Topologies on linear ordered fields

In this file we prove that a linear ordered field with order topology has continuous multiplication and division (apart from zero in the denominator). We also prove theorems like Filter.Tendsto.mul_atTop: if f tends to a positive number and g tends to positive infinity, then f * g tends to positive infinity.

theorem TopologicalRing.of_norm {R : Type u_1} {𝕜 : Type u_2} [NonUnitalNonAssocRing R] [LinearOrderedField 𝕜] [TopologicalSpace R] [TopologicalAddGroup R] (norm : R → 𝕜) (norm_nonneg : ∀ (x : R), 0 ≤ norm x) (norm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y) (nhds_basis : (nhds 0).HasBasis (fun (x : 𝕜) => 0 < x) fun (ε : 𝕜) => {x : R | norm x < ε}) : TopologicalRing R

If a (possibly non-unital and/or non-associative) ring R admits a submultiplicative nonnegative norm norm : R → 𝕜, where 𝕜 is a linear ordered field, and the open balls { x | norm x < ε }, ε > 0, form a basis of neighborhoods of zero, then R is a topological ring.

@[instance 100]

instance LinearOrderedField.topologicalRing {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : TopologicalRing 𝕜

Equations

• ⋯ = ⋯

theorem Filter.Tendsto.atTop_mul {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l (nhds C)) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atTop

In a linearly ordered field with the order topology, if f tends to Filter.atTop and g tends to a positive constant C then f * g tends to Filter.atTop.

theorem Filter.Tendsto.mul_atTop {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l (nhds C)) (hg : Filter.Tendsto g l Filter.atTop) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atTop

In a linearly ordered field with the order topology, if f tends to a positive constant C and g tends to Filter.atTop then f * g tends to Filter.atTop.

theorem Filter.Tendsto.atTop_mul_neg {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l (nhds C)) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atBot

In a linearly ordered field with the order topology, if f tends to Filter.atTop and g tends to a negative constant C then f * g tends to Filter.atBot.

theorem Filter.Tendsto.neg_mul_atTop {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l (nhds C)) (hg : Filter.Tendsto g l Filter.atTop) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atBot

In a linearly ordered field with the order topology, if f tends to a negative constant C and g tends to Filter.atTop then f * g tends to Filter.atBot.

theorem Filter.Tendsto.atBot_mul {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l (nhds C)) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atBot

In a linearly ordered field with the order topology, if f tends to Filter.atBot and g tends to a positive constant C then f * g tends to Filter.atBot.

theorem Filter.Tendsto.atBot_mul_neg {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l (nhds C)) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atTop

In a linearly ordered field with the order topology, if f tends to Filter.atBot and g tends to a negative constant C then f * g tends to Filter.atTop.

theorem Filter.Tendsto.mul_atBot {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l (nhds C)) (hg : Filter.Tendsto g l Filter.atBot) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atBot

In a linearly ordered field with the order topology, if f tends to a positive constant C and g tends to Filter.atBot then f * g tends to Filter.atBot.

theorem Filter.Tendsto.neg_mul_atBot {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l (nhds C)) (hg : Filter.Tendsto g l Filter.atBot) : Filter.Tendsto (fun (x : α) => f x * g x) l Filter.atTop

In a linearly ordered field with the order topology, if f tends to a negative constant C and g tends to Filter.atBot then f * g tends to Filter.atTop.

@[simp]

theorem inv_atTop₀ {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : Filter.atTop⁻¹ = nhdsWithin 0 (Set.Ioi 0)

@[simp]

theorem inv_nhdsWithin_Ioi_zero {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : (nhdsWithin 0 (Set.Ioi 0))⁻¹ = Filter.atTop

theorem tendsto_inv_zero_atTop {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : Filter.Tendsto (fun (x : 𝕜) => x⁻¹) (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop

The function x ↦ x⁻¹ tends to +∞ on the right of 0.

theorem tendsto_inv_atTop_zero' {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : Filter.Tendsto (fun (r : 𝕜) => r⁻¹) Filter.atTop (nhdsWithin 0 (Set.Ioi 0))

The function r ↦ r⁻¹ tends to 0 on the right as r → +∞.

