A remark on a theorem of Solecki
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 43-54
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S. Solecki proved that if $\Cal F$ 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 $\Cal F$ contains a $\boldsymbol G_{\delta}$ set which cannot be covered by countably many members of $\Cal F$. 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 $\Cal F$ 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.
@article{CMUC_2005__46_1_a4,
author = {Holick\'y, P. and Zaj{\'\i}\v{c}ek, L. and Zelen\'y, M.},
title = {A remark on a theorem of {Solecki}},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {43--54},
publisher = {mathdoc},
volume = {46},
number = {1},
year = {2005},
mrnumber = {2175858},
zbl = {1121.03058},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2005__46_1_a4/}
}
TY - JOUR AU - Holický, P. AU - Zajíček, L. AU - Zelený, M. TI - A remark on a theorem of Solecki JO - Commentationes Mathematicae Universitatis Carolinae PY - 2005 SP - 43 EP - 54 VL - 46 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CMUC_2005__46_1_a4/ LA - en ID - CMUC_2005__46_1_a4 ER -
Holický, P.; Zajíček, L.; Zelený, M. A remark on a theorem of Solecki. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 43-54. http://geodesic.mathdoc.fr/item/CMUC_2005__46_1_a4/