# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Definable hereditary families in the projective hierarchy

### Tom 140 / 1992

Fundamenta Mathematicae 140 (1992), 183-189 DOI: 10.4064/fm-140-2-183-189

#### Streszczenie

We show that if ℱ is a hereditary family of subsets of $ω^ω$ satisfying certain definable conditions, then the $Δ_1^1$ reals are precisely the reals α such that ${β:α ∈ Δ_1^1(β)} ∉ ∈ ℱ$. This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the $Q_{2n+1}$-encodable reals is also shown.

• R. Barua
• V. Srivatsa

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