theorem tendsto_inv_atTop_zero {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : Filter.Tendsto (fun (r : 𝕜) => r⁻¹) Filter.atTop (nhds 0)

theorem Filter.Tendsto.div_atTop {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {a : 𝕜} (h : Filter.Tendsto f l (nhds a)) (hg : Filter.Tendsto g l Filter.atTop) : Filter.Tendsto (fun (x : α) => f x / g x) l (nhds 0)

theorem Filter.Tendsto.inv_tendsto_atTop {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} (h : Filter.Tendsto f l Filter.atTop) : Filter.Tendsto f⁻¹ l (nhds 0)

theorem Filter.Tendsto.inv_tendsto_zero {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} (h : Filter.Tendsto f l (nhdsWithin 0 (Set.Ioi 0))) : Filter.Tendsto f⁻¹ l Filter.atTop

theorem bdd_le_mul_tendsto_zero' {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} (C : 𝕜) (hf : ∀ᶠ (x : α) in l, |f x| ≤ C) (hg : Filter.Tendsto g l (nhds 0)) : Filter.Tendsto (fun (x : α) => f x * g x) l (nhds 0)

If g tends to zero and there exists a constant C : 𝕜 such that eventually |f x| ≤ C, then the product f * g tends to zero.

theorem bdd_le_mul_tendsto_zero {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {b : 𝕜} {B : 𝕜} (hb : ∀ᶠ (x : α) in l, b ≤ f x) (hB : ∀ᶠ (x : α) in l, f x ≤ B) (hg : Filter.Tendsto g l (nhds 0)) : Filter.Tendsto (fun (x : α) => f x * g x) l (nhds 0)

If g tends to zero and there exist constants b B : 𝕜 such that eventually b ≤ f x| ≤ B, then the product f * g tends to zero.

theorem tendsto_bdd_div_atTop_nhds_zero {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} {g : α → 𝕜} {b : 𝕜} {B : 𝕜} (hb : ∀ᶠ (x : α) in l, b ≤ f x) (hB : ∀ᶠ (x : α) in l, f x ≤ B) (hg : Filter.Tendsto g l Filter.atTop) : Filter.Tendsto (fun (x : α) => f x / g x) l (nhds 0)

If g tends to atTop and there exist constants b B : 𝕜 such that eventually b ≤ f x| ≤ B, then the quotient f / g tends to zero.

theorem tendsto_pow_neg_atTop {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℕ} (hn : n ≠ 0) : Filter.Tendsto (fun (x : 𝕜) => x ^ (-↑n)) Filter.atTop (nhds 0)

The function x^(-n) tends to 0 at +∞ for any positive natural n. A version for positive real powers exists as tendsto_rpow_neg_atTop.

theorem tendsto_zpow_atTop_zero {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℤ} (hn : n < 0) : Filter.Tendsto (fun (x : 𝕜) => x ^ n) Filter.atTop (nhds 0)

theorem tendsto_const_mul_zpow_atTop_zero {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℤ} {c : 𝕜} (hn : n < 0) : Filter.Tendsto (fun (x : 𝕜) => c * x ^ n) Filter.atTop (nhds 0)

theorem tendsto_const_mul_pow_nhds_iff' {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℕ} {c : 𝕜} {d : 𝕜} : Filter.Tendsto (fun (x : 𝕜) => c * x ^ n) Filter.atTop (nhds d) ↔ (c = 0 ∨ n = 0) ∧ c = d

theorem tendsto_const_mul_pow_nhds_iff {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℕ} {c : 𝕜} {d : 𝕜} (hc : c ≠ 0) : Filter.Tendsto (fun (x : 𝕜) => c * x ^ n) Filter.atTop (nhds d) ↔ n = 0 ∧ c = d

theorem tendsto_const_mul_zpow_atTop_nhds_iff {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℤ} {c : 𝕜} {d : 𝕜} (hc : c ≠ 0) : Filter.Tendsto (fun (x : 𝕜) => c * x ^ n) Filter.atTop (nhds d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0

@[instance 100]

instance LinearOrderedSemifield.toHasContinuousInv₀ {𝕜 : Type u_3} [LinearOrderedSemifield 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] [ContinuousMul 𝕜] : HasContinuousInv₀ 𝕜

Equations

• ⋯ = ⋯

@[instance 100]

instance LinearOrderedField.toTopologicalDivisionRing {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] : TopologicalDivisionRing 𝕜

Equations

• ⋯ = ⋯

theorem nhdsWithin_pos_comap_mul_left {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {x : 𝕜} (hx : 0 < x) : Filter.comap (fun (x_1 : 𝕜) => x * x_1) (nhdsWithin 0 (Set.Ioi 0)) = nhdsWithin 0 (Set.Ioi 0)

theorem eventually_nhdsWithin_pos_mul_left {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {x : 𝕜} (hx : 0 < x) {p : 𝕜 → Prop} (h : ∀ᶠ (ε : 𝕜) in nhdsWithin 0 (Set.Ioi 0), p ε) : ∀ᶠ (ε : 𝕜) in nhdsWithin 0 (Set.Ioi 0), p (x * ε)