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stringclasses 32
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| filename
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powOrderIso (n : ℕ) (hn : n ≠ 0) : ℝ≥0 ≃o ℝ≥0 :=
StrictMono.orderIsoOfSurjective (fun x ↦ x ^ n) (fun x y h =>
pow_left_strictMonoOn₀ hn (zero_le x) (zero_le y) h) <|
(continuous_id.pow _).surjective (tendsto_pow_atTop hn) <| by
simpa [OrderBot.atBot_eq, pos_iff_ne_zero]
|
def
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
powOrderIso
|
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. -/
nonrec theorem tendsto_tsum_compl_atTop_zero {α : Type*} (f : α → ℝ≥0) :
Tendsto (fun s : Finset α => ∑' b : { x // x ∉ s }, f b) atTop (𝓝 0) := by
simp_rw [← tendsto_coe, coe_tsum, NNReal.coe_zero]
exact tendsto_tsum_compl_atTop_zero fun a : α => (f a : ℝ)
/-- `x ↦ x ^ n` as an order isomorphism of `ℝ≥0`.
|
_root_.Real.tendsto_of_bddAbove_monotone {f : ℕ → ℝ} (h_bdd : BddAbove (Set.range f))
(h_mon : Monotone f) : ∃ r : ℝ, Tendsto f atTop (𝓝 r) := by
obtain ⟨B, hB⟩ := Real.exists_isLUB (Set.range_nonempty f) h_bdd
exact ⟨B, tendsto_atTop_isLUB h_mon hB⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.Real.tendsto_of_bddAbove_monotone
|
A monotone, bounded above sequence `f : ℕ → ℝ` has a finite limit.
|
_root_.Real.tendsto_of_bddBelow_antitone {f : ℕ → ℝ} (h_bdd : BddBelow (Set.range f))
(h_ant : Antitone f) : ∃ r : ℝ, Tendsto f atTop (𝓝 r) := by
obtain ⟨B, hB⟩ := Real.exists_isGLB (Set.range_nonempty f) h_bdd
exact ⟨B, tendsto_atTop_isGLB h_ant hB⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
_root_.Real.tendsto_of_bddBelow_antitone
|
An antitone, bounded below sequence `f : ℕ → ℝ` has a finite limit.
|
tendsto_of_antitone {f : ℕ → ℝ≥0} (h_ant : Antitone f) :
∃ r : ℝ≥0, Tendsto f atTop (𝓝 r) := by
have h_bdd_0 : (0 : ℝ) ∈ lowerBounds (Set.range fun n : ℕ => (f n : ℝ)) := by
rintro r ⟨n, hn⟩
simp_rw [← hn]
exact NNReal.coe_nonneg _
obtain ⟨L, hL⟩ := Real.tendsto_of_bddBelow_antitone ⟨0, h_bdd_0⟩ h_ant
have hL0 : 0 ≤ L :=
haveI h_glb : IsGLB (Set.range fun n => (f n : ℝ)) L := isGLB_of_tendsto_atTop h_ant hL
(le_isGLB_iff h_glb).mpr h_bdd_0
exact ⟨⟨L, hL0⟩, NNReal.tendsto_coe.mp hL⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
tendsto_of_antitone
|
An antitone sequence `f : ℕ → ℝ≥0` has a finite limit.
|
iSup_pow_of_ne_zero (hn : n ≠ 0) (f : ι → ℝ≥0) : (⨆ i, f i) ^ n = ⨆ i, f i ^ n :=
(NNReal.powOrderIso n hn).map_ciSup' _
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
iSup_pow_of_ne_zero
| null |
iSup_pow [Nonempty ι] (f : ι → ℝ≥0) (n : ℕ) : (⨆ i, f i) ^ n = ⨆ i, f i ^ n := by
by_cases hn : n = 0
· simp [hn]
· exact iSup_pow_of_ne_zero hn _
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
iSup_pow
| null |
powOrderIso (n : ℕ) (hn : n ≠ 0) : ℝ≥0∞ ≃o ℝ≥0∞ :=
(NNReal.powOrderIso n hn).withTopCongr.copy (· ^ n) _
(by cases n; (· cases hn rfl); · ext (_ | _) <;> rfl) rfl
|
def
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
powOrderIso
|
`x ↦ x ^ n` as an order isomorphism of `ℝ≥0∞`.
See also `ENNReal.orderIsoRpow`.
|
iSup_pow_of_ne_zero (hn : n ≠ 0) (f : ι → ℝ≥0∞) : (⨆ i, f i) ^ n = ⨆ i, f i ^ n :=
(powOrderIso n hn).map_iSup _
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
iSup_pow_of_ne_zero
| null |
iSup_pow [Nonempty ι] (f : ι → ℝ≥0∞) (n : ℕ) : (⨆ i, f i) ^ n = ⨆ i, f i ^ n := by
by_cases hn : n = 0
· simp [hn]
· exact iSup_pow_of_ne_zero hn _
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
iSup_pow
| null |
iSup₂_pow_of_ne_zero {κ : ι → Sort*} (f : (i : ι) → κ i → ℝ≥0∞) {n : ℕ} (hn : n ≠ 0) :
(⨆ i, ⨆ j, f i j) ^ n = ⨆ i, ⨆ j, f i j ^ n :=
(powOrderIso n hn).map_iSup₂ f
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
iSup₂_pow_of_ne_zero
| null |
Real.iSup_pow [Nonempty ι] {f : ι → ℝ} (hf : ∀ i, 0 ≤ f i) (n : ℕ) :
(⨆ i, f i) ^ n = ⨆ i, f i ^ n := by
lift f to ι → ℝ≥0 using hf; dsimp; exact mod_cast NNReal.iSup_pow f n
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
Real.iSup_pow
| null |
Real.iSup_pow_of_ne_zero {f : ι → ℝ} (hf : ∀ i, 0 ≤ f i) (hn : n ≠ 0) :
(⨆ i, f i) ^ n = ⨆ i, f i ^ n := by
cases isEmpty_or_nonempty ι
· simp [hn]
· exact iSup_pow hf _
|
lemma
|
Topology
|
[
"Mathlib.Data.NNReal.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Ring.Real",
"Mathlib.Topology.ContinuousMap.Basic"
] |
Mathlib/Topology/Instances/NNReal/Lemmas.lean
|
Real.iSup_pow_of_ne_zero
| null |
Real.isTopologicalBasis_Ioo_rat :
@IsTopologicalBasis ℝ _ (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) b}) :=
isTopologicalBasis_of_isOpen_of_nhds (by simp +contextual [isOpen_Ioo])
fun a _ hav hv =>
let ⟨_, _, ⟨hl, hu⟩, h⟩ := mem_nhds_iff_exists_Ioo_subset.mp (IsOpen.mem_nhds hv hav)
let ⟨q, hlq, hqa⟩ := exists_rat_btwn hl
let ⟨p, hap, hpu⟩ := exists_rat_btwn hu
⟨Ioo q p, by
simp only [mem_iUnion]
exact ⟨q, p, Rat.cast_lt.1 <| hqa.trans hap, rfl⟩, ⟨hqa, hap⟩, fun _ ⟨hqa', ha'p⟩ =>
h ⟨hlq.trans hqa', ha'p.trans hpu⟩⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.isTopologicalBasis_Ioo_rat
| null |
Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop := by
simp only [← comap_dist_right_atTop (0 : ℝ), Real.dist_eq, sub_zero, comap_abs_atTop]
/- TODO(Mario): Prove that these are uniform isomorphisms instead of uniform embeddings
