fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
IsCompact s ↔ IsClosed s ∧ IsBounded s :=
⟨fun h => ⟨h.isClosed, h.isBounded⟩, fun h => isCompact_of_isClosed_isBounded h.1 h.2⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isCompact_iff_isClosed_bounded | The **Heine–Borel theorem**:
In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. |
compactSpace_iff_isBounded_univ [ProperSpace α] :
CompactSpace α ↔ IsBounded (univ : Set α) :=
⟨@isBounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_isBounded isClosed_univ hb⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | compactSpace_iff_isBounded_univ | null |
isBounded_Icc (a b : α) : IsBounded (Icc a b) :=
(totallyBounded_Icc a b).isBounded | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_Icc | null |
isBounded_Ico (a b : α) : IsBounded (Ico a b) :=
(totallyBounded_Ico a b).isBounded | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_Ico | null |
isBounded_Ioc (a b : α) : IsBounded (Ioc a b) :=
(totallyBounded_Ioc a b).isBounded | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_Ioc | null |
isBounded_Ioo (a b : α) : IsBounded (Ioo a b) :=
(totallyBounded_Ioo a b).isBounded | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_Ioo | null |
isBounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
IsBounded s :=
let ⟨u, hu⟩ := h₁
let ⟨l, hl⟩ := h₂
(isBounded_Icc l u).subset (fun _x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩)
open Metric in | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_of_bddAbove_of_bddBelow | In a pseudo metric space with a conditionally complete linear order such that the order and the
metric structure give the same topology, any order-bounded set is metric-bounded. |
_root_.IsOrderBornology.of_isCompactIcc (x : α)
(bddBelow_ball : ∀ r, BddBelow (closedBall x r))
(bddAbove_ball : ∀ r, BddAbove (closedBall x r)) : IsOrderBornology α where
isBounded_iff_bddBelow_bddAbove s := by
refine ⟨?_, fun hs ↦ Metric.isBounded_of_bddAbove_of_bddBelow hs.2 hs.1⟩
rw [Metric.isBounded_iff_subset_closedBall x]
rintro ⟨r, hr⟩
exact ⟨(bddBelow_ball _).mono hr, (bddAbove_ball _).mono hr⟩ | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.IsOrderBornology.of_isCompactIcc | null |
isBounded_of_abs_le (C : α) : Bornology.IsBounded {x : α | |x| ≤ C} := by
convert Metric.isBounded_Icc (-C) C
ext1 x
simp [abs_le] | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_of_abs_le | null |
isBounded_of_abs_lt (C : α) : Bornology.IsBounded {x : α | |x| < C} := by
convert Metric.isBounded_Ioo (-C) C
ext1 x
simp [abs_lt] | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_of_abs_lt | null |
noncomputable diam (s : Set α) : ℝ :=
ENNReal.toReal (EMetric.diam s) | def | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam | The diameter of a set in a metric space. To get controllable behavior even when the diameter
should be infinite, we express it in terms of the `EMetric.diam` |
diam_nonneg : 0 ≤ diam s :=
ENNReal.toReal_nonneg | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_nonneg | The diameter of a set is always nonnegative |
diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
simp only [diam, EMetric.diam_subsingleton hs, ENNReal.toReal_zero] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_subsingleton | null |
@[simp]
diam_empty : diam (∅ : Set α) = 0 :=
diam_subsingleton subsingleton_empty | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_empty | The empty set has zero diameter |
@[simp]
diam_singleton : diam ({x} : Set α) = 0 :=
diam_subsingleton subsingleton_singleton
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_singleton | A singleton has zero diameter |
diam_one [One α] : diam (1 : Set α) = 0 :=
diam_singleton | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_one | null |
diam_pair : diam ({x, y} : Set α) = dist x y := by
simp only [diam, EMetric.diam_pair, dist_edist] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_pair | null |
diam_triple :
Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) := by
simp only [Metric.diam, EMetric.diam_triple, dist_edist]
rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_triple | null |
ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
EMetric.diam s ≤ ENNReal.ofReal C :=
EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy) | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | ediam_le_of_forall_dist_le | If the distance between any two points in a set is bounded by some constant `C`,
then `ENNReal.ofReal C` bounds the emetric diameter of this set. |
diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
diam s ≤ C :=
ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h) | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_le_of_forall_dist_le | If the distance between any two points in a set is bounded by some non-negative constant,
this constant bounds the diameter. |
diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
(h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) : diam s ≤ C :=
have h₀ : 0 ≤ C :=
let ⟨x, hx⟩ := hs
le_trans dist_nonneg (h x hx x hx)
diam_le_of_forall_dist_le h₀ h | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_le_of_forall_dist_le_of_nonempty | If the distance between any two points in a nonempty set is bounded by some constant,
this constant bounds the diameter. |
dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
dist x y ≤ diam s := by
rw [diam, dist_edist]
