is said to be "standard Borel" if there exists a metric on
that makes it a complete separable metric space in such a way that
[1] Standard Borel spaces have several useful properties that do not hold for general measurable spaces.
a complete separable metric space.
(This result is reminiscent of Maharam's theorem.)
It follows that a standard Borel space is characterized up to isomorphism by its cardinality,[2] and that any uncountable standard Borel space has the cardinality of the continuum.
Borel isomorphisms on standard Borel 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 only Borel measurable.