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.cobounded_eq
| null |
uniform_embedding_add_rat {r : ℚ} : uniform_embedding (fun p : ℚ => p + r) :=
_
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
uniform_embedding_add_rat
| null |
uniform_embedding_mul_rat {q : ℚ} (hq : q ≠ 0) : uniform_embedding ((*) q) :=
_ -/
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
uniform_embedding_mul_rat
| null |
Real.mem_closure_iff {s : Set ℝ} {x : ℝ} :
x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε := by
simp [mem_closure_iff_nhds_basis nhds_basis_ball, Real.dist_eq]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.mem_closure_iff
| null |
Real.uniformContinuous_inv (s : Set ℝ) {r : ℝ} (r0 : 0 < r) (H : ∀ x ∈ s, r ≤ |x|) :
UniformContinuous fun p : s => p.1⁻¹ :=
Metric.uniformContinuous_iff.2 fun _ε ε0 =>
let ⟨δ, δ0, Hδ⟩ := rat_inv_continuous_lemma abs ε0 r0
⟨δ, δ0, fun {a b} h => Hδ (H _ a.2) (H _ b.2) h⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.uniformContinuous_inv
| null |
Real.uniformContinuous_abs : UniformContinuous (abs : ℝ → ℝ) :=
Metric.uniformContinuous_iff.2 fun ε ε0 =>
⟨ε, ε0, fun _ _ ↦ lt_of_le_of_lt (abs_abs_sub_abs_le_abs_sub _ _)⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.uniformContinuous_abs
| null |
Real.continuous_inv : Continuous fun a : { r : ℝ // r ≠ 0 } => a.val⁻¹ :=
continuousOn_inv₀.restrict
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.continuous_inv
| null |
Real.uniformContinuous_mul (s : Set (ℝ × ℝ)) {r₁ r₂ : ℝ}
(H : ∀ x ∈ s, |(x : ℝ × ℝ).1| < r₁ ∧ |x.2| < r₂) :
UniformContinuous fun p : s => p.1.1 * p.1.2 :=
Metric.uniformContinuous_iff.2 fun _ε ε0 =>
let ⟨δ, δ0, Hδ⟩ := rat_mul_continuous_lemma abs ε0
⟨δ, δ0, fun {a b} h =>
let ⟨h₁, h₂⟩ := max_lt_iff.1 h
Hδ (H _ a.2).1 (H _ b.2).2 h₁ h₂⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.uniformContinuous_mul
| null |
Real.totallyBounded_ball (x ε : ℝ) : TotallyBounded (ball x ε) := by
rw [Real.ball_eq_Ioo]; apply totallyBounded_Ioo
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.totallyBounded_ball
| null |
Real.subfield_eq_of_closed {K : Subfield ℝ} (hc : IsClosed (K : Set ℝ)) : K = ⊤ := by
rw [SetLike.ext'_iff, Subfield.coe_top, ← hc.closure_eq]
refine Rat.denseRange_cast.mono ?_ |>.closure_eq
rintro - ⟨_, rfl⟩
exact SubfieldClass.ratCast_mem K _
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.subfield_eq_of_closed
| null |
Real.exists_seq_rat_strictMono_tendsto (x : ℝ) :
∃ u : ℕ → ℚ, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto (u · : ℕ → ℝ) atTop (𝓝 x) :=
Rat.denseRange_cast.exists_seq_strictMono_tendsto Rat.cast_strictMono.monotone x
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.exists_seq_rat_strictMono_tendsto
| null |
Real.exists_seq_rat_strictAnti_tendsto (x : ℝ) :
∃ u : ℕ → ℚ, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto (u · : ℕ → ℝ) atTop (𝓝 x) :=
Rat.denseRange_cast.exists_seq_strictAnti_tendsto Rat.cast_strictMono.monotone x
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Real.exists_seq_rat_strictAnti_tendsto
| null |
closure_ordConnected_inter_rat {s : Set ℝ} (conn : s.OrdConnected) (nt : s.Nontrivial) :
closure (s ∩ .range Rat.cast) = closure s :=
(closure_mono inter_subset_left).antisymm <| isClosed_closure.closure_subset_iff.mpr fun x hx ↦
Real.mem_closure_iff.mpr fun ε ε_pos ↦ by
have ⟨z, hz, ne⟩ := nt.exists_ne x
refine ne.lt_or_gt.elim (fun lt ↦ ?_) fun lt ↦ ?_
· have ⟨q, h₁, h₂⟩ := exists_rat_btwn (max_lt lt (sub_lt_self x ε_pos))
rw [max_lt_iff] at h₁
refine ⟨q, ⟨conn.out hz hx ⟨h₁.1.le, h₂.le⟩, q, rfl⟩, ?_⟩
simpa only [abs_sub_comm, abs_of_pos (sub_pos.mpr h₂), sub_lt_comm] using h₁.2
· have ⟨q, h₁, h₂⟩ := exists_rat_btwn (lt_min lt (lt_add_of_pos_right x ε_pos))
rw [lt_min_iff] at h₂
refine ⟨q, ⟨conn.out hx hz ⟨h₁.le, h₂.1.le⟩, q, rfl⟩, ?_⟩
simpa only [abs_of_pos (sub_pos.2 h₁), sub_lt_iff_lt_add'] using h₂.2
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
closure_ordConnected_inter_rat
| null |
closure_of_rat_image_lt {q : ℚ} :
closure (((↑) : ℚ → ℝ) '' { x | q < x }) = { r | ↑q ≤ r } := by
convert closure_ordConnected_inter_rat (ordConnected_Ioi (a := (q : ℝ))) _ using 1
· congr!; aesop
· exact (closure_Ioi _).symm
· exact ⟨q + 1, show (q : ℝ) < _ by linarith, q + 2, show (q : ℝ) < _ by linarith, by simp⟩
/- TODO(Mario): Put these back only if needed later
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
closure_of_rat_image_lt
| null |
closure_of_rat_image_le_eq {q : ℚ} : closure ((coe : ℚ → ℝ) '' {x | q ≤ x}) = {r | ↑q ≤ r} :=
_
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
closure_of_rat_image_le_eq
| null |
closure_of_rat_image_le_le_eq {a b : ℚ} (hab : a ≤ b) :
closure (of_rat '' {q:ℚ | a ≤ q ∧ q ≤ b}) = {r:ℝ | of_rat a ≤ r ∧ r ≤ of_rat b} :=
_
-/
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
closure_of_rat_image_le_le_eq
| null |
Periodic.compact_of_continuous [TopologicalSpace α] {f : ℝ → α} {c : ℝ} (hp : Periodic f c)
(hc : c ≠ 0) (hf : Continuous f) : IsCompact (range f) := by
rw [← hp.image_uIcc hc 0]
exact isCompact_uIcc.image hf
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Periodic.compact_of_continuous
|
A continuous, periodic function has compact range.
