Topology on ℝ≥0

The natural topology on ℝ≥0 (the one induced from ℝ), and a basic API.

Main definitions

Instances for the following typeclasses are defined:

• TopologicalSpace ℝ≥0
• TopologicalSemiring ℝ≥0
• SecondCountableTopology ℝ≥0
• OrderTopology ℝ≥0
• ProperSpace ℝ≥0
• ContinuousSub ℝ≥0
• HasContinuousInv₀ ℝ≥0 (continuity of x⁻¹ away from 0)
• ContinuousSMul ℝ≥0 α (whenever α has a continuous MulAction ℝ α)

Everything is inherited from the corresponding structures on the reals.

Main statements

Various mathematically trivial lemmas are proved about the compatibility of limits and sums in ℝ≥0 and ℝ. For example

• tendsto_coe {f : Filter α} {m : α → ℝ≥0} {x : ℝ≥0} : Filter.Tendsto (fun a, (m a : ℝ)) f (𝓝 (x : ℝ)) ↔ Filter.Tendsto m f (𝓝 x)

says that the limit of a filter along a map to ℝ≥0 is the same in ℝ and ℝ≥0, and

• coe_tsum {f : α → ℝ≥0} : ((∑'a, f a) : ℝ) = (∑'a, (f a : ℝ))

says that says that a sum of elements in ℝ≥0 is the same in ℝ and ℝ≥0.

Similarly, some mathematically trivial lemmas about infinite sums are proved, a few of which rely on the fact that subtraction is continuous.

instance NNReal.instTopologicalSpace : TopologicalSpace NNReal

Equations

• NNReal.instTopologicalSpace = inferInstance

instance NNReal.instTopologicalSemiring : TopologicalSemiring NNReal

Equations

• NNReal.instTopologicalSemiring = ⋯

instance NNReal.instSecondCountableTopology : SecondCountableTopology NNReal

Equations

• NNReal.instSecondCountableTopology = ⋯

instance NNReal.instOrderTopology : OrderTopology NNReal

Equations

• NNReal.instOrderTopology = ⋯

instance NNReal.instCompleteSpace : CompleteSpace NNReal

Equations

• NNReal.instCompleteSpace = ⋯

instance NNReal.instContinuousStar : ContinuousStar NNReal

Equations

• NNReal.instContinuousStar = ⋯

theorem NNReal.isOpen_Ico_zero {x : NNReal} : IsOpen (Set.Ico 0 x)

theorem continuous_real_toNNReal : Continuous Real.toNNReal

noncomputable def ContinuousMap.realToNNReal : C(ℝ, NNReal)

Real.toNNReal bundled as a continuous map for convenience.

Equations

• ContinuousMap.realToNNReal = { toFun := Real.toNNReal, continuous_toFun := continuous_real_toNNReal }

@[simp]

theorem ContinuousMap.realToNNReal_apply : ⇑ContinuousMap.realToNNReal = Real.toNNReal

theorem NNReal.continuous_coe : Continuous NNReal.toReal

theorem ContinuousOn.ofReal_map_toNNReal {f : NNReal → NNReal} {s : Set ℝ} {t : Set NNReal} (hf : ContinuousOn f t) (h : Set.MapsTo Real.toNNReal s t) : ContinuousOn (fun (x : ℝ) => ↑(f x.toNNReal)) s

def ContinuousMap.coeNNRealReal : C(NNReal, ℝ)

Embedding of ℝ≥0 to ℝ as a bundled continuous map.

Equations

• ContinuousMap.coeNNRealReal = { toFun := NNReal.toReal, continuous_toFun := NNReal.continuous_coe }

@[simp]

theorem ContinuousMap.coeNNRealReal_apply : ⇑ContinuousMap.coeNNRealReal = NNReal.toReal

instance NNReal.ContinuousMap.canLift {X : Type u_2} [TopologicalSpace X] : CanLift C(X, ℝ) C(X, NNReal) ContinuousMap.coeNNRealReal.comp fun (f : C(X, ℝ)) => ∀ (x : X), 0 ≤ f x

