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Left and right translations and inversion in a topological group are homeomorphisms
Statement
For in a topological group, , , and are homeomorphisms.
Facts & Assumptions
Given: A topological group and .
Multiplication and inversion are continuous (Topological group: multiplication and inversion are continuous).
and inversion is its own inverse (In a group , and , the order of the last product being essential).
A homeomorphism is a continuous bijection with continuous inverse (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).
Proof
The maps and are continuous as composites of multiplication with constant maps, and their inverses are and , also continuous.
Inversion is continuous and its own continuous inverse by [L1] and [L2].
Thus all three maps are homeomorphisms by [L3].
Depends on
Used by
- Involution on L1 of a locally compact group Definition
- A compact identity neighbourhood in the dual Lemma
- A compactly generated LCA group with no open subgroup of infinite index is Euclidean times compact Lemma
- A locally compact subgroup of a Hausdorff topological group is closed Lemma
- Compact-open neighbourhoods on the dual give a neighbourhood basis on the group Lemma
- Continuous characters separate points of an LCA group Lemma
- Continuous finite-dimensional representations of profinite groups factor through finite quotients Lemma
- Dual homomorphisms: continuity, and the annihilator of a closed subgroup Lemma
- Evaluation of characters is jointly continuous Lemma
- Every LCA group has an open compactly generated subgroup with no open subgroup of infinite index Lemma
- Every locally compact Hausdorff group has an open sigma-compact subgroup Lemma
- Matrix coefficients of finite-dimensional representations separate points of a compact group Lemma
- The L1 involution is isometric, involutive and reverses convolution Lemma
- The quotient of an LCA group by a closed subgroup is LCA Lemma
- Totally disconnected LCA groups have bases of compact open subgroups Lemma
- Compact groups have discrete duals and discrete groups have compact duals Theorem
- Compact groups have property (T) by Haar averaging Theorem
- L1 group algebras have a contractively bounded approximate identity Theorem
- Pontryagin biduality: the evaluation map is a topological isomorphism Theorem
- Property (T) passes to Hausdorff quotients Theorem
- The dual of a locally compact abelian group is locally compact abelian Theorem
- The left and right uniformities of a topological group induce its topology, and inversion interchanges them Theorem
Dependency tree · two levels
13 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)