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The rationals are discrete in the adeles of the rationals #266

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Dec 2, 2024
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The rationals are discrete in the adeles of the rationals #266

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14 changes: 13 additions & 1 deletion FLT/NumberField/AdeleRing.lean
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Expand Up @@ -94,8 +94,20 @@ theorem Rat.AdeleRing.zero_discrete : ∃ U : Set (AdeleRing ℚ),

-- Maybe this discreteness isn't even stated in the best way?
-- I'm ambivalent about how it's stated
open Pointwise in
theorem Rat.AdeleRing.discrete : ∀ q : ℚ, ∃ U : Set (AdeleRing ℚ),
IsOpen U ∧ (algebraMap ℚ (AdeleRing ℚ)) ⁻¹' U = {q} := sorry -- issue #256
IsOpen U ∧ (algebraMap ℚ (AdeleRing ℚ)) ⁻¹' U = {q} := by
obtain ⟨V, hV, hV0⟩ := zero_discrete
intro q
set ι := algebraMap ℚ (AdeleRing ℚ) with hι
set qₐ := ι q with hqₐ
set f := Homeomorph.subLeft qₐ with hf
use f ⁻¹' V, f.isOpen_preimage.mpr hV
have : f ∘ ι = ι ∘ Homeomorph.subLeft q := by ext; simp [hf, hqₐ]
rw [← Set.preimage_comp, this, Set.preimage_comp, hV0]
ext
simp only [Set.mem_preimage, Homeomorph.subLeft_apply, Set.mem_singleton_iff, sub_eq_zero, eq_comm]


variable (K : Type*) [Field K] [NumberField K]

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