $X$, then each Suslin set $S\subset X$ which cannot be covered by countably many members of $\mathcal \{F\}$ contains a $G_\{\delta \}$ set which cannot be covered by countably many members of $\mathcal \{F\}$. We show that the assumption of separability of $X$ cannot be removed from this theorem. On the other hand it can be removed under an extra assumption that the $\sigma $-ideal generated by $\mathcal \{F\}$ is locally determined. Using Solecki’s arguments, our result can be used to reprove a Hurewicz type theorem due to Michalewski and Pol, and a nonseparable version of Feng’s theorem due to Chaber and Pol.},
author = {Holický, Petr, Zajíček, Luděk, Zelený, Miroslav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Solecki’s theorem; Suslin set; $\sigma $-ideal; Suslin set; -ideal},
language = {eng},
number = {1},
pages = {43-54},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A remark on a theorem of Solecki},
url = {http://eudml.org/doc/249555},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Holický, Petr
AU - Zajíček, Luděk
AU - Zelený, Miroslav
TI - A remark on a theorem of Solecki
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 43
EP - 54
AB - S. Solecki proved that if $\mathcal {F}$ is a system of closed subsets of a complete separable metric space $X$, then each Suslin set $S\subset X$ which cannot be covered by countably many members of $\mathcal {F}$ contains a $G_{\delta }$ set which cannot be covered by countably many members of $\mathcal {F}$. We show that the assumption of separability of $X$ cannot be removed from this theorem. On the other hand it can be removed under an extra assumption that the $\sigma $-ideal generated by $\mathcal {F}$ is locally determined. Using Solecki’s arguments, our result can be used to reprove a Hurewicz type theorem due to Michalewski and Pol, and a nonseparable version of Feng’s theorem due to Chaber and Pol.
LA - eng
KW - Solecki’s theorem; Suslin set; $\sigma $-ideal; Suslin set; -ideal
UR - http://eudml.org/doc/249555
ER -