|
Periodic.isBounded_of_continuous [PseudoMetricSpace α] {f : ℝ → α} {c : ℝ}
(hp : Periodic f c) (hc : c ≠ 0) (hf : Continuous f) : IsBounded (range f) :=
(hp.compact_of_continuous hc hf).isBounded
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Field.Periodic",
"Mathlib.Algebra.Field.Subfield.Basic",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Algebra.Ring.Real"
] |
Mathlib/Topology/Instances/Real/Lemmas.lean
|
Periodic.isBounded_of_continuous
|
A continuous, periodic function is bounded.
|
@[mk_iff isProperMap_iff_clusterPt, fun_prop]
IsProperMap (f : X → Y) : Prop extends Continuous f where
/-- By definition, if `f` is a proper map and `ℱ` is any filter on `X`, then any cluster point of
`map f ℱ` is the image by `f` of some cluster point of `ℱ`. -/
clusterPt_of_mapClusterPt :
∀ ⦃ℱ : Filter X⦄, ∀ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
|
structure
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap
|
A map `f : X → Y` between two topological spaces is said to be **proper** if it is continuous
and, for all `ℱ : Filter X`, any cluster point of `map f ℱ` is the image by `f` of a cluster point
of `ℱ`.
|
@[fun_prop]
IsProperMap.continuous (h : IsProperMap f) : Continuous f := h.toContinuous
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.continuous
|
Definition of proper maps. See also `isClosedMap_iff_clusterPt` for a related criterion
for closed maps. -/
add_decl_doc isProperMap_iff_clusterPt
/-- By definition, a proper map is continuous.
|
IsProperMap.isClosedMap (h : IsProperMap f) : IsClosedMap f := by
rw [isClosedMap_iff_clusterPt]
exact fun s y ↦ h.clusterPt_of_mapClusterPt (ℱ := 𝓟 s) (y := y)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.isClosedMap
|
A proper map is closed.
|
isProperMap_iff_ultrafilter : IsProperMap f ↔ Continuous f ∧
∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x := by
rw [isProperMap_iff_clusterPt]
refine and_congr_right (fun _ ↦ ?_)
constructor <;> intro H
· intro 𝒰 y (hY : (Ultrafilter.map f 𝒰 : Filter Y) ≤ _)
simp_rw [← Ultrafilter.clusterPt_iff] at hY ⊢
exact H hY
· simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, ← exists_ultrafilter_iff,
forall_exists_index]
intro ℱ y 𝒰 hy
rcases H (tendsto_iff_comap.mpr <| hy.trans inf_le_left) with ⟨x, hxy, hx⟩
exact ⟨x, hxy, 𝒰, le_inf hx (hy.trans inf_le_right)⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_iff_ultrafilter
|
Characterization of proper maps by ultrafilters.
|
isProperMap_iff_ultrafilter_of_t2 [T2Space Y] : IsProperMap f ↔ Continuous f ∧
∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, 𝒰.1 ≤ 𝓝 x :=
isProperMap_iff_ultrafilter.trans <| and_congr_right fun hc ↦ forall₃_congr fun _𝒰 _y hy ↦
exists_congr fun x ↦ and_iff_right_of_imp fun h ↦
tendsto_nhds_unique ((hc.tendsto x).mono_left h) hy
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_iff_ultrafilter_of_t2
| null |
IsProperMap.ultrafilter_le_nhds_of_tendsto (h : IsProperMap f) ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄
(hy : Tendsto f 𝒰 (𝓝 y)) : ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x :=
(isProperMap_iff_ultrafilter.mp h).2 hy
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.ultrafilter_le_nhds_of_tendsto
|
If `f` is proper and converges to `y` along some ultrafilter `𝒰`, then `𝒰` converges to some
`x` such that `f x = y`.
|
IsProperMap.comp (hf : IsProperMap f) (hg : IsProperMap g) :
IsProperMap (g ∘ f) := by
refine ⟨by fun_prop, fun ℱ z h ↦ ?_⟩
rw [mapClusterPt_comp] at h
rcases hg.clusterPt_of_mapClusterPt h with ⟨y, rfl, hy⟩
rcases hf.clusterPt_of_mapClusterPt hy with ⟨x, rfl, hx⟩
use x, rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.comp
|
The composition of two proper maps is proper.