Equations

• ⋯ = ⋯

@[simp]

theorem NNReal.tendsto_coe {α : Type u_1} {f : Filter α} {m : α → NNReal} {x : NNReal} : Filter.Tendsto (fun (a : α) => ↑(m a)) f (nhds ↑x) ↔ Filter.Tendsto m f (nhds x)

theorem NNReal.tendsto_coe' {α : Type u_1} {f : Filter α} [f.NeBot] {m : α → NNReal} {x : ℝ} : Filter.Tendsto (fun (a : α) => ↑(m a)) f (nhds x) ↔ ∃ (hx : 0 ≤ x), Filter.Tendsto m f (nhds ⟨x, hx⟩)

@[simp]

theorem NNReal.map_coe_atTop : Filter.map NNReal.toReal Filter.atTop = Filter.atTop

@[simp]

theorem NNReal.comap_coe_atTop : Filter.comap NNReal.toReal Filter.atTop = Filter.atTop

@[simp]

theorem NNReal.tendsto_coe_atTop {α : Type u_1} {f : Filter α} {m : α → NNReal} : Filter.Tendsto (fun (a : α) => ↑(m a)) f Filter.atTop ↔ Filter.Tendsto m f Filter.atTop

theorem tendsto_real_toNNReal {α : Type u_1} {f : Filter α} {m : α → ℝ} {x : ℝ} (h : Filter.Tendsto m f (nhds x)) : Filter.Tendsto (fun (a : α) => (m a).toNNReal) f (nhds x.toNNReal)

@[simp]

theorem Real.map_toNNReal_atTop : Filter.map Real.toNNReal Filter.atTop = Filter.atTop

theorem tendsto_real_toNNReal_atTop : Filter.Tendsto Real.toNNReal Filter.atTop Filter.atTop

@[simp]

theorem Real.comap_toNNReal_atTop : Filter.comap Real.toNNReal Filter.atTop = Filter.atTop

@[simp]

theorem Real.tendsto_toNNReal_atTop_iff {α : Type u_1} {l : Filter α} {f : α → ℝ} : Filter.Tendsto (fun (x : α) => (f x).toNNReal) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop

theorem Real.tendsto_toNNReal_atTop : Filter.Tendsto Real.toNNReal Filter.atTop Filter.atTop

theorem NNReal.nhds_zero : nhds 0 = ⨅ (a : NNReal), ⨅ (_ : a ≠ 0), Filter.principal (Set.Iio a)

theorem NNReal.nhds_zero_basis : (nhds 0).HasBasis (fun (a : NNReal) => 0 < a) fun (a : NNReal) => Set.Iio a

instance NNReal.instContinuousSub : ContinuousSub NNReal

Equations

• NNReal.instContinuousSub = ⋯

instance NNReal.instHasContinuousInv₀ : HasContinuousInv₀ NNReal

Equations

• NNReal.instHasContinuousInv₀ = ⋯

instance NNReal.instContinuousSMulOfReal {α : Type u_1} [TopologicalSpace α] [MulAction ℝ α] [ContinuousSMul ℝ α] : ContinuousSMul NNReal α

Equations

• ⋯ = ⋯

theorem NNReal.hasSum_coe {α : Type u_1} {f : α → NNReal} {r : NNReal} : HasSum (fun (a : α) => ↑(f a)) ↑r ↔ HasSum f r

theorem HasSum.toNNReal {α : Type u_1} {f : α → ℝ} {y : ℝ} (hf₀ : ∀ (n : α), 0 ≤ f n) (hy : HasSum f y) : HasSum (fun (x : α) => (f x).toNNReal) y.toNNReal

theorem NNReal.hasSum_real_toNNReal_of_nonneg {α : Type u_1} {f : α → ℝ} (hf_nonneg : ∀ (n : α), 0 ≤ f n) (hf : Summable f) : HasSum (fun (n : α) => (f n).toNNReal) (∑' (n : α), f n).toNNReal

theorem NNReal.summable_coe {α : Type u_1} {f : α → NNReal} : (Summable fun (a : α) => ↑(f a)) ↔ Summable f

theorem NNReal.summable_mk {α : Type u_1} {f : α → ℝ} (hf : ∀ (n : α), 0 ≤ f n) : (Summable fun (n : α) => ⟨f n, ⋯⟩) ↔ Summable f

theorem NNReal.coe_tsum {α : Type u_1} {f : α → NNReal} : ↑(∑' (a : α), f a) = ∑' (a : α), ↑(f a)

