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chore(topology/algebra/ordered): move squeeze_zero from metric_space #1828

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chore(topology/algebra/ordered): move squeeze_zero from metric_space #1828

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8 changes: 8 additions & 0 deletions src/topology/algebra/ordered.lean
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Expand Up @@ -286,6 +286,14 @@ tendsto_orderable.2
have {b : β | h b < a'} ∈ b, from (tendsto_orderable.1 hh).right a' h',
by filter_upwards [this, hfh] assume a h₁ h₂, lt_of_le_of_lt h₂ h₁⟩

/-- Also known as squeeze or sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
for the general case. -/
lemma squeeze_zero [has_zero α] {f g : β → α} {b : filter β} (hf : ∀t, 0 ≤ f t)
(hft : ∀t, f t ≤ g t) (g0 : tendsto g b (𝓝 0)) :
tendsto f b (𝓝 0) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds g0
(univ_mem_sets' hf) (univ_mem_sets' hft)

lemma nhds_orderable_unbounded {a : α} (hu : ∃u, a < u) (hl : ∃l, l < a) :
𝓝 a = (⨅l (h₂ : l < a) u (h₂ : a < u), principal (Ioo l u)) :=
let ⟨u, hu⟩ := hu, ⟨l, hl⟩ := hl in
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7 changes: 0 additions & 7 deletions src/topology/metric_space/basic.lean
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Expand Up @@ -661,13 +661,6 @@ lemma closed_ball_Icc {x r : ℝ} : closed_ball x r = Icc (x-r) (x+r) :=
by ext y; rw [mem_closed_ball, dist_comm, real.dist_eq,
abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add', sub_le]

lemma squeeze_zero {α} {f g : α → ℝ} {t₀ : filter α} (hf : ∀t, 0 ≤ f t) (hft : ∀t, f t ≤ g t)
(g0 : tendsto g t₀ (𝓝 0)) : tendsto f t₀ (𝓝 0) :=
begin
apply tendsto_of_tendsto_of_tendsto_of_le_of_le (tendsto_const_nhds) g0;
simp [*]; exact filter.univ_mem_sets
end

theorem metric.uniformity_eq_comap_nhds_zero :
𝓤 α = comap (λp:α×α, dist p.1 p.2) (𝓝 (0 : ℝ)) :=
begin
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