Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Left and right translations and inversion in a topological group are homeomorphisms

Statement

For gg in a topological group, Lg(x)=gxL_g(x)=gx, Rg(x)=xgR_g(x)=xg, and ι(x)=x−1\iota(x)=x^{-1} are homeomorphisms.

Facts & Assumptions

Given: A topological group GG and g∈Gg\in G.

[L1]

Multiplication and inversion are continuous (Topological group: multiplication and inversion are continuous).

[L2]

(gh)−1=h−1g−1(gh)^{-1}=h^{-1}g^{-1} and inversion is its own inverse (In a group e−1=ee^{-1} = e, (g−1)−1=g(g^{-1})^{-1} = g and (gh)−1=h−1g−1(gh)^{-1} = h^{-1}g^{-1}, the order of the last product being essential).

[L3]

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

technique · direct
1.1

The maps LgL_g and RgR_g are continuous as composites of multiplication with constant maps, and their inverses are Lg−1L_{g^{-1}} and Rg−1R_{g^{-1}}, also continuous.

L1L2
1.2

Inversion is continuous and its own continuous inverse by [L1] and [L2].

L1L2
2.1

Thus all three maps are homeomorphisms by [L3].

step 1.1step 1.2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources