If X is Polish then any closed subset of X can be written as the disjoint union of a perfect set and a countable set. Further, if the Polish space X is uncountable, it can be written as the disjoint union of a perfect set and a countable open set.
A subspace Q of a Polish space P is Polish if and only ifQ is the intersection of a sequence of open subsets of P.
Every Polish space is homeomorphic to a Gδ-subset of the Hilbert cube.
The following spaces are Polish:
closed subsets of a Polish space,
open subsets of a Polish space,
products and disjoint unions of countable families of Polish spaces,
There are numerous characterizations that tell when a second-countable topological space is metrizable, such as Urysohn's metrization theorem. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval can be given both complete metrics and incomplete metrics generating their topology. There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game. A separable metric space is completely metrizable if and only if the second player has a winning strategy in this game. A second characterization follows from Alexandrov's theorem. It states that a separable metric space is completely metrizable if and only if it is a subset of its completion in the original metric.
Polish metric spaces
Although Polish spaces are metrizable, they are not in and of themselves metric spaces; each Polish space admits many complete metrics giving rise to the same topology, but no one of these is singled out or distinguished. A Polish space with a distinguished complete metric is called a Polish metric space. An alternative approach, equivalent to the one given here, is first to define "Polish metric space" to mean "complete separable metric space", and then to define a "Polish space" as the topological space obtained from a Polish metric space by forgetting the metric.
A subspace of a Lusin space is Lusin if and only if it is a Borel set.
Any countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.
The product of a countable number of Lusin spaces is Lusin.
The disjoint union of a countable number of Lusin spaces is Lusin.
Suslin spaces
A Suslin space is the image of a Polish space under a continuous mapping. So every Lusin space is Suslin. In a Polish space, a subset is a Suslin space if and only if it is a Suslin set. The following are Suslin spaces:
closed or open subsets of a Suslin space,
countable products and disjoint unions of Suslin spaces,
countable intersections or countable unions of Suslin subspaces of a Hausdorff topological space,
A Polish group is a topological groupG that is also a Polish space, in other words homeomorphic to a separable complete metric space. A remarkable fact about Polish groups is that Baire-measurable homomorphisms between them are automatically continuous. The group of homeomorphisms of the Hilbert cube N is a universal Polish group, in the sense that every Polish group is isomorphic to a closed subgroup of it. Examples:
All finite dimensional Lie groups with a countable number of components are Polish groups.
The unitary group of a separable Hilbert space is a Polish group.
The group of homeomorphisms of a compact metric space is a Polish group.
The product of a countable number of Polish groups is a Polish group.
The group of isometries of a separable complete metric space is a Polish group.