|
isProperMap_of_comp_of_surj (hf : Continuous f)
(hg : Continuous g) (hgf : IsProperMap (g ∘ f)) (f_surj : f.Surjective) : IsProperMap g := by
refine ⟨hg, fun ℱ z h ↦ ?_⟩
rw [← ℱ.map_comap_of_surjective f_surj, ← mapClusterPt_comp] at h
rcases hgf.clusterPt_of_mapClusterPt h with ⟨x, rfl, hx⟩
rw [← ℱ.map_comap_of_surjective f_surj]
exact ⟨f x, rfl, hx.map hf.continuousAt tendsto_map⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_of_comp_of_surj
|
If the composition of two continuous functions `g ∘ f` is proper and `f` is surjective,
then `g` is proper.
|
isProperMap_of_comp_of_inj {f : X → Y} {g : Y → Z} (hf : Continuous f) (hg : Continuous g)
(hgf : IsProperMap (g ∘ f)) (g_inj : g.Injective) : IsProperMap f := by
refine ⟨hf, fun ℱ y h ↦ ?_⟩
rcases hgf.clusterPt_of_mapClusterPt (h.map hg.continuousAt tendsto_map) with ⟨x, hx1, hx2⟩
exact ⟨x, g_inj hx1, hx2⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_of_comp_of_inj
|
If the composition of two continuous functions `g ∘ f` is proper and `g` is injective,
then `f` is proper.
|
isProperMap_of_comp_of_t2 [T2Space Y] (hf : Continuous f) (hg : Continuous g)
(hgf : IsProperMap (g ∘ f)) : IsProperMap f := by
rw [isProperMap_iff_ultrafilter_of_t2]
refine ⟨hf, fun 𝒰 y h ↦ ?_⟩
rw [isProperMap_iff_ultrafilter] at hgf
rcases hgf.2 ((hg.tendsto y).comp h) with ⟨x, -, hx⟩
exact ⟨x, hx⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_of_comp_of_t2
|
If the composition of two continuous functions `f : X → Y` and `g : Y → Z` is proper
and `Y` is T2, then `f` is proper.
|
IsProperMap.prodMap {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :
IsProperMap (Prod.map f g) := by
simp_rw [isProperMap_iff_ultrafilter] at hf hg ⊢
constructor
· exact hf.1.prodMap hg.1
· intro 𝒰 ⟨y, w⟩ hyw
simp_rw [nhds_prod_eq, tendsto_prod_iff'] at hyw
rcases hf.2 (show Tendsto f (Ultrafilter.map fst 𝒰) (𝓝 y) by simpa using hyw.1) with
⟨x, hxy, hx⟩
rcases hg.2 (show Tendsto g (Ultrafilter.map snd 𝒰) (𝓝 w) by simpa using hyw.2) with
⟨z, hzw, hz⟩
refine ⟨⟨x, z⟩, Prod.ext hxy hzw, ?_⟩
rw [nhds_prod_eq, le_prod]
exact ⟨hx, hz⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.prodMap
|
A binary product of proper maps is proper.
|
IsProperMap.pi_map {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (h : ∀ i, IsProperMap (f i)) :
IsProperMap (fun (x : ∀ i, X i) i ↦ f i (x i)) := by
simp_rw [isProperMap_iff_ultrafilter] at h ⊢
constructor
· exact continuous_pi fun i ↦ (h i).1.comp (continuous_apply i)
· intro 𝒰 y hy
have : ∀ i, Tendsto (f i) (Ultrafilter.map (eval i) 𝒰) (𝓝 (y i)) := by
simpa [tendsto_pi_nhds] using hy
choose x hxy hx using fun i ↦ (h i).2 (this i)
refine ⟨x, funext hxy, ?_⟩
rwa [nhds_pi, le_pi]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.pi_map
|
Any product of proper maps is proper.
|
IsProperMap.isCompact_preimage (h : IsProperMap f) {K : Set Y} (hK : IsCompact K) :
IsCompact (f ⁻¹' K) := by
rw [isCompact_iff_ultrafilter_le_nhds]
intro 𝒰 h𝒰
rw [← comap_principal, ← map_le_iff_le_comap, ← Ultrafilter.coe_map] at h𝒰
rcases hK.ultrafilter_le_nhds _ h𝒰 with ⟨y, hyK, hy⟩
rcases h.ultrafilter_le_nhds_of_tendsto hy with ⟨x, rfl, hx⟩
exact ⟨x, hyK, hx⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.isCompact_preimage
|
The preimage of a compact set by a proper map is again compact. See also
`isProperMap_iff_isCompact_preimage` which proves that this property completely characterizes
proper map when the codomain is compactly generated and Hausdorff.
|
isProperMap_iff_isClosedMap_and_compact_fibers :
IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ ∀ y, IsCompact (f ⁻¹' {y}) := by
constructor <;> intro H
· exact ⟨H.continuous, H.isClosedMap, fun y ↦ H.isCompact_preimage isCompact_singleton⟩
· rw [isProperMap_iff_clusterPt]
refine ⟨H.1, fun ℱ y hy ↦ ?_⟩
rw [H.2.1.mapClusterPt_iff_lift'_closure H.1] at hy
rcases H.2.2 y (f := Filter.lift' ℱ closure ⊓ 𝓟 (f ⁻¹' {y})) inf_le_right with ⟨x, hxy, hx⟩
refine ⟨x, hxy, ?_⟩
rw [← clusterPt_lift'_closure_iff]
exact hx.mono inf_le_left
|
theorem
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_iff_isClosedMap_and_compact_fibers
|
A map is proper if and only if it is closed and its fibers are compact.
|
isProperMap_iff_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) :
IsProperMap f ↔ IsClosedMap f := by
refine ⟨fun h ↦ h.isClosedMap, fun h ↦ ?_⟩
rw [isProperMap_iff_isClosedMap_and_compact_fibers]
exact ⟨f_cont, h, fun y ↦ (subsingleton_singleton.preimage f_inj).isCompact⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_iff_isClosedMap_of_inj
|
An injective and continuous function is proper if and only if it is closed.
|
isProperMap_of_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective)
(f_closed : IsClosedMap f) : IsProperMap f :=
(isProperMap_iff_isClosedMap_of_inj f_cont f_inj).2 f_closed
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_of_isClosedMap_of_inj
|
A injective continuous and closed map is proper.
|
@[simp] Homeomorph.isProperMap (e : X ≃ₜ Y) : IsProperMap e :=
isProperMap_of_isClosedMap_of_inj e.continuous e.injective e.isClosedMap
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
Homeomorph.isProperMap
|
A homeomorphism is proper.
