# Borel space

###### Definition 0.1.

A $(X;\mathcal{B}(X))$ is defined as a set $X$, together with a Borel $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) $\mathcal{B}(X)$ of subsets of $X$, called Borel sets. The Borel algebra on $X$ is the smallest $\sigma$-algebra containing all open sets (or, equivalently, all closed sets if the topology on closed sets is selected).

###### Remark 0.1.

Borel sets were named after the French mathematician Emile Borel.

###### Remark 0.2.

A subspace of a Borel space $(X;\mathcal{B}(X))$ is a subset $S\subset X$ endowed with the relative Borel structure, that is the $\sigma$-algebra of all subsets of $S$ of the form $S\bigcap E$, where $E$ is a Borel subset of $X$.

###### Definition 0.2.

A rigid Borel space $(X_{r};\mathcal{B}(X_{r}))$ is defined as a Borel space whose only automorphism $f:X_{r}\to X_{r}$ (that is, with $f$ being a bijection, and also with $f(A)=f^{-1}(A)$ for any $A\in\mathcal{B}(X_{r})$) is the identity function $1_{(X_{r};\mathcal{B}(X_{r}))}$ (ref.[2]).

###### Remark 0.3.

R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a ‘set of large cardinality’.

## References

• 1 M.R. Buneci. 2006., http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71–98.
• 2 B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS., 113 (4):1013-1015., http://www.jstor.org/pss/2048777available online.
• 3 A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.
 Title Borel space Canonical name BorelSpace Date of creation 2013-03-22 18:23:02 Last modified on 2013-03-22 18:23:02 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 22 Author bci1 (20947) Entry type Definition Classification msc 60A10 Classification msc 28C15 Classification msc 28A12 Classification msc 54H05 Classification msc 28A05 Synonym measurable space Related topic BorelSet Related topic SigmaAlgebra Related topic MeasurableSpace Related topic BorelMeasure Related topic BorelGroupoid Related topic BorelMorphism Defines rigid Borel space Defines Borel subset space