In mathematics, Borel isomorphism is a Borel measurable bijective function from one Polish space

Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...

 to another Polish space. Borel isomorphisms are closed under composition and under taking of inverses. The set of Borel isomorphisms from a Polish space to itself apparently forms a group

Group

A group is a number of things or persons being in some relation to one another.- Mathematics :* Group , a set together with a binary operation satisfying certain algebraic conditions** Group theory, the study of groups...

 under composition. Borel isomorphisms on Polish spaces are analogous to homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

s on topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being Borel measurable.

External links

Borel Spaces, by S. K. Berberian http://www.ma.utexas.edu/mp_arc/c/02/02-156.pdf

Real Analysis and Probability, page 487, Second edition, by R. M. Dudley
http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511755347&cid=CBO9780511755347A091

A course on Borel sets by Sashi Mohan Srivastava http://books.google.com/books?id=FhYGYJtMwcUC&pg=PA169