|
protected IsHomeomorph.isProperMap (hf : IsHomeomorph f) : IsProperMap f :=
isProperMap_of_isClosedMap_of_inj hf.continuous hf.injective hf.isClosedMap
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsHomeomorph.isProperMap
| null |
@[simp] isProperMap_id : IsProperMap (id : X → X) := IsHomeomorph.id.isProperMap
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_id
|
The identity is proper.
|
Topology.IsClosedEmbedding.isProperMap (hf : IsClosedEmbedding f) : IsProperMap f :=
isProperMap_of_isClosedMap_of_inj hf.continuous hf.injective hf.isClosedMap
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
Topology.IsClosedEmbedding.isProperMap
|
A closed embedding is proper.
|
IsClosed.isProperMap_subtypeVal {C : Set X} (hC : IsClosed C) : IsProperMap ((↑) : C → X) :=
hC.isClosedEmbedding_subtypeVal.isProperMap
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsClosed.isProperMap_subtypeVal
|
The coercion from a closed subset is proper.
|
IsProperMap.restrict {C : Set X} (hf : IsProperMap f) (hC : IsClosed C) :
IsProperMap fun x : C ↦ f x := hC.isProperMap_subtypeVal.comp hf
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.restrict
|
The restriction of a proper map to a closed subset is proper.
|
IsProperMap.isClosed_range (hf : IsProperMap f) : IsClosed (range f) :=
hf.isClosedMap.isClosed_range
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.isClosed_range
|
The range of a proper map is closed.
|
isProperMap_iff_isClosedMap_and_tendsto_cofinite [T1Space Y] :
IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ Tendsto f (cocompact X) cofinite := by
simp_rw [isProperMap_iff_isClosedMap_and_compact_fibers, Tendsto,
le_cofinite_iff_compl_singleton_mem, mem_map, preimage_compl]
refine and_congr_right fun f_cont ↦ and_congr_right fun _ ↦
⟨fun H y ↦ (H y).compl_mem_cocompact, fun H y ↦ ?_⟩
rcases mem_cocompact.mp (H y) with ⟨K, hK, hKy⟩
exact hK.of_isClosed_subset (isClosed_singleton.preimage f_cont)
(compl_le_compl_iff_le.mp hKy)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_iff_isClosedMap_and_tendsto_cofinite
|
Version of `isProperMap_iff_isClosedMap_and_compact_fibers` in terms of `cofinite` and
`cocompact`. Only works when the codomain is `T1`.
|
Continuous.isProperMap [CompactSpace X] [T2Space Y] (hf : Continuous f) : IsProperMap f :=
isProperMap_iff_isClosedMap_and_tendsto_cofinite.2 ⟨hf, hf.isClosedMap, by simp⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
Continuous.isProperMap
|
A continuous map from a compact space to a T₂ space is a proper map.
|
IsProperMap.universally_closed (Z) [TopologicalSpace Z] (h : IsProperMap f) :
IsClosedMap (Prod.map f id : X × Z → Y × Z) :=
(h.prodMap isProperMap_id).isClosedMap
|
theorem
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
IsProperMap.universally_closed
|
A proper map `f : X → Y` is **universally closed**: for any topological space `Z`, the map
`Prod.map f id : X × Z → Y × Z` is closed. We will prove in `isProperMap_iff_universally_closed`
that proper maps are exactly continuous maps which have this property, but this result should be
easier to use because it allows `Z` to live in any universe.
|
isProperMap_iff_isClosedMap_filter {X : Type u} {Y : Type v} [TopologicalSpace X]
[TopologicalSpace Y] {f : X → Y} :
IsProperMap f ↔ Continuous f ∧ IsClosedMap
(Prod.map f id : X × Filter X → Y × Filter X) := by
constructor <;> intro H
· exact ⟨H.continuous, H.universally_closed _⟩
· rw [isProperMap_iff_ultrafilter]
refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩
let F : Set (X × Filter X) := closure {xℱ | xℱ.2 = pure xℱ.1}
have := H.2 F isClosed_closure
have : (y, ↑𝒰) ∈ Prod.map f id '' F :=
this.mem_of_tendsto (hy.prodMk_nhds (Filter.tendsto_pure_self (𝒰 : Filter X)))
(Eventually.of_forall fun x ↦ ⟨⟨x, pure x⟩, subset_closure rfl, rfl⟩)
rcases this with ⟨⟨x, _⟩, hx, ⟨_, _⟩⟩
refine ⟨x, rfl, fun U hU ↦ Ultrafilter.compl_notMem_iff.mp fun hUc ↦ ?_⟩
rw [mem_closure_iff_nhds] at hx
rcases hx (U ×ˢ {𝒢 | Uᶜ ∈ 𝒢}) (prod_mem_nhds hU (isOpen_setOf_mem.mem_nhds hUc)) with
⟨⟨z, 𝒢⟩, ⟨⟨hz : z ∈ U, hz' : Uᶜ ∈ 𝒢⟩, rfl : 𝒢 = pure z⟩⟩
exact hz' hz
|
theorem
|
Topology
|
[
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Filter"
] |
Mathlib/Topology/Maps/Proper/Basic.lean
|
isProperMap_iff_isClosedMap_filter
|
A map `f : X → Y` is proper if and only if it is continuous and the map
`(Prod.map f id : X × Filter X → Y × Filter X)` is closed. This is stronger than
`isProperMap_iff_universally_closed` since it shows that there's only one space to check to get
properness, but in most cases it doesn't matter.
|
isProperMap_iff_isCompact_preimage :
IsProperMap f ↔ Continuous f ∧ ∀ ⦃K⦄, IsCompact K → IsCompact (f ⁻¹' K) where
mp hf := ⟨hf.continuous, fun _ ↦ hf.isCompact_preimage⟩
mpr := fun ⟨hf, h⟩ ↦ isProperMap_iff_isClosedMap_and_compact_fibers.2
⟨hf, fun _ hs ↦ CompactlyGeneratedSpace.isClosed
fun _ hK ↦ image_inter_preimage .. ▸ (((h hK).inter_left hs).image hf).isClosed,
fun _ ↦ h isCompact_singleton⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Compactness.CompactlyGeneratedSpace",
"Mathlib.Topology.Maps.Proper.Basic"
] |
Mathlib/Topology/Maps/Proper/CompactlyGenerated.lean
|
isProperMap_iff_isCompact_preimage
|
If `Y` is Hausdorff and compactly generated, then proper maps `X → Y` are exactly
continuous maps such that the preimage of any compact set is compact. This is in particular true
if `Y` is Hausdorff and sequential or locally compact.
