English
A compact set K is contained in a finite union of left translates of a set with nonempty interior.
Русский
Компактное множество K лежит в конечной объединении левых транспозиций множества с непустымInterior.
LaTeX
$$$\exists t : Finset G,\; K \subseteq \bigcup_{g\in t} g^{-1} V$ при некоторых условиях.$$
Lean4
/-- A compact set is covered by finitely many left multiplicative translates of a set
with non-empty interior. -/
@[to_additive /-- A compact set is covered by finitely many left additive translates of a set
with non-empty interior. -/
]
theorem compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty) :
∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V :=
by
obtain ⟨t, ht⟩ : ∃ t : Finset G, K ⊆ ⋃ x ∈ t, interior ((x * ·) ⁻¹' V) :=
by
refine hK.elim_finite_subcover (fun x => interior <| (x * ·) ⁻¹' V) (fun x => isOpen_interior) ?_
obtain ⟨g₀, hg₀⟩ := hV
refine fun g _ => mem_iUnion.2 ⟨g₀ * g⁻¹, ?_⟩
refine preimage_interior_subset_interior_preimage (continuous_const.mul continuous_id) ?_
rwa [mem_preimage, Function.id_def, inv_mul_cancel_right]
exact ⟨t, Subset.trans ht <| iUnion₂_mono fun g _ => interior_subset⟩
