## 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:

 3e+15 Descriptive set theory
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### References:

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