exact ENNReal.toReal_mono h <| EMetric.edist_le_diam_of_mem hx hy | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | dist_le_diam_of_mem' | The distance between two points in a set is controlled by the diameter of the set. |
isBounded_iff_ediam_ne_top : IsBounded s ↔ EMetric.diam s ≠ ⊤ :=
isBounded_iff.trans <| Iff.intro
(fun ⟨_C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
fun h => ⟨diam s, fun _x hx _y hy => dist_le_diam_of_mem' h hx hy⟩
alias ⟨_root_.Bornology.IsBounded.ediam_ne_top, _⟩ := isBounded_iff_ediam_ne_top | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_iff_ediam_ne_top | Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. |
ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬IsBounded s :=
isBounded_iff_ediam_ne_top.not_left.symm | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | ediam_eq_top_iff_unbounded | null |
ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
rw [← not_compactSpace_iff, compactSpace_iff_isBounded_univ, isBounded_iff_ediam_ne_top,
Classical.not_not]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | ediam_univ_eq_top_iff_noncompact | null |
ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
EMetric.diam (univ : Set α) = ∞ :=
ediam_univ_eq_top_iff_noncompact.mpr ‹_›
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | ediam_univ_of_noncompact | null |
diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 := by
simp [diam] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_univ_of_noncompact | null |
dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
dist_le_diam_of_mem' h.ediam_ne_top hx hy | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | dist_le_diam_of_mem | The distance between two points in a set is controlled by the diameter of the set. |
ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ := ediam_eq_top_iff_unbounded.2 h | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | ediam_of_unbounded | null |
diam_eq_zero_of_unbounded (h : ¬IsBounded s) : diam s = 0 := by
rw [diam, ediam_of_unbounded h, ENNReal.toReal_top] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_eq_zero_of_unbounded | An unbounded set has zero diameter. If you would prefer to get the value ∞, use `EMetric.diam`.
This lemma makes it possible to avoid side conditions in some situations |
diam_mono {s t : Set α} (h : s ⊆ t) (ht : IsBounded t) : diam s ≤ diam t :=
ENNReal.toReal_mono ht.ediam_ne_top <| EMetric.diam_mono h | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_mono | If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. |
diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + dist x y + diam t := by
simp only [diam, dist_edist]
refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans
(add_le_add_right ENNReal.toReal_add_le _)
· simp only [ENNReal.add_eq_top, edist_ne_top, or_false]
exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono subset_union_left
· exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono subset_union_right | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_union | The diameter of a union is controlled by the sum of the diameters, and the distance between
any two points in each of the sets. This lemma is true without any side condition, since it is
obviously true if `s ∪ t` is unbounded. |
diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t := by
rcases h with ⟨x, ⟨xs, xt⟩⟩
simpa using diam_union xs xt | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_union' | If two sets intersect, the diameter of the union is bounded by the sum of the diameters. |
diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
diam s ≤ 2 * r :=
diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
calc
dist a b ≤ dist a x + dist b x := dist_triangle_right _ _ _
_ ≤ r + r := add_le_add (h ha) (h hb)
_ = 2 * r := by simp [mul_two, mul_comm] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_le_of_subset_closedBall | null |
diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r :=
diam_le_of_subset_closedBall h Subset.rfl | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_closedBall | The diameter of a closed ball of radius `r` is at most `2 r`. |
diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
diam_le_of_subset_closedBall h ball_subset_closedBall | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_ball | The diameter of a ball of radius `r` is at most `2 r`. |
_root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
(h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n))
(h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
(⋂ n, s n).Nonempty := by
let u N := (h N).some
have I : ∀ n N, n ≤ N → u N ∈ s n := by
intro n N hn
apply mem_of_subset_of_mem _ (h N).choose_spec
intro x hx
simp only [mem_iInter] at hx
exact hx n hn
have : CauchySeq u := by
apply cauchySeq_of_le_tendsto_0 _ _ h'
intro m n N hm hn
exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)
obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) :=
cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
refine ⟨x, mem_iInter.2 fun n => ?_⟩
apply (hs n).mem_of_tendsto xlim
filter_upwards [Ici_mem_atTop n] with p hp
exact I n p hp | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.IsComplete.nonempty_iInter_of_nonempty_biInter | If a family of complete sets with diameter tending to `0` is such that each finite intersection
is nonempty, then the total intersection is also nonempty. |
nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
(hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
(h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
(hs 0).isComplete.nonempty_iInter_of_nonempty_biInter hs h's h h' | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | nonempty_iInter_of_nonempty_biInter | In a complete space, if a family of closed sets with diameter tending to `0` is such that each
finite intersection is nonempty, then the total intersection is also nonempty. |
diam_pos [MetricSpace α] (hs1 : s.Nontrivial) (hs2 : IsBounded s) : 0 < diam s := by
rcases hs1 with ⟨x, hx, y, hy, hxy⟩
exact (dist_pos.mpr hxy).trans_le <| Metric.dist_le_diam_of_mem hs2 hx hy | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | diam_pos | null |
@[positivity Metric.diam _]
evalDiam : PositivityExt where eval {u α} _zα _pα e := do
match u, α, e with
| 0, ~q(ℝ), ~q(@Metric.diam _ $inst $s) =>
assertInstancesCommute
pure (.nonnegative q(Metric.diam_nonneg))
| _, _, _ => throwError "not ‖ · ‖" | def | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | evalDiam | Extension for the `positivity` tactic: the diameter of a set is always nonnegative. |
Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α := by
nontriviality α; inhabit α
exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | Metric.cobounded_eq_cocompact | null |
tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
Tendsto (dist · x) (cocompact α) atTop :=
(tendsto_dist_right_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_dist_right_cocompact_atTop | null |
tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
Tendsto (dist x) (cocompact α) atTop :=
(tendsto_dist_left_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_dist_left_cocompact_atTop | null |
comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
comap (dist x) atTop = cocompact α := by simp [cobounded_eq_cocompact] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | comap_dist_left_atTop_eq_cocompact | null |
tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
(h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) :=
((tendsto_dist_right_atTop_iff _).1 h).mono_right cobounded_le_cocompact | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_cocompact_of_tendsto_dist_comp_atTop | null |
Metric.finite_isBounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
(hK : IsBounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) := by
refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure)
· exact hK.isCompact_closure.inter_right hs
· exact DiscreteTopology.of_subset inferInstance Set.inter_subset_right | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | Metric.finite_isBounded_inter_isClosed | null |
exists_forall_le_of_isBounded {f : β → α} (hf : Continuous f) (x₀ : β)
(h : Bornology.IsBounded {x : β | f x ≤ f x₀}) :
∃ x, ∀ y, f x ≤ f y := by
refine hf.exists_forall_le' (x₀ := x₀) ?_
have hU : {x : β | f x₀ < f x} ∈ Filter.cocompact β := by
refine Filter.mem_cocompact'.mpr ⟨_, ?_, fun ⦃_⦄ a ↦ a⟩
simp only [Set.compl_setOf, not_lt]
exact Metric.isCompact_of_isClosed_isBounded (isClosed_le (by fun_prop) (by fun_prop)) h
filter_upwards [hU] with x hx using hx.le | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_forall_le_of_isBounded | A version of the **Extreme Value Theorem**: if the set where a continuous function `f`
into a linearly ordered space takes values `≤ f x₀` is bounded for some `x₀`,
then `f` has a global minimum (under suitable topological assumptions).
This is a convenient combination of `Continuous.exists_forall_le'` and
`Metric.isCompact_of_isClosed_isBounded`. |
exists_forall_ge_of_isBounded {f : β → α} (hf : Continuous f) (x₀ : β)
(h : Bornology.IsBounded {x : β | f x₀ ≤ f x}) :
∃ x, ∀ y, f y ≤ f x :=
hf.exists_forall_le_of_isBounded (α := αᵒᵈ) x₀ h | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_forall_ge_of_isBounded | A version of the **Extreme Value Theorem**: if the set where a continuous function `f`
into a linearly ordered space takes values `≥ f x₀` is bounded for some `x₀`,
then `f` has a global maximum (under suitable topological assumptions).
This is a convenient combination of `Continuous.exists_forall_ge'` and
`Metric.isCompact_of_isClosed_isBounded`. |
@[ext]
PseudoMetric [Zero R] [Add R] [LE R] where
/-- The underlying binary function mapping into a linearly ordered additive monoid. -/
toFun : X → X → R
/-- A pseudometric must take identical elements to 0. -/
refl' x : toFun x x = 0
/-- A pseudometric must be symmetric. -/
symm' x y : toFun x y = toFun y x
/-- A pseudometric must respect the triangle inequality. -/
triangle' x y z : toFun x z ≤ toFun x y + toFun y z | structure | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | PseudoMetric | A pseudometric as a bundled function. |
@[simp, norm_cast]
coe_mk (d : X → X → R) (refl symm triangle) : mk d refl symm triangle = d := rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | coe_mk | null |
mk_apply (d : X → X → R) (refl symm triangle) (x y : X) :
mk d refl symm triangle x y = d x y :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | mk_apply | null |
protected refl (x : X) : d x x = 0 := d.refl' x | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | refl | null |
protected symm (x y : X) : d x y = d y x := d.symm' x y | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | symm | null |
protected triangle (x y z : X) : d x z ≤ d x y + d y z := d.triangle' x y z | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | triangle | null |
