## Journal of Symbolic Logic

### Louveau's Theorem for the Descriptive Set Theory of Internal Sets

#### Abstract

We give positive answers to two open questions from [15]. (1) For every set $C$ countably determined over $\mathscr{A}$, if $C$ is $\Pi^0_\alpha (\Sigma^0_\alpha)$ then it must be $\Pi^0_\alpha (\Sigma^0_\alpha)$ over $\mathscr{A}$, and (2) every Borel subset of the product of two internal sets $X$ and $Y$ all of whose vertical sections are $\Pi^0_\alpha (\Sigma^0_\alpha)$ can be represented as an intersection (union) of Borel sets with vertical sections of lower Borel rank. We in fact show that (2) is a consequence of the analogous result in the case when $X$ is a measurable space and $Y$ a complete separable metric space (Polish space) which was proved by A. Louveau and that (1) is equivalent to the property shared by the inverse standard part map in Polish spaces of preserving almost all levels of the Borel hierarchy.

#### Article information

Source
J. Symbolic Logic, Volume 62, Issue 2 (1997), 595-607.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745244

Mathematical Reviews number (MathSciNet)
MR1464115

Zentralblatt MATH identifier
0886.03045

JSTOR