There was an older version of this theorem which was changed to this one to make use
of the `CompactlyGeneratedSpace` typeclass. (since 2024-11-10)
|
isProperMap_iff_tendsto_cocompact :
IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y) := by
simp_rw [isProperMap_iff_isCompact_preimage,
hasBasis_cocompact.tendsto_right_iff, ← mem_preimage, eventually_mem_set, preimage_compl]
refine and_congr_right fun f_cont ↦
⟨fun H K hK ↦ (H hK).compl_mem_cocompact, fun H K hK ↦ ?_⟩
rcases mem_cocompact.mp (H K hK) with ⟨K', hK', hK'y⟩
exact hK'.of_isClosed_subset (hK.isClosed.preimage f_cont)
(compl_le_compl_iff_le.mp hK'y)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Compactness.CompactlyGeneratedSpace",
"Mathlib.Topology.Maps.Proper.Basic"
] |
Mathlib/Topology/Maps/Proper/CompactlyGenerated.lean
|
isProperMap_iff_tendsto_cocompact
|
Version of `isProperMap_iff_isCompact_preimage` in terms of `cocompact`.
There was an older version of this theorem which was changed to this one to make use
of the `CompactlyGeneratedSpace` typeclass. (since 2024-11-10)
|
isProperMap_iff_isClosedMap_ultrafilter {X : Type u} {Y : Type v} [TopologicalSpace X]
[TopologicalSpace Y] {f : X → Y} :
IsProperMap f ↔ Continuous f ∧ IsClosedMap
(Prod.map f id : X × Ultrafilter X → Y × Ultrafilter X) := by
constructor <;> intro H
· exact ⟨H.continuous, H.universally_closed _⟩
· rw [isProperMap_iff_ultrafilter]
refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩
let F : Set (X × Ultrafilter X) := closure {xℱ | xℱ.2 = pure xℱ.1}
have := H.2 F isClosed_closure
have : (y, 𝒰) ∈ Prod.map f id '' F :=
this.mem_of_tendsto (hy.prodMk_nhds (Ultrafilter.tendsto_pure_self 𝒰))
(Eventually.of_forall fun x ↦ ⟨⟨x, pure x⟩, subset_closure rfl, rfl⟩)
rcases this with ⟨⟨x, _⟩, hx, ⟨_, _⟩⟩
refine ⟨x, rfl, fun U hU ↦ Ultrafilter.compl_notMem_iff.mp fun hUc ↦ ?_⟩
rw [mem_closure_iff_nhds] at hx
rcases hx (U ×ˢ {𝒢 | Uᶜ ∈ 𝒢}) (prod_mem_nhds hU ((ultrafilter_isOpen_basic _).mem_nhds hUc))
with ⟨⟨y, 𝒢⟩, ⟨⟨hy : y ∈ U, hy' : Uᶜ ∈ 𝒢⟩, rfl : 𝒢 = pure y⟩⟩
exact hy' hy
|
theorem
|
Topology
|
[
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Maps/Proper/UniversallyClosed.lean
|
isProperMap_iff_isClosedMap_ultrafilter
|
A map `f : X → Y` is proper if and only if it is continuous and the map
`(Prod.map f id : X × Ultrafilter X → Y × Ultrafilter X)` is closed. This is stronger than
`isProperMap_iff_universally_closed` since it shows that there's only one space to check to get
properness, but in most cases it doesn't matter.
|
isProperMap_iff_universally_closed {X : Type u} {Y : Type v} [TopologicalSpace X]
[TopologicalSpace Y] {f : X → Y} :
IsProperMap f ↔ Continuous f ∧ ∀ (Z : Type u) [TopologicalSpace Z],
IsClosedMap (Prod.map f id : X × Z → Y × Z) := by
constructor <;> intro H
· exact ⟨H.continuous, fun Z ↦ H.universally_closed _⟩
· rw [isProperMap_iff_isClosedMap_ultrafilter]
exact ⟨H.1, H.2 _⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.Compactification.StoneCech"
] |
Mathlib/Topology/Maps/Proper/UniversallyClosed.lean
|
isProperMap_iff_universally_closed
|
A map `f : X → Y` is proper if and only if it is continuous and **universally closed**, in the
sense that for any topological space `Z`, the map `Prod.map f id : X × Z → Y × Z` is closed. Note
that `Z` lives in the same universe as `X` here, but `IsProperMap.universally_closed` does not
have this restriction.
This is taken as the definition of properness in
[N. Bourbaki, *General Topology*][bourbaki1966].
|
exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
∃ r' ∈ Ioo 0 r, s ⊆ ball x r' := by
rcases eq_empty_or_nonempty s with (rfl | hne)
· exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩
have : IsCompact s :=
(isCompact_closedBall x r).of_isClosed_subset hs (h.trans ball_subset_closedBall)
obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closedBall x (dist y x) :=
this.exists_isMaxOn (β := α) (α := ℝ) hne (continuous_id.dist continuous_const).continuousOn
have hyr : dist y x < r := h hys
rcases exists_between hyr with ⟨r', hyr', hrr'⟩
exact ⟨r', ⟨dist_nonneg.trans_lt hyr', hrr'⟩, hy.trans <| closedBall_subset_ball hyr'⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.LocalExtr"
] |
Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean
|
exists_pos_lt_subset_ball
|
If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
ball with the same center and a strictly smaller radius that includes `s`.
|
exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' := by
rcases le_or_gt r 0 with hr | hr
· rw [ball_eq_empty.2 hr, subset_empty_iff] at h
subst s
exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
· exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.1.2, hr'.2⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.LocalExtr"
] |
Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean
|
exists_lt_subset_ball
|
If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
center and a strictly smaller radius that includes `s`.
