## Journal of Symbolic Logic

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

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

Kenneth Schilling and Bosko Zivaljevic

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR1464115

**Zentralblatt MATH identifier**

0886.03045

**JSTOR**

links.jstor.org

**Subjects**

Primary: 03H04

Secondary: 03E15: Descriptive set theory [See also 28A05, 54H05] 04A15 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 54J05: Nonstandard topology [See also 03H05]

#### Citation

Schilling, Kenneth; Zivaljevic, Bosko. Louveau's Theorem for the Descriptive Set Theory of Internal Sets. J. Symbolic Logic 62 (1997), no. 2, 595--607. https://projecteuclid.org/euclid.jsl/1183745244