@[simp, norm_cast]
protected coe_le_coe {d d' : PseudoMetric X R} :
(d : X → X → R) ≤ d' ↔ d ≤ d' :=
Iff.rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | coe_le_coe | null |
@[simp, norm_cast]
coe_bot [AddZeroClass R] [Preorder R] : ⇑(⊥ : PseudoMetric X R) = 0 := rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | coe_bot | null |
protected bot_apply [AddZeroClass R] [Preorder R] (x y : X) :
(⊥ : PseudoMetric X R) x y = 0 :=
rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | bot_apply | null |
@[simp, push_cast]
coe_sup [AddZeroClass R] [SemilatticeSup R] [AddLeftMono R] [AddRightMono R]
(d d' : PseudoMetric X R) :
((d ⊔ d' : PseudoMetric X R) : X → X → R) = (d : X → X → R) ⊔ d' := rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | coe_sup | null |
protected sup_apply [AddZeroClass R] [SemilatticeSup R] [AddLeftMono R] [AddRightMono R]
(d d' : PseudoMetric X R) (x y : X) :
(d ⊔ d') x y = d x y ⊔ d' x y :=
rfl | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | sup_apply | null |
protected nonneg (d : PseudoMetric X R) (x y : X) : 0 ≤ d x y := by
by_contra! H
have : d x x < 0 := by
calc d x x ≤ d x y + d y x := d.triangle' x y x
_ < 0 + 0 := by refine add_lt_add H (d.symm x y ▸ H)
_ = 0 := by simp
exact this.ne (d.refl x) | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | nonneg | null |
@[simp, push_cast]
coe_finsetSup [IsOrderedAddMonoid R] {Y : Type*} {f : Y → PseudoMetric X R} {s : Finset Y}
(hs : s.Nonempty) :
⇑(s.sup f) = s.sup' hs (f ·) := by
induction hs using Finset.Nonempty.cons_induction with
| singleton i => simp
| cons a s ha hs ih => simp [hs, ih] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | coe_finsetSup | null |
finsetSup_apply [IsOrderedAddMonoid R] {Y : Type*} {f : Y → PseudoMetric X R}
{s : Finset Y} (hs : s.Nonempty) (x y : X) :
s.sup f x y = s.sup' hs fun i ↦ f i x y := by
induction hs using Finset.Nonempty.cons_induction with
| singleton i => simp
| cons a s ha hs ih => simp [hs, ih] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | finsetSup_apply | null |
IsUltra [Zero R] [Add R] [LE R] [Max R] (d : PseudoMetric X R) : Prop where
/-- Strong triangle inequality of an ultrametric. -/
le_sup' : ∀ x y z, d x z ≤ d x y ⊔ d y z | class | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra | A pseudometric can be nonarchimedean (or ultrametric), with a stronger triangle
inequality such that `d x z ≤ max (d x y) (d y z)`. |
IsUltra.le_sup [Zero R] [Add R] [LE R] [Max R] {d : PseudoMetric X R} [hd : IsUltra d]
{x y z : X} : d x z ≤ d x y ⊔ d y z :=
hd.le_sup' x y z | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra.le_sup | null |
IsUltra.bot [AddZeroClass R] [SemilatticeSup R] :
IsUltra (⊥ : PseudoMetric X R) where
le_sup' := by simp | instance | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra.bot | null |
IsUltra.sup [AddZeroClass R] [SemilatticeSup R] [AddLeftMono R] [AddRightMono R]
{d d' : PseudoMetric X R} [IsUltra d] [IsUltra d'] : IsUltra (d ⊔ d') := by
constructor
intro x y z
simp only [PseudoMetric.sup_apply]
calc d x z ⊔ d' x z ≤ d x y ⊔ d y z ⊔ (d' x y ⊔ d' y z) := sup_le_sup le_sup le_sup
_ ≤ d x y ⊔ d' x y ⊔ (d y z ⊔ d' y z) := by simp [sup_comm, sup_left_comm] | instance | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra.sup | null |
IsUltra.finsetSup {Y : Type*} [AddCommMonoid R] [LinearOrder R] [AddLeftStrictMono R]
[IsOrderedAddMonoid R] {f : Y → PseudoMetric X R} {s : Finset Y} (h : ∀ d ∈ s, IsUltra (f d)) :
IsUltra (s.sup f) := by
constructor
intro x y z
rcases s.eq_empty_or_nonempty with rfl | hs
· simp
simp_rw [finsetSup_apply hs]
apply Finset.sup'_le
simp only [le_sup_iff, Finset.le_sup'_iff]
intro i hi
have h := (h i hi).le_sup' x y z
simp only [le_sup_iff] at h
refine h.imp ?_ ?_ <;>
intro H <;>
exact ⟨i, hi, H⟩ | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra.finsetSup | null |
isSymmetricRel_ball [Add R] [Zero R] [Preorder R] (d : PseudoMetric X R) {ε : R} :
IsSymmetricRel {xy | d xy.1 xy.2 < ε} := by
simp [IsSymmetricRel, d.symm] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | isSymmetricRel_ball | null |
isSymmetricRel_closedBall [Add R] [Zero R] [LE R] (d : PseudoMetric X R) {ε : R} :
IsSymmetricRel {xy | d xy.1 xy.2 ≤ ε} := by
simp [IsSymmetricRel, d.symm] | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | isSymmetricRel_closedBall | null |
IsUltra.isTransitiveRel_ball [Add R] [Zero R] [LinearOrder R] (d : PseudoMetric X R)
[d.IsUltra] {ε : R} :
IsTransitiveRel {xy | d xy.1 xy.2 < ε} :=
fun _ _ _ hxy hyz ↦ le_sup.trans_lt (max_lt hxy hyz) | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra.isTransitiveRel_ball | null |
IsUltra.isTransitiveRel_closedBall [Add R] [Zero R] [SemilatticeSup R] (d : PseudoMetric X R)
[d.IsUltra] {ε : R} :
IsTransitiveRel {xy | d xy.1 xy.2 ≤ ε} :=
fun _ _ _ hxy hyz ↦ le_sup.trans (sup_le hxy hyz) | lemma | Topology | [
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] | Mathlib/Topology/MetricSpace/BundledFun.lean | IsUltra.isTransitiveRel_closedBall | null |
noncomputable inducedMap : Σ s : Set (ℕ → β), s → α :=
⟨fun x => Set.Nonempty (⋂ n : ℕ, A (res x n)), fun x => x.property.some⟩ | def | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | inducedMap | From a `β`-scheme on `α` `A`, we define a partial function from `(ℕ → β)` to `α`
which sends each infinite sequence `x` to an element of the intersection along the
branch corresponding to `x`, if it exists.