theorem NNReal.coe_tsum_of_nonneg {α : Type u_1} {f : α → ℝ} (hf₁ : ∀ (n : α), 0 ≤ f n) : ⟨∑' (n : α), f n, ⋯⟩ = ∑' (n : α), ⟨f n, ⋯⟩

theorem NNReal.tsum_mul_left {α : Type u_1} (a : NNReal) (f : α → NNReal) : ∑' (x : α), a * f x = a * ∑' (x : α), f x

theorem NNReal.tsum_mul_right {α : Type u_1} (f : α → NNReal) (a : NNReal) : ∑' (x : α), f x * a = (∑' (x : α), f x) * a

theorem NNReal.summable_comp_injective {α : Type u_1} {β : Type u_2} {f : α → NNReal} (hf : Summable f) {i : β → α} (hi : Function.Injective i) : Summable (f ∘ i)

theorem NNReal.summable_nat_add (f : ℕ → NNReal) (hf : Summable f) (k : ℕ) : Summable fun (i : ℕ) => f (i + k)

theorem NNReal.summable_nat_add_iff {f : ℕ → NNReal} (k : ℕ) : (Summable fun (i : ℕ) => f (i + k)) ↔ Summable f

theorem NNReal.hasSum_nat_add_iff {f : ℕ → NNReal} (k : ℕ) {a : NNReal} : HasSum (fun (n : ℕ) => f (n + k)) a ↔ HasSum f (a + ∑ i ∈ Finset.range k, f i)

theorem NNReal.sum_add_tsum_nat_add {f : ℕ → NNReal} (k : ℕ) (hf : Summable f) : ∑' (i : ℕ), f i = ∑ i ∈ Finset.range k, f i + ∑' (i : ℕ), f (i + k)

theorem NNReal.iInf_real_pos_eq_iInf_nnreal_pos {α : Type u_1} [CompleteLattice α] {f : ℝ → α} : ⨅ (n : ℝ), ⨅ (_ : 0 < n), f n = ⨅ (n : NNReal), ⨅ (_ : 0 < n), f ↑n

theorem NNReal.tendsto_cofinite_zero_of_summable {α : Type u_1} {f : α → NNReal} (hf : Summable f) : Filter.Tendsto f Filter.cofinite (nhds 0)

theorem NNReal.tendsto_atTop_zero_of_summable {f : ℕ → NNReal} (hf : Summable f) : Filter.Tendsto f Filter.atTop (nhds 0)

theorem NNReal.tendsto_tsum_compl_atTop_zero {α : Type u_1} (f : α → NNReal) : Filter.Tendsto (fun (s : Finset α) => ∑' (b : { x : α // x ∉ s }), f ↑b) Filter.atTop (nhds 0)

The sum over the complement of a finset tends to 0 when the finset grows to cover the whole space. This does not need a summability assumption, as otherwise all sums are zero.

def NNReal.powOrderIso (n : ℕ) (hn : n ≠ 0) : NNReal ≃o NNReal

x ↦ x ^ n as an order isomorphism of ℝ≥0.

Equations

• NNReal.powOrderIso n hn = StrictMono.orderIsoOfSurjective (fun (x : NNReal) => x ^ n) ⋯ ⋯

theorem Real.tendsto_of_bddAbove_monotone {f : ℕ → ℝ} (h_bdd : BddAbove (Set.range f)) (h_mon : Monotone f) : ∃ (r : ℝ), Filter.Tendsto f Filter.atTop (nhds r)

A monotone, bounded above sequence f : ℕ → ℝ has a finite limit.

theorem Real.tendsto_of_bddBelow_antitone {f : ℕ → ℝ} (h_bdd : BddBelow (Set.range f)) (h_ant : Antitone f) : ∃ (r : ℝ), Filter.Tendsto f Filter.atTop (nhds r)

An antitone, bounded below sequence f : ℕ → ℝ has a finite limit.

theorem NNReal.tendsto_of_antitone {f : ℕ → NNReal} (h_ant : Antitone f) : ∃ (r : NNReal), Filter.Tendsto f Filter.atTop (nhds r)

An antitone sequence f : ℕ → ℝ≥0 has a finite limit.

instance NNReal.instProperSpace : ProperSpace NNReal

Equations

• NNReal.instProperSpace = ⋯