|
Metric.exists_isLocalMin_mem_ball [TopologicalSpace β]
[ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
(hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
(hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z := by
simp_rw [← closedBall_diff_ball] at hf1
exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1
isOpen_ball
|
theorem
|
Topology
|
[
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.LocalExtr"
] |
Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean
|
Metric.exists_isLocalMin_mem_ball
| null |
instProperSpace : ProperSpace ℝ≥0 where
isCompact_closedBall x r := by
have emb : IsClosedEmbedding ((↑) : ℝ≥0 → ℝ) := Isometry.isClosedEmbedding fun _ ↦ congrFun rfl
exact emb.isCompact_preimage (K := Metric.closedBall x r) (isCompact_closedBall _ _)
|
instance
|
Topology
|
[
"Mathlib.Data.Rat.Encodable",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/MetricSpace/ProperSpace/Real.lean
|
instProperSpace
| null |
dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1)) := by
induction n, h using Nat.le_induction with
| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]
| succ n hle ihn =>
calc
dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _
_ ≤ (∑ i ∈ Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
_ = ∑ i ∈ Finset.Ico m (n + 1), _ := by
rw [← Finset.insert_Ico_right_eq_Ico_add_one hle, Finset.sum_insert, add_comm]; simp
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
dist_le_Ico_sum_dist
|
The triangle (polygon) inequality for sequences of points; `Finset.Ico` version.
|
dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, dist (f i) (f (i + 1)) :=
Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (Nat.zero_le n)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
dist_le_range_sum_dist
|
The triangle (polygon) inequality for sequences of points; `Finset.range` version.
|
dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ}
(hd : ∀ {k}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i :=
le_trans (dist_le_Ico_sum_dist f hmn) <|
Finset.sum_le_sum fun _k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
dist_le_Ico_sum_of_dist_le
|
A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
with an upper estimate.
|
dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ}
(hd : ∀ {k}, k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, d i :=
Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_of_dist_le (zero_le n) fun _ => hd
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
dist_le_range_sum_of_dist_le
|
A version of `dist_le_range_sum_dist` with each intermediate distance replaced
with an upper estimate.
|
controlled_of_isUniformEmbedding [PseudoMetricSpace β] {f : α → β}
(h : IsUniformEmbedding f) :
(∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
⟨uniformContinuous_iff.1 h.uniformContinuous, (isUniformEmbedding_iff.1 h).2.2⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
controlled_of_isUniformEmbedding
|
If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y`.
|
totallyBounded_iff {s : Set α} :
TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
uniformity_basis_dist.totallyBounded_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
totallyBounded_iff
| null |
totallyBounded_of_finite_discretization {s : Set α}
(H : ∀ ε > (0 : ℝ),
∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
TotallyBounded s := by
rcases s.eq_empty_or_nonempty with hs | hs
· rw [hs]
exact totallyBounded_empty
rcases hs with ⟨x0, hx0⟩
haveI : Inhabited s := ⟨⟨x0, hx0⟩⟩
refine totallyBounded_iff.2 fun ε ε0 => ?_
rcases H ε ε0 with ⟨β, fβ, F, hF⟩
let Finv := Function.invFun F
refine ⟨range (Subtype.val ∘ Finv), finite_range _, fun x xs => ?_⟩
let x' := Finv (F ⟨x, xs⟩)
have : F x' = F ⟨x, xs⟩ := Function.invFun_eq ⟨⟨x, xs⟩, rfl⟩
simp only [Set.mem_iUnion, Set.mem_range]
exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
totallyBounded_of_finite_discretization
|
A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
space from finitely many data.
|
finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
∀ ε > 0, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε := by
intro ε ε_pos
rw [totallyBounded_iff_subset] at hs
exact hs _ (dist_mem_uniformity ε_pos)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
finite_approx_of_totallyBounded
| null |
tendstoUniformlyOnFilter_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} :
TendstoUniformlyOnFilter F f p p' ↔
∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε := by
refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
exact (H ε εpos).mono fun n hn => hε hn
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
tendstoUniformlyOnFilter_iff
|
Expressing uniform convergence using `dist`
|
tendstoLocallyUniformlyOn_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} {s : Set β} :
TendstoLocallyUniformlyOn F f p s ↔
∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu x hx => ?_⟩
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
rcases H ε εpos x hx with ⟨t, ht, Ht⟩
exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
tendstoLocallyUniformlyOn_iff
|
Expressing locally uniform convergence on a set using `dist`.
|
tendstoUniformlyOn_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε := by
refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
exact (H ε εpos).mono fun n hs x hx => hε (hs x hx)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
tendstoUniformlyOn_iff
|
Expressing uniform convergence on a set using `dist`.
|
tendstoLocallyUniformly_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} :
TendstoLocallyUniformly F f p ↔
∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by
simp only [← tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, nhdsWithin_univ,
mem_univ, forall_const]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
tendstoLocallyUniformly_iff
|
Expressing locally uniform convergence using `dist`.
|
tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} :
TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by
rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]
simp
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
tendstoUniformly_iff
|
Expressing uniform convergence using `dist`.
|
protected cauchy_iff {f : Filter α} :
Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε :=
uniformity_basis_dist.cauchy_iff
variable {s : Set α}
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
cauchy_iff
| null |
exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
∃ ε > 0, Metric.ball x ε ∩ s = {x} :=
nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
exists_ball_inter_eq_singleton_of_mem_discrete
|
Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
centered at `x` and intersecting `s` only at `x`.
|
exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) :
∃ ε > 0, Metric.closedBall x ε ∩ s = {x} :=
nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hx
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
exists_closedBall_inter_eq_singleton_of_discrete
|
Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
of positive radius centered at `x` and intersecting `s` only at `x`.
|
Metric.inseparable_iff_nndist {x y : α} : Inseparable x y ↔ nndist x y = 0 := by
rw [EMetric.inseparable_iff, edist_nndist, ENNReal.coe_eq_zero]
alias ⟨Inseparable.nndist_eq_zero, _⟩ := Metric.inseparable_iff_nndist
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
Metric.inseparable_iff_nndist
| null |
Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
rw [Metric.inseparable_iff_nndist, dist_nndist, NNReal.coe_eq_zero]
alias ⟨Inseparable.dist_eq_zero, _⟩ := Metric.inseparable_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
Metric.inseparable_iff
| null |
tendsto_nhds_unique_dist {f : β → α} {l : Filter β} {x y : α} [NeBot l]
(ha : Tendsto f l (𝓝 x)) (hb : Tendsto f l (𝓝 y)) : dist x y = 0 :=
(tendsto_nhds_unique_inseparable ha hb).dist_eq_zero
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
tendsto_nhds_unique_dist
|
A weaker version of `tendsto_nhds_unique` for `PseudoMetricSpace`.
