PlanetPhysics/Categories of Polish Groups and Polish Spaces

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Introduction[edit | edit source]

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 groups[edit | edit source]

The \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.