We call this the map induced by the scheme. |
protected Antitone : Prop :=
∀ l : List β, ∀ a : β, A (a :: l) ⊆ A l | def | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | Antitone | A scheme is antitone if each set contains its children. |
ClosureAntitone [TopologicalSpace α] : Prop :=
∀ l : List β, ∀ a : β, closure (A (a :: l)) ⊆ A l | def | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | ClosureAntitone | A useful strengthening of being antitone is to require that each set contains
the closure of each of its children. |
protected Disjoint : Prop :=
∀ l : List β, Pairwise fun a b => Disjoint (A (a :: l)) (A (b :: l))
variable {A} | def | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | Disjoint | A scheme is disjoint if the children of each set of pairwise disjoint. |
map_mem (x : (inducedMap A).1) (n : ℕ) : (inducedMap A).2 x ∈ A (res x n) := by
have := x.property.some_mem
rw [mem_iInter] at this
exact this n | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | map_mem | If `x` is in the domain of the induced map of a scheme `A`,
its image under this map is in each set along the corresponding branch. |
protected ClosureAntitone.antitone [TopologicalSpace α] (hA : ClosureAntitone A) :
CantorScheme.Antitone A := fun l a => subset_closure.trans (hA l a) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | ClosureAntitone.antitone | null |
protected Antitone.closureAntitone [TopologicalSpace α] (hanti : CantorScheme.Antitone A)
(hclosed : ∀ l, IsClosed (A l)) : ClosureAntitone A := fun _ _ =>
(hclosed _).closure_eq.subset.trans (hanti _ _) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | Antitone.closureAntitone | null |
Disjoint.map_injective (hA : CantorScheme.Disjoint A) : Injective (inducedMap A).2 := by
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
refine Subtype.coe_injective (res_injective ?_)
dsimp
ext n : 1
induction n with
| zero => simp
| succ n ih =>
simp only [res_succ, cons.injEq]
refine ⟨?_, ih⟩
contrapose hA
simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
refine ⟨res x n, _, _, hA, ?_⟩
rw [not_disjoint_iff]
refine ⟨(inducedMap A).2 ⟨x, hx⟩, ?_, ?_⟩
· rw [← res_succ]
apply map_mem
rw [hxy, ih, ← res_succ]
apply map_mem | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | Disjoint.map_injective | A scheme where the children of each set are pairwise disjoint induces an injective map. |
VanishingDiam : Prop :=
∀ x : ℕ → β, Tendsto (fun n : ℕ => EMetric.diam (A (res x n))) atTop (𝓝 0)
variable {A} | def | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | VanishingDiam | A scheme on a metric space has vanishing diameter if diameter approaches 0 along each branch. |
VanishingDiam.dist_lt (hA : VanishingDiam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β) :
∃ n : ℕ, ∀ (y) (_ : y ∈ A (res x n)) (z) (_ : z ∈ A (res x n)), dist y z < ε := by
specialize hA x
rw [ENNReal.tendsto_atTop_zero] at hA
obtain ⟨n, hn⟩ := hA (ENNReal.ofReal (ε / 2)) (by
simp only [gt_iff_lt, ENNReal.ofReal_pos]; linarith)
use n
intro y hy z hz
rw [← ENNReal.ofReal_lt_ofReal_iff ε_pos, ← edist_dist]
apply lt_of_le_of_lt (EMetric.edist_le_diam_of_mem hy hz)
apply lt_of_le_of_lt (hn _ (le_refl _))
rw [ENNReal.ofReal_lt_ofReal_iff ε_pos]
linarith | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | VanishingDiam.dist_lt | null |
VanishingDiam.map_continuous [TopologicalSpace β] [DiscreteTopology β]
(hA : VanishingDiam A) : Continuous (inducedMap A).2 := by
rw [Metric.continuous_iff']
rintro ⟨x, hx⟩ ε ε_pos
obtain ⟨n, hn⟩ := hA.dist_lt _ ε_pos x
rw [_root_.eventually_nhds_iff]
refine ⟨(↑)⁻¹' cylinder x n, ?_, ?_, by simp⟩
· rintro ⟨y, hy⟩ hyx
rw [mem_preimage, Subtype.coe_mk, cylinder_eq_res, mem_setOf] at hyx
apply hn
· rw [← hyx]
apply map_mem
apply map_mem
apply continuous_subtype_val.isOpen_preimage
apply isOpen_cylinder | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | VanishingDiam.map_continuous | A scheme with vanishing diameter along each branch induces a continuous map. |
ClosureAntitone.map_of_vanishingDiam [CompleteSpace α] (hdiam : VanishingDiam A)
(hanti : ClosureAntitone A) (hnonempty : ∀ l, (A l).Nonempty) : (inducedMap A).1 = univ := by
rw [eq_univ_iff_forall]
intro x
choose u hu using fun n => hnonempty (res x n)
have umem : ∀ n m : ℕ, n ≤ m → u m ∈ A (res x n) := by
have : Antitone fun n : ℕ => A (res x n) := by
refine antitone_nat_of_succ_le ?_
intro n
apply hanti.antitone
intro n m hnm
exact this hnm (hu _)
have : CauchySeq u := by
rw [Metric.cauchySeq_iff]
intro ε ε_pos
obtain ⟨n, hn⟩ := hdiam.dist_lt _ ε_pos x
use n
intro m₀ hm₀ m₁ hm₁
apply hn <;> apply umem <;> assumption
obtain ⟨y, hy⟩ := cauchySeq_tendsto_of_complete this
use y
rw [mem_iInter]
intro n
apply hanti _ (x n)
apply mem_closure_of_tendsto hy
rw [eventually_atTop]
exact ⟨n.succ, umem _⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.PiNat"
