Math 2.0

Topological groups, Lie groups Definition

Completion of a topological group

Builds on Hausdorff topological groupUniform structure of a topological groupIsomorphism of topological groupsCauchy filter

Definition

Let GG be a Hausdorff topological group. The two-sided uniformity of GG is the coarsest uniform structure finer than both the left and the right uniform structures of GG. GG is complete if every Cauchy filter for the two-sided uniformity converges. A completion of GG is a pair (G^,ι)(\hat G, \iota) such that:

  1. G^\hat G is a complete Hausdorff topological group;
  2. ι ⁣:G→G^\iota \colon G \to \hat G is an isomorphism of topological groups from GG onto the subgroup ι(G)\iota(G) with the subspace topology;
  3. ι(G)\iota(G) is dense in G^\hat G.

Remarks

Every Hausdorff topological group has a completion, and any two completions are isomorphic by a unique isomorphism compatible with the maps ι\iota. Example: (ℝ,+)(\R, +) with the inclusion of (ℚ,+)(\Q, +). The group ℝ\R is complete, ℚ\Q carries the subspace topology, and ℚ\Q is dense in ℝ\R. Non-example: (ℝ,+)(\R, +) with the inclusion of (ℤ,+)(\Z, +) fails clause 3, since ℤ\Z is closed in ℝ\R and not dense. For an abelian group the left, right and two-sided uniformities agree. In general the completion for the left uniformity alone need not be a group.

Reading path

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

Show the path as a graph
Cauchy filter, concept, no evidence levelCauchy filterconceptHausdorff topological group, concept, no evidence levelHausdorff topological…conceptIsomorphism of topological groups, concept, no evidence levelIsomorphism of topolo…conceptUniform structure of a topological group, concept, no evidence levelUniform structure of …conceptCompletion of a topological group, definition, no evidence levelCompletion of a topol…definition
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.
  • no evidence level
  1. Cauchy filter concept
  2. Hausdorff topological group concept
  3. Isomorphism of topological groups concept
  4. Uniform structure of a topological group concept
  5. Completion of a topological group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_topological_vector_space
About this page
TagY63847Z06N
URIm2://Y63847Z06N
Kinddefinition
WritingW0 stub
MSC22A05, 54E15
DefinesCompletion of a topological group
UsesHausdorff topological group, Uniform structure of a topological group, Isomorphism of topological groups, Cauchy filter

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.