# nLab Polish space

A Polish space is a topological space that’s homeomorphic to a separable complete metric space. Every second countable locally compact Hausdorff space is a Polish space, among others.

Polish spaces provide a useful framework for doing measure theory. As with any topological space, we can take a Polish space and regard it as a measurable space with its sigma-algebra of Borel sets. Then, there is a very nice classification of Polish spaces up to measurable bijection: there is one for each countable cardinality, one whose cardinality is that of the continuum, and no others.

Why are Polish spaces ‘not very big’? In other words, why are there none with cardinality exceeding the continuum? As with any separable metric space, it’s because any Polish space has a countable dense subset and you can write any point as a limit of a sequence of points in this subset. So, you only need a sequence of integers to specify any point in a Polish space.

## References

Revised on May 11, 2010 10:21:55 by Tim van Beek (192.76.162.8)