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[Merged by Bors] - feat(topology/stone_cech): add stone_cech_hom_ext #13472

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[Merged by Bors] - feat(topology/stone_cech): add stone_cech_hom_ext #13472

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8324ab0
unique_stone_cech_extend
mapehe Apr 15, 2022
dd295d1
reviewer suggestions
mapehe Apr 15, 2022
f6b244a
fix lint error
mapehe Apr 16, 2022
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9 changes: 9 additions & 0 deletions src/topology/stone_cech.lean
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Expand Up @@ -262,6 +262,15 @@ ultrafilter_extend_extends f
lemma continuous_stone_cech_extend : continuous (stone_cech_extend hf) :=
continuous_quot_lift _ (continuous_ultrafilter_extend f)

lemma stone_cech_hom_ext {g₁ g₂ : stone_cech α → γ}
(h₁ : continuous g₁) (h₂ : continuous g₂)
(h : g₁ ∘ stone_cech_unit = g₂ ∘ stone_cech_unit) : g₁ = g₂ :=
begin
apply continuous.ext_on dense_range_stone_cech_unit h₁ h₂,
rintros x ⟨x, rfl⟩,
apply (congr_fun h x)
end

end extension

lemma convergent_eqv_pure {u : ultrafilter α} {x : α} (ux : ↑u ≤ 𝓝 x) : u ≈ pure x :=
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