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 $A^\sharp$ 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 of "big" $\Sigma^1_1$ and $\Sigma^1_2$ sets are consequences of our results.