## Abstract

The familiar theorem that any Σ _{2} ^{1} (*a*)-set *X* of real numbers (where *a* is a fixed real parameter) not containing a perfect kernel necessarily satisfies the condition *X*\( \subseteq \)**L**[*a*] is extended to a wider class of sets, with countable ordinals allowed as additional parameters in Σ _{2} ^{1} (*a*)-definitions.

## Key words

perfect kernel property perfect subset forcing descriptive set theory*CA*-set

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