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. More sharply, see Lemma 1 below.
The primordial example (and in practice, one of the most convenient) is Baire space , viz. the space of irrational real numbers between and , with the subspace topology inherited from the real line. This is obviously not complete with respect to the metric induced from the real line, but it is homeomorphic to the product space via regular continued fraction expansions, and the latter is metrizable by a complete metric where the distance between two sequences and of positive integers is given by the formula where is the least integer such that . A countable dense subset is given by continued fractions that eventually repeat (quadratic surds).
Much of the material here is adapted from Marker, who in turn cites the text by Kechris as a main source.
The metric topology on coincides with the product topology.
In particular, the Hilbert cube is Polish.
Any Polish space is homeomorphic to a subspace of the Hilbert cube.
In fact, this is true of any separable metrizable space. Note that mere separability does not suffice, as there are separable spaces that are not first-countable, such as the Stone-Čech compactification of , and hence cannot be subspaces of any metrizable space. What distinguishes Polish spaces is that they are, up to homeomorphism, precisely the subsets of the Hilbert cube.
Every inhabited Polish space admits a continuous surjection from Baire space.
Construct by induction a collection of closed sets (balls) indexed over finite sequences of positive integers, with the following properties:
For the empty sequence , ;
For nonempty sequences , where ;
Letting denote the extension of obtained by appending to the final element ,
If extends , then .
Then define where for each infinite sequence ,
One may check that is continuous and surjective.
For a subset , recall that is a limit point if . A point that is not a limit point of is called an isolated point of . Clearly each isolated point is open relative to , as is therefore the set of isolated points.
The Cantor-Bendixson derivative of is the set of limit points relative to . For each ordinal the iterated derivative is defined by recursion: , , and if is a limit ordinal.
Since implies , it is clear that there is a least ordinal for which . This ordinal is called the Cantor-Bendixson rank of . A perfect set is a closed set such that . (Some people insist that a perfect set also be nonempty; we do not.)
Let be the complete infinite binary tree, whose infinite paths from the root correspond to points in Cantor space . To each node in (a finite sequence of ‘s and ’s) we construct by induction an open set with the following properties:
for the empty sequence ,
if is an initial segment of ,
For the two children and of , the sets and are disjoint,
where is the length of (nonempty) ,
for each .
Indeed, if has been constructed, and given , there are (at least!) two points since is perfect. We can easily find disjoint neighborhoods of these points respectively with the required properties.
Then for each path , define
where means is an initial segment of . The intersection consists of a single point because it equals the intersection of a decreasing chain of closed sets with shrinking diameter (thus closing in on a limit of a Cauchy sequence). The map is clearly injective by the disjointness of open sets of children, and it is easy to see is continuous.
For a closed subset of a Polish space , the Cantor-Bendixson rank is a countable ordinal . The complement is at most countable.
Let be a countable basis of . For each , each point is an isolated point, so we can find an in the basis such that . It is then clear that is injective, so each is countable. Similarly, whenever is nonempty, we can find a basis element that isolates one of its points (say ), and this same cannot isolate any point of an earlier since is a limit point of . It follows that is an injective map, so that must be a countable ordinal, and the collection is (at most) countable.
A closed set in a Polish space is the disjoint union of a perfect set and a finite or countable set . Hence an infinite closed subset in a Polish space has cardinality either or the continuum (continuum hypothesis for closed sets).
Recall that a function between topological spaces is Borel if is a Borel set in for every open in . Topological spaces and Borel functions form a category.
If are countably infinite -spaces, then any bijection is a Borel isomorphism.
For, the inverse image of an open set, being countable, is an set. In particular, any two denumerable Polish spaces are Borel isomorphic.
The unit interval and Cantor space are Borel isomorphic.
Let be the set of -sequences that are eventually constant. Then
maps homeomorphically onto the space of non-(dyadic rational) numbers in . Pick any bijection . Then the union of and defines a Borel isomorphism .
Any Polish space is Borel isomorphic to a Borel subset of Cantor space.
There is a sequence of maps
where the first map is an inclusion of a set, the second is induced from a Borel isomorphism , and the third is a homeomorphism.
Any two Polish spaces of the same cardinality are Borel isomorphic.
It suffices to prove this for uncountable Polish spaces (which have continuum cardinality, as we saw in Corollary 1). We show that any such is Borel isomorphic to Cantor space. By Proposition 3, we have an inclusion that maps Borel isomorphically onto its image, and by Proposition 2, we have an inclusion that maps Cantor space Borel isomorphically (even homeomorphically) onto its image. The rest is just a matter of checking that at least one of the proofs of the Cantor-Schroeder-Bernstein theorem applies to this Borel context.
Indeed, as explained here, we may consider
Each of the direct image maps and takes Borel sets to Borel sets, since their inverses are Borel functions on their domains. So each of the iterates is a Borel set and so is their countable intersection . The set is a fixed point of and so we construct a Borel isomorphism as
following the proof of the Cantor-Schroeder-Bernstein theorem.
For another proof, see theorem 3.1.1 of Berberian.
The classical -spaces for are Polish spaces.
A locally compact Hausdorff space is Polish iff it is second-countable.
Spaces of structures and models (in the model theory sense), and spaces of -types? (again in the model theory sense), quite often provide examples of Polish spaces. For example, if is a countable language (a countable signature), then the collection of possible -structures on the countable universe , topologized by taking as basic opens
where is a quantifier-free sentence, is a Polish space homeomorphic to the product space
(taking constants to be functions of arity in the signature).
As an example of the last principle, we have a kind of continuum hypothesis for substructures:
Let be a countable structure of a language; then the number of substructures of is either countable or the continuum.
A subset of is specified by its characteristic function , where is regarded as a Polish space. In order for the subset not to support a substructure, then must be some function symbol of the language, say of arity , and elements such that for all and . The basic open
would thus contain and also exclude any substructure; we conclude that the collection of substructures forms a closed subset of . Then the Cantor-Bendixson theorem (i.e., Corollary 1) shows that is either countable or the continuum.
Internal groupoids in Polish spaces are considered in