PlanetPhysics/Categories of Polish Groups and Polish Spaces
Introduction
editLet us recall that a Polish space is a separable, completely metrizable topological space, and that Polish groups are metrizable (topological) groups whose topology is Polish, and thus they admit a compatible metric which is left-invariant; (a topological group is metrizable iff is Hausdorff, and the identity of has a countable neighborhood basis).
Polish spaces can be classified up to a (Borel) isomorphism according to the following provable results:
- All uncountable Polish spaces are Borel isomorphic to Failed to parse (syntax error): {\displaystyle \mathbb{R } equipped with the standard topology;} This also implies that all uncountable Polish space have the cardinality of the continuum.
- Two Polish spaces are Borel isomorphic if and only if they have the same cardinality.
Furthermore, the subcategory of Polish spaces that are Borel isomorphic is, in fact, a Borel groupoid.
Category of Polish groups
editThe \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} of Polish groups} has, as its objects, all Polish groups and, as its morphisms the group homomorphisms between Polish groups, compatible with the Polish topology on .
is obviously a subcategory of the category of topological groups; moreover, is a subcategory of Failed to parse (unknown function "\grp"): {\displaystyle \mathcal{T}_{\grp}} -the category of topological groupoids and topological groupoid homomorphisms.