Math 2.0

Topological groups, Lie groups Definition

Polish group

Builds on Topological groupPolish spaceSymmetric groupBaire space

Definition

A Polish group is a topological group GG whose underlying topological space is a Polish space. That is, the topology of GG is separable and is induced by some complete metric.

Remarks

Example: the symmetric group S∞S_\infty of all bijections ℕ→ℕ\N \to \N, with the topology of pointwise convergence. It is a GδG_\delta subset of the Polish space ℕℕ\N^\N, hence Polish, and composition and inversion are continuous for this topology. Non-example: (ℚ,+)(\Q, +) with the topology from ℝ\R. It is separable and metrizable, but it is countable without isolated points. So it is a countable union of nowhere dense singletons and is not a Baire space; hence no complete metric induces its topology. The metric need not be invariant: a Polish group always has a compatible left-invariant metric, but this metric may fail to be complete.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Baire space, concept, no evidence levelBaire spaceconceptPolish space, concept, no evidence levelPolish spaceconceptSymmetric group, concept, no evidence levelSymmetric groupconceptTopological group, concept, no evidence levelTopological groupconceptPolish group, definition, no evidence levelPolish groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Baire space concept
  2. Polish space concept
  3. Symmetric group concept
  4. Topological group concept
  5. Polish group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Polish_space
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MSC22A05, 54H11, 54H05
DefinesPolish group
UsesTopological group, Polish space, Symmetric group, Baire space

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