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# Borel space

###### Definition 0.1.

A Borel space $(X;\mathcal{B}(X))$ is defined as a set $X$, together with a Borel $\sigma$-algebra $\mathcal{B}(X)$ 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., Groupoid 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., available online.
- 3 A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.

## Mathematics Subject Classification

60A10*no label found*28C15

*no label found*28A12

*no label found*54H05

*no label found*28A05

*no label found*

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