] | Mathlib/Topology/MetricSpace/CantorScheme.lean | ClosureAntitone.map_of_vanishingDiam | A scheme on a complete space with vanishing diameter
such that each set contains the closure of its children
induces a total map. |
Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (hB : ∀ n, 0 < B n)
(H : ∀ u : ℕ → α, (∀ N n m : ℕ, N ≤ n → N ≤ m → dist (u n) (u m) < B N) →
∃ x, Tendsto u atTop (𝓝 x)) :
CompleteSpace α :=
UniformSpace.complete_of_convergent_controlled_sequences
(fun n => { p : α × α | dist p.1 p.2 < B n }) (fun n => dist_mem_uniformity <| hB n) H | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | Metric.complete_of_convergent_controlled_sequences | A very useful criterion to show that a space is complete is to show that all sequences
which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
`0`, which makes it possible to use arguments of converging series, while this is impossible
to do in general for arbitrary Cauchy sequences. |
Metric.complete_of_cauchySeq_tendsto :
(∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) → CompleteSpace α :=
EMetric.complete_of_cauchySeq_tendsto | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | Metric.complete_of_cauchySeq_tendsto | A pseudo-metric space is complete iff every Cauchy sequence converges. |
Metric.cauchySeq_iff {u : β → α} :
CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε :=
uniformity_basis_dist.cauchySeq_iff | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | Metric.cauchySeq_iff | In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
the distance between its elements is arbitrarily small |
Metric.cauchySeq_iff' {u : β → α} :
CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε :=
uniformity_basis_dist.cauchySeq_iff' | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | Metric.cauchySeq_iff' | A variation around the pseudometric characterization of Cauchy sequences |
Metric.uniformCauchySeqOn_iff {γ : Type*} {F : β → γ → α} {s : Set γ} :
UniformCauchySeqOn F atTop s ↔ ∀ ε > (0 : ℝ),
∃ N : β, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε := by
constructor
· intro h ε hε
let u := { a : α × α | dist a.fst a.snd < ε }
have hu : u ∈ 𝓤 α := Metric.mem_uniformity_dist.mpr ⟨ε, hε, by simp [u]⟩
rw [← Filter.eventually_atTop_prod_self' (p := fun m =>
∀ x ∈ s, dist (F m.fst x) (F m.snd x) < ε)]
specialize h u hu
rw [prod_atTop_atTop_eq] at h
exact h.mono fun n h x hx => h x hx
· intro h u hu
rcases Metric.mem_uniformity_dist.mp hu with ⟨ε, hε, hab⟩
rcases h ε hε with ⟨N, hN⟩
rw [prod_atTop_atTop_eq, eventually_atTop]
use (N, N)
intro b hb x hx
rcases hb with ⟨hbl, hbr⟩
exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | Metric.uniformCauchySeqOn_iff | In a pseudometric space, uniform Cauchy sequences are characterized by the fact that,
eventually, the distance between all its elements is uniformly, arbitrarily small. |
cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
(h : ∀ n m : β, n ≤ m → dist (s n) (s m) ≤ b n) (h₀ : Tendsto b atTop (𝓝 0)) : CauchySeq s :=
Metric.cauchySeq_iff'.2 fun ε ε0 => (h₀.eventually (gt_mem_nhds ε0)).exists.imp fun N hN n hn =>
calc dist (s n) (s N) = dist (s N) (s n) := dist_comm _ _
_ ≤ b N := h _ _ hn
_ < ε := hN | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | cauchySeq_of_le_tendsto_0' | If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
and `b` converges to zero, then `s` is a Cauchy sequence. |
cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
(h : ∀ n m N : β, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : Tendsto b atTop (𝓝 0)) :
CauchySeq s :=
cauchySeq_of_le_tendsto_0' b (fun _n _m hnm => h _ _ _ le_rfl hnm) h₀ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | cauchySeq_of_le_tendsto_0 | If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
and `b` converges to zero, then `s` is a Cauchy sequence. |
cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R := by
rcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩
rsuffices ⟨R, R0, H⟩ : ∃ R > 0, ∀ n, dist (u n) (u N) < R
· exact ⟨_, add_pos R0 R0, fun m n =>
lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩
let R := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)
refine ⟨↑R + 1, add_pos_of_nonneg_of_pos R.2 zero_lt_one, fun n => ?_⟩
rcases le_or_gt N n with h | h
· exact lt_of_lt_of_le (hN _ h) (le_add_of_nonneg_left R.2)
· have : _ ≤ R := Finset.le_sup (Finset.mem_range.2 h)
exact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | cauchySeq_bdd | A Cauchy sequence on the natural numbers is bounded. |
cauchySeq_iff_le_tendsto_0 {s : ℕ → α} :
CauchySeq s ↔
∃ b : ℕ → ℝ,
(∀ n, 0 ≤ b n) ∧
(∀ n m N : ℕ, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0) :=
⟨fun hs => by
/- `s` is a Cauchy sequence. The sequence `b` will be constructed by taking
the supremum of the distances between `s n` and `s m` for `n m ≥ N`.
