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Keywords:
Solecki's theorem; Suslin set; $\sigma$-ideal
Solecki's theorem; Suslin set; $\sigma$-ideal
Summary:
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.
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