## Abstract and Applied Analysis

### Local inverses of Borel homomorphisms and analytic P-ideals

Sławomir Solecki

#### Abstract

We present a theorem on the existence of local continuous homomorphic inverses of surjective Borel homomorphisms with countable kernels from Borel groups onto Polish groups. We also associate in a canonical way subgroups of $\mathbb{R}$ with certain analytic P-ideals of subsets of $\mathbb{N}$. These groups, with appropriate topologies, provide examples of Polish, nonlocally compact, totally disconnected groups for which global continuous homomorphic inverses exist in the situation described above. The method of producing these groups generalizes constructions of Stevens and Hjorth and, just as those constructions, yields examples of Polish groups which are totally disconnected and yet are generated by each neighborhood of the identity.

#### Article information

Source
Abstr. Appl. Anal., Volume 2005, Number 3 (2005), 207-219.

Dates
First available in Project Euclid: 25 July 2005

https://projecteuclid.org/euclid.aaa/1122298424

Digital Object Identifier
doi:10.1155/AAA.2005.207

Mathematical Reviews number (MathSciNet)
MR2197115

Zentralblatt MATH identifier
1096.22003

#### Citation

Solecki, Sławomir. Local inverses of Borel homomorphisms and analytic P-ideals. Abstr. Appl. Anal. 2005 (2005), no. 3, 207--219. doi:10.1155/AAA.2005.207. https://projecteuclid.org/euclid.aaa/1122298424

#### References

• D. Bump, Lie Groups, Graduate Texts in Mathematics, vol. 225, Springer, New York, 2004.
• J. R. P. Christensen, V. Kanovei, and M. Reeken, On Borel orderable groups, Topology Appl. 109 (2001), no. 3, 285--299.
• R. Engelking, General Topology, Monografie Matematyczne, vol. 60, PWN---Polish Scientific Publishers, Warsaw, 1977.
• I. Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, p. xvi+177.
• G. Hjorth, Classification and Orbit Equivalence Relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Rhode Island, 2000.
• --------, A new zero-dimensional Polish group, preprint, 1998.
• A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer, New York, 1995.
• L. Pontrjagin, Topological Groups, Princeton Mathematical Series, vol. 2, Princeton University Press, New Jersey, 1939.
• S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), no. 1--3, 51--72.
• T. C. Stevens, Connectedness of complete metric groups, Colloq. Math. 50 (1986), no. 2, 233--240.