PlanetPhysics/Categories of Polish Groups and Polish Spaces


Let 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 groupsEdit

The \htmladdnormallink{category {} 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.