PlanetPhysics/Borel Space

A Borel space $(X;ℬ(X))$ is defined as a set $X$, together with a Borel [[../SigmaAlgebra/|$\sigma$-algebra]] $ℬ(X)$ of subsets of $X$, called [[../InvariantBorelSet/|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).

Borel sets were named after the French mathematician Emile Borel.

A subspace of a Borel space $(X;ℬ(X))$ 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 $X$.

A rigid Borel space $(X_r;ℬ(X_r))$ 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 $A\in ℬ(X_r)$) is the [[../Cod/|identity]] [[../Bijective/|function]] $1_{(X_r;ℬ(X_r))}$ (ref.).

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