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Topological group: multiplication and inversion are continuous
Definition
A topological group is a group (Group and abelian group) with a topology such that multiplication , , and inversion , , are continuous (Continuity of a map of topological spaces at a point and globally) for the product topology (The product set of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).
Depends on
- Group and abelian group
- Continuity of a map of topological spaces at a point and globally
- The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
Used by
- Topological abelian groups are additive but not abelian Counterexample
- A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups Definition
- Principal g bundle and associated fiber bundle Definition
- The left and right uniformities of a topological group Definition
- The left, right, upper and Roelcke uniformities of the additive topological group ℝ all equal its metric uniformity Example
- Left and right translations and inversion in a topological group are homeomorphisms Lemma
- Pointwise multiplication and concatenation of loops in a topological group agree up to homotopy Lemma
- The inverse limit of finite discrete groups is a closed topological subgroup of the full product Lemma
- Assuming Choice, a topological group is profinite exactly when it is compact, Hausdorff, and totally disconnected Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)