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values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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dist_pi_le_iff {f g : ∀ b, X b} {r : ℝ} (hr : 0 ≤ r) :
dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r | by
lift r to ℝ≥0 using hr
exact nndist_pi_le_iff | lemma | dist_pi_le_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"nndist_pi_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_pi_eq_iff {f g : ∀ b, X b} {r : ℝ} (hr : 0 < r) :
dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r | by
lift r to ℝ≥0 using hr.le
simp_rw [← coe_nndist, NNReal.coe_inj, nndist_pi_eq_iff hr, NNReal.coe_le_coe] | lemma | dist_pi_eq_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"NNReal.coe_inj",
"NNReal.coe_le_coe",
"coe_nndist",
"nndist_pi_eq_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_pi_le_iff' [Nonempty β] {f g : ∀ b, X b} {r : ℝ} :
dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r | by
by_cases hr : 0 ≤ r
· exact dist_pi_le_iff hr
· exact iff_of_false (fun h => hr <| dist_nonneg.trans h) fun h =>
hr <| dist_nonneg.trans <| h <| Classical.arbitrary _ | lemma | dist_pi_le_iff' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Classical.arbitrary",
"by_cases",
"dist_pi_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b | (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl | lemma | dist_pi_const_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"dist_nonneg",
"dist_pi_le_iff",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b | nndist_pi_le_iff.2 fun _ => le_rfl | lemma | nndist_pi_const_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_pi_const [Nonempty β] (a b : α) : (dist (fun _ : β => a) fun _ => b) = dist a b | by
simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b) | lemma | dist_pi_const | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"ENNReal.toReal",
"dist_edist",
"edist_pi_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_pi_const [Nonempty β] (a b : α) : (nndist (fun _ : β => a) fun _ => b) = nndist a b | NNReal.eq <| dist_pi_const a b | lemma | nndist_pi_const | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"NNReal.eq",
"dist_pi_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_le_pi_nndist (f g : ∀ b, X b) (b : β) : nndist (f b) (g b) ≤ nndist f g | by
rw [← ENNReal.coe_le_coe, ← edist_nndist, ← edist_nndist]
exact edist_le_pi_edist f g b | lemma | nndist_le_pi_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"ENNReal.coe_le_coe",
"edist_le_pi_edist",
"edist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_pi_dist (f g : ∀ b, X b) (b : β) : dist (f b) (g b) ≤ dist f g | by
simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b] | lemma | dist_le_pi_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"NNReal.coe_le_coe",
"dist_nndist",
"nndist_le_pi_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_pi (x : ∀ b, X b) {r : ℝ} (hr : 0 < r) :
ball x r = Set.pi univ fun b => ball (x b) r | by
ext p
simp [dist_pi_lt_iff hr] | lemma | ball_pi | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Set.pi",
"dist_pi_lt_iff"
] | An open ball in a product space is a product of open balls. See also `ball_pi'`
for a version assuming `Nonempty β` instead of `0 < r`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ball_pi' [Nonempty β] (x : ∀ b, X b) (r : ℝ) :
ball x r = Set.pi univ fun b => ball (x b) r | (lt_or_ge 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr] | lemma | ball_pi' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Set.pi",
"ball_pi",
"lt_or_ge"
] | An open ball in a product space is a product of open balls. See also `ball_pi`
for a version assuming `0 < r` instead of `Nonempty β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closedBall_pi (x : ∀ b, X b) {r : ℝ} (hr : 0 ≤ r) :
closedBall x r = Set.pi univ fun b => closedBall (x b) r | by
ext p
simp [dist_pi_le_iff hr] | lemma | closedBall_pi | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Set.pi",
"dist_pi_le_iff"
] | A closed ball in a product space is a product of closed balls. See also `closedBall_pi'`
for a version assuming `Nonempty β` instead of `0 ≤ r`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closedBall_pi' [Nonempty β] (x : ∀ b, X b) (r : ℝ) :
closedBall x r = Set.pi univ fun b => closedBall (x b) r | (le_or_gt 0 r).elim (closedBall_pi x) fun hr => by simp [closedBall_eq_empty.2 hr] | lemma | closedBall_pi' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Set.pi",
"closedBall_pi",
"le_or_gt"
] | A closed ball in a product space is a product of closed balls. See also `closedBall_pi`
for a version assuming `0 ≤ r` instead of `Nonempty β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
sphere_pi (x : ∀ b, X b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r | by
obtain hr | rfl | hr := lt_trichotomy r 0
· simp [hr]
· rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]
letI := h.resolve_left (lt_irrefl _)
inhabit β
refine subset_iUnion_of_subset default ?_
intro x hx
replace hx := hx.le
rw [dist_pi_le_iff le_rfl] at hx
ex... | lemma | sphere_pi | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Function.eval",
"Set.inter_eq_right",
"dist_nonneg",
"dist_pi_eq_iff",
"dist_pi_le_iff",
"le_antisymm",
"le_rfl",
"lt_irrefl",
"lt_trichotomy"
] | A sphere in a product space is a union of spheres on each component restricted to the closed
ball. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
[∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) | eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succAbove] | lemma | Fin.nndist_insertNth_insertNth | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"PseudoMetricSpace",
"nndist_pi_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
[∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) | by
simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max] | lemma | Fin.dist_insertNth_insertNth | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"Fin.nndist_insertNth_insertNth",
"NNReal.coe_max",
"PseudoMetricSpace",
"dist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_single_single {Y : Type*} [PseudoMetricSpace Y] [Zero Y] [DecidableEq β]
(i j : β) (a b : Y) (h : i ≠ j) :
nndist (Pi.single i a : β → Y) (Pi.single j b) = max (nndist a 0) (nndist b 0) | by
refine le_antisymm (nndist_pi_le_iff.2 fun k ↦ ?_) (max_le ?_ ?_)
· simp only [Pi.single_apply]
by_cases hki : k = i <;> by_cases hkj : k = j <;> simp_all [nndist_comm]
· simpa [h] using nndist_le_pi_nndist (Pi.single i a : β → Y) (Pi.single j b) i
· simpa [h, nndist_comm] using nndist_le_pi_nndist (Pi.s... | lemma | nndist_single_single | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"PseudoMetricSpace",
"by_cases",
"le_antisymm",
"