English
There is a canonical homeomorphism between the group of units G₀ˣ and the set {g ∈ G₀ | g ≠ 0}, given by identifying a unit with its underlying element.
Русский
Существует каноническое домашеморфизм между группой единиц G₀ˣ и множеством {g ∈ G₀ | g ≠ 0}, которому соответствует единица как элемент G₀.
LaTeX
$$$G_0^{\\times} \\;\\cong_t\\; \\{ g \\in G_0 \\mid g \\neq 0 \\}$$$
Lean4
/-- If a group with zero has continuous inversion, then its group of units is homeomorphic to
the set of nonzero elements. -/
noncomputable def unitsHomeomorphNeZero : G₀ˣ ≃ₜ { g : G₀ // g ≠ 0 } :=
Units.isEmbedding_val₀.toHomeomorph.trans <|
show _ ≃ₜ {g | _} from .setCongr <| Set.ext fun x ↦ (Units.exists_iff_ne_zero (p := (· = x))).trans <| by simp
