PlanetPhysics/Borel Space

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).

Borel sets were named after the French mathematician Emile Borel.

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 $$X$$.

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. ).

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