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**Covering analytic sets by families of closed sets.**
*(English)*
Zbl 0808.03031

Summary: We prove that for every family \(I\) of closed subsets of a Polish space each \(\Sigma^ 1_ 1\) set can be covered by countably many members of \(I\) or else contains a nonempty \(\Pi^ 0_ 2\) set which cannot be covered by countably many members of \(I\). We prove an analogous result for \(\kappa\)-Souslin sets and show that if \(A^ \#\) exists for any \(A\subset \omega^ \omega\), then the above result is true for \(\Sigma^ 1_ 2\) sets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz’s theorem due to Kechris, Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding “big” closed sets inside “big” \(\Sigma^ 1_ 1\) and \(\Sigma^ 1_ 2\) sets are consequences of our results.

### MSC:

03E15 | Descriptive set theory |

Full Text:
DOI

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