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In mathematics, Borel isomorphism is a bijective Borel function from one Polish space to another Polish space.[clarification needed] (Clarification: A subset of the domain is Borel if and only if its image under the Borel Isomorphism is Borel.) Borel isomorphisms are closed under composition and under taking of inverses. The set of Borel isomorphisms from a Polish space to itself clearly forms a group under composition. Borel isomorphisms on Polish spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being Borel measurable.
- Alexander S. Kechris (1995) Classical Descriptive Set Theory, Springer-Verlag.