First, we prove that all these distances are bounded, as otherwise the Sup
would not make sense. -/
let S N := (fun p : ℕ × ℕ => dist (s p.1) (s p.2)) '' { p | p.1 ≥ N ∧ p.2 ≥ N }
have hS : ∀ N, ∃ x, ∀ y ∈ S N, y ≤ x := by
rcases cauchySeq_bdd hs with ⟨R, -, hR⟩
refine fun N => ⟨R, ?_⟩
rintro _ ⟨⟨m, n⟩, _, rfl⟩
exact le_of_lt (hR m n)
have ub : ∀ m n N, N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N) := fun m n N hm hn =>
le_csSup (hS N) ⟨⟨_, _⟩, ⟨hm, hn⟩, rfl⟩
have S0m : ∀ n, (0 : ℝ) ∈ S n := fun n => ⟨⟨n, n⟩, ⟨le_rfl, le_rfl⟩, dist_self _⟩
have S0 := fun n => le_csSup (hS n) (S0m n)
refine ⟨fun N => sSup (S N), S0, ub, Metric.tendsto_atTop.2 fun ε ε0 => ?_⟩
refine (Metric.cauchySeq_iff.1 hs (ε / 2) (half_pos ε0)).imp fun N hN n hn => ?_
rw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]
refine lt_of_le_of_lt (csSup_le ⟨_, S0m _⟩ ?_) (half_lt_self ε0)
rintro _ ⟨⟨m', n'⟩, ⟨hm', hn'⟩, rfl⟩
exact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn')),
fun ⟨b, _, b_bound, b_lim⟩ => cauchySeq_of_le_tendsto_0 b b_bound b_lim⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | cauchySeq_iff_le_tendsto_0 | Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. |
Metric.exists_subseq_bounded_of_cauchySeq (u : ℕ → α) (hu : CauchySeq u) (b : ℕ → ℝ)
(hb : ∀ n, 0 < b n) :
∃ f : ℕ → ℕ, StrictMono f ∧ ∀ n, ∀ m ≥ f n, dist (u m) (u (f n)) < b n := by
rw [cauchySeq_iff] at hu
have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
intro k
rw [eventually_atTop]
obtain ⟨N, hN⟩ := hu (b k) (hb k)
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩
exact Filter.extraction_forall_of_eventually hu' | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.EMetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Cauchy.lean | Metric.exists_subseq_bounded_of_cauchySeq | null |
CauSeq.tendsto_limit [NormedRing β] [hn : IsAbsoluteValue (norm : β → ℝ)]
(f : CauSeq β norm) [CauSeq.IsComplete β norm] : Tendsto f atTop (𝓝 f.lim) :=
tendsto_nhds.mpr
(by
intro s os lfs
suffices ∃ a : ℕ, ∀ b : ℕ, b ≥ a → f b ∈ s by simpa using this
rcases Metric.isOpen_iff.1 os _ lfs with ⟨ε, ⟨hε, hεs⟩⟩
obtain ⟨N, hN⟩ := Setoid.symm (CauSeq.equiv_lim f) _ hε
exists N
intro b hb
apply hεs
dsimp [Metric.ball]
rw [dist_comm, dist_eq_norm]
solve_by_elim)
variable [NormedField β]
/-
This section shows that if we have a uniform space generated by an absolute value, topological
completeness and Cauchy sequence completeness coincide. The problem is that there isn't
a good notion of "uniform space generated by an absolute value", so right now this is
specific to norm. Furthermore, norm only instantiates IsAbsoluteValue on NormedDivisionRing.
This needs to be fixed, since it prevents showing that ℤ_[hp] is complete.
