Covering analytic sets by families of closed set

Abstract

Abstract We prove that for every family I of closed subsets of a Polish space each set can be covered by countably many members of I or else contains a nonempty set which cannot be covered by countably many members of I . We prove an analogous result for κ -Souslin sets and show that if A # exists for any A ⊂ ω ω , then the above result is true for 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 of “big” and are consequences of our results.