|
cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) := by
rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff,
Function.comp_def]
simp_rw [Prod.map_fst, Prod.map_snd]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
cauchySeq_iff_tendsto_dist_atTop_0
| null |
protected IsInducing.isSeparable_preimage {f : β → α} [TopologicalSpace β]
(hf : IsInducing f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) := by
have : SeparableSpace s := hs.separableSpace
have : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _
have : IsInducing ((mapsTo_preimage f s).restrict _ _ _) :=
(hf.comp IsInducing.subtypeVal).codRestrict _
have := this.secondCountableTopology
exact .of_subtype _
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
IsInducing.isSeparable_preimage
|
The preimage of a separable set by an inducing map is separable.
|
protected IsEmbedding.isSeparable_preimage {f : β → α} [TopologicalSpace β]
(hf : IsEmbedding f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) :=
hf.isInducing.isSeparable_preimage hs
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
IsEmbedding.isSeparable_preimage
| null |
IsCompact.isSeparable {s : Set α} (hs : IsCompact s) : IsSeparable s :=
haveI : CompactSpace s := isCompact_iff_compactSpace.mp hs
.of_subtype s
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
IsCompact.isSeparable
|
A compact set is separable.
|
secondCountable_of_almost_dense_set
(H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
SecondCountableTopology α := by
refine EMetric.secondCountable_of_almost_dense_set fun ε ε0 => ?_
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
choose s hsc y hys hyx using H ε' (mod_cast ε'0)
refine ⟨s, hsc, iUnion₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans ?_ ε'ε.le⟩⟩
exact mod_cast hyx x
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
secondCountable_of_almost_dense_set
|
A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
is `ε`-dense.
|
finite_cover_balls_of_compact (hs : IsCompact s) {e : ℝ} (he : 0 < e) :
∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, ball x e :=
let ⟨t, hts, ht⟩ := hs.elim_nhds_subcover _ (fun x _ => ball_mem_nhds x he)
⟨t, hts, t.finite_toSet, ht⟩
alias IsCompact.finite_cover_balls := finite_cover_balls_of_compact
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
finite_cover_balls_of_compact
|
Any compact set in a pseudometric space can be covered by finitely many balls of a given
positive radius
|
exists_finite_cover_balls_of_isCompact_closure (hs : IsCompact (closure s)) (hε : 0 < ε) :
∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, ball x ε := by
obtain ⟨t, hst⟩ := hs.elim_finite_subcover (fun x : s ↦ ball x ε) (fun _ ↦ isOpen_ball) fun x hx ↦
let ⟨y, hy, hxy⟩ := Metric.mem_closure_iff.1 hx _ hε; mem_iUnion.2 ⟨⟨y, hy⟩, hxy⟩
refine ⟨t.map ⟨Subtype.val, Subtype.val_injective⟩, by simp, Finset.finite_toSet _, ?_⟩
simpa using subset_closure.trans hst
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
exists_finite_cover_balls_of_isCompact_closure
|
Any relatively compact set in a pseudometric space can be covered by finitely many balls of a
given positive radius.
|
ContinuousOn.isSeparable_image [TopologicalSpace β] {f : α → β} {s : Set α}
(hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s) := by
rw [image_eq_range, ← image_univ]
exact (isSeparable_univ_iff.2 hs.separableSpace).image hf.restrict
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
ContinuousOn.isSeparable_image
|
If a map is continuous on a separable set `s`, then the image of `s` is also separable.
|
PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β) :
PseudoMetricSpace α where
dist x y := dist (f x) (f y)
dist_self _ := dist_self _
dist_comm _ _ := dist_comm _ _
dist_triangle _ _ _ := dist_triangle _ _ _
edist x y := edist (f x) (f y)
edist_dist _ _ := edist_dist _ _
toUniformSpace := UniformSpace.comap f m.toUniformSpace
uniformity_dist := (uniformity_basis_dist.comap _).eq_biInf
toBornology := Bornology.induced f
cobounded_sets := Set.ext fun s => mem_comap_iff_compl.trans <| by
simp only [← isBounded_def, isBounded_iff, forall_mem_image, mem_setOf]
|
abbrev
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
PseudoMetricSpace.induced
|
Pseudometric space structure pulled back by a function.
|
Topology.IsInducing.comapPseudoMetricSpace {α β : Type*} [TopologicalSpace α]
[m : PseudoMetricSpace β] {f : α → β} (hf : IsInducing f) : PseudoMetricSpace α :=
.replaceTopology (.induced f m) hf.eq_induced
|
def
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Topology.IsInducing.comapPseudoMetricSpace
|
Pull back a pseudometric space structure by an inducing map. This is a version of
`PseudoMetricSpace.induced` useful in case if the domain already has a `TopologicalSpace`
structure.
|
IsUniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [m : PseudoMetricSpace β]
(f : α → β) (h : IsUniformInducing f) : PseudoMetricSpace α :=
.replaceUniformity (.induced f m) h.comap_uniformity.symm
|
def
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
IsUniformInducing.comapPseudoMetricSpace
|
Pull back a pseudometric space structure by a uniform inducing map. This is a version of
`PseudoMetricSpace.induced` useful in case if the domain already has a `UniformSpace`
structure.
|
Subtype.pseudoMetricSpace {p : α → Prop} : PseudoMetricSpace (Subtype p) :=
PseudoMetricSpace.induced Subtype.val ‹_›
|
instance
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Subtype.pseudoMetricSpace
| null |
Subtype.dist_eq {p : α → Prop} (x y : Subtype p) : dist x y = dist (x : α) y := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Subtype.dist_eq
| null |
Subtype.nndist_eq {p : α → Prop} (x y : Subtype p) : nndist x y = nndist (x : α) y := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Subtype.nndist_eq
| null |
@[to_additive]
instPseudoMetricSpace : PseudoMetricSpace αᵐᵒᵖ :=
PseudoMetricSpace.induced MulOpposite.unop ‹_›
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
instPseudoMetricSpace
| null |
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