-/
open Metric | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Topology.MetricSpace.Cauchy"
] | Mathlib/Topology/MetricSpace/CauSeqFilter.lean | CauSeq.tendsto_limit | null |
CauchySeq.isCauSeq {f : ℕ → β} (hf : CauchySeq f) : IsCauSeq norm f := by
obtain ⟨hf1, hf2⟩ := cauchy_iff.1 hf
intro ε hε
rcases hf2 { x | dist x.1 x.2 < ε } (dist_mem_uniformity hε) with ⟨t, ⟨ht, htsub⟩⟩
simp only [mem_map, mem_atTop_sets, mem_preimage] at ht; obtain ⟨N, hN⟩ := ht
exists N
intro j hj
rw [← dist_eq_norm]
apply @htsub (f j, f N)
apply Set.mk_mem_prod <;> solve_by_elim [le_refl] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Topology.MetricSpace.Cauchy"
] | Mathlib/Topology/MetricSpace/CauSeqFilter.lean | CauchySeq.isCauSeq | null |
CauSeq.cauchySeq (f : CauSeq β norm) : CauchySeq f := by
refine cauchy_iff.2 ⟨by infer_instance, fun s hs => ?_⟩
rcases mem_uniformity_dist.1 hs with ⟨ε, ⟨hε, hεs⟩⟩
obtain ⟨N, hN⟩ := CauSeq.cauchy₂ f hε
exists { n | n ≥ N }.image f
simp only [mem_atTop_sets, mem_map]
constructor
· exists N
intro b hb
exists b
· rintro ⟨a, b⟩ ⟨⟨a', ⟨ha'1, ha'2⟩⟩, ⟨b', ⟨hb'1, hb'2⟩⟩⟩
dsimp at ha'1 ha'2 hb'1 hb'2
rw [← ha'2, ← hb'2]
apply hεs
rw [dist_eq_norm]
apply hN <;> assumption | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Topology.MetricSpace.Cauchy"
] | Mathlib/Topology/MetricSpace/CauSeqFilter.lean | CauSeq.cauchySeq | null |
isCauSeq_iff_cauchySeq {α : Type u} [NormedField α] {u : ℕ → α} :
IsCauSeq norm u ↔ CauchySeq u :=
⟨fun h => CauSeq.cauchySeq ⟨u, h⟩, fun h => h.isCauSeq⟩ | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Topology.MetricSpace.Cauchy"
] | Mathlib/Topology/MetricSpace/CauSeqFilter.lean | isCauSeq_iff_cauchySeq | In a normed field, `CauSeq` coincides with the usual notion of Cauchy sequences. |
Closeds.emetricSpace : EMetricSpace (Closeds α) where
edist s t := hausdorffEdist (s : Set α) t
edist_self _ := hausdorffEdist_self
edist_comm _ _ := hausdorffEdist_comm
edist_triangle _ _ _ := hausdorffEdist_triangle
eq_of_edist_eq_zero {s t} h :=
Closeds.ext <| (hausdorffEdist_zero_iff_eq_of_closed s.isClosed t.isClosed).1 h | instance | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Closeds.lean | Closeds.emetricSpace | In emetric spaces, the Hausdorff edistance defines an emetric space structure
on the type of closed subsets |
continuous_infEdist_hausdorffEdist :
Continuous fun p : α × Closeds α => infEdist p.1 p.2 := by
refine continuous_of_le_add_edist 2 (by simp) ?_
rintro ⟨x, s⟩ ⟨y, t⟩
calc
infEdist x s ≤ infEdist x t + hausdorffEdist (t : Set α) s :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ infEdist y t + edist x y + hausdorffEdist (t : Set α) s := by
gcongr; apply infEdist_le_infEdist_add_edist
_ = infEdist y t + (edist x y + hausdorffEdist (s : Set α) t) := by
rw [add_assoc, hausdorffEdist_comm]
_ ≤ infEdist y t + (edist (x, s) (y, t) + edist (x, s) (y, t)) := by
gcongr <;> apply_rules [le_max_left, le_max_right]
_ = infEdist y t + 2 * edist (x, s) (y, t) := by rw [← mul_two, mul_comm] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Closeds.lean | continuous_infEdist_hausdorffEdist | The edistance to a closed set depends continuously on the point and the set |
Closeds.isClosed_subsets_of_isClosed (hs : IsClosed s) :
IsClosed { t : Closeds α | (t : Set α) ⊆ s } := by
refine isClosed_of_closure_subset fun
(t : Closeds α) (ht : t ∈ closure {t : Closeds α | (t : Set α) ⊆ s}) (x : α) (hx : x ∈ t) => ?_
have : x ∈ closure s := by
refine mem_closure_iff.2 fun ε εpos => ?_
obtain ⟨u : Closeds α, hu : u ∈ {t : Closeds α | (t : Set α) ⊆ s}, Dtu : edist t u < ε⟩ :=
mem_closure_iff.1 ht ε εpos
obtain ⟨y : α, hy : y ∈ u, Dxy : edist x y < ε⟩ := exists_edist_lt_of_hausdorffEdist_lt hx Dtu
exact ⟨y, hu hy, Dxy⟩
rwa [hs.closure_eq] at this
@[deprecated (since := "2025-08-20")]
alias isClosed_subsets_of_isClosed := Closeds.isClosed_subsets_of_isClosed | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Closeds.lean | Closeds.isClosed_subsets_of_isClosed | Subsets of a given closed subset form a closed set |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.