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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
namespace OAI
noncomputable section
namespace Problem340
theorem exists_compact_set_small_fiber_labels {X : Type*} [PseudoMetricSpace X]
{s : Set X} (hs : IsCompact s) {ε : ℝ} (hε : 0 < ε) :
∃ K : ℕ, ∃ b : s → Fin K,
∀ x y, b x = b y → dist (x : X) (y : X) < ε := by
classical
obtain ⟨t, _hts, ht, hcover⟩ := hs.finite_cover_balls (half_pos hε)
let := ht.fintype
have hex (x : s) : ∃ c : t, dist (x : X) (c : X) < ε / 2 := by
have hx := hcover x.property
simp only [Set.mem_iUnion, Metric.mem_ball] at hx
obtain ⟨c, hc, hxc⟩ := hx
exact ⟨⟨c, hc⟩, hxc⟩
choose c hc using hex
refine ⟨Fintype.card t, fun x => (Fintype.equivFin t) (c x), ?_⟩
intro x y hxy
have heq : c x = c y := (Fintype.equivFin t).injective hxy
have hx := hc x
have hy := hc y
rw [← heq, dist_comm (y : X) (c x : X)] at hy
have htriangle := dist_triangle (x : X) (c x : X) (y : X)
linarith
theorem exists_small_fiber_labels {X : Type*} [PseudoMetricSpace X] [CompactSpace X]
{ε : ℝ} (hε : 0 < ε) :
∃ K : ℕ, ∃ b : X → Fin K, ∀ x y, b x = b y → dist x y < ε := by
obtain ⟨K, b, hb⟩ := exists_compact_set_small_fiber_labels
(s := (Set.univ : Set X)) isCompact_univ hε
exact ⟨K, fun x => b ⟨x, Set.mem_univ x⟩,
fun x y hxy => hb ⟨x, Set.mem_univ x⟩ ⟨y, Set.mem_univ y⟩ hxy⟩
theorem exists_pos_small_fiber_labels {X : Type*} [PseudoMetricSpace X] [CompactSpace X]
[Nonempty X] {ε : ℝ} (hε : 0 < ε) :
∃ K : ℕ, 0 < K ∧ ∃ b : X → Fin K, ∀ x y, b x = b y → dist x y < ε := by
obtain ⟨K, b, hb⟩ := exists_small_fiber_labels (X := X) hε
obtain ⟨x⟩ := ‹Nonempty X›
exact ⟨K, Nat.zero_lt_of_lt (b x).isLt, b, hb⟩
end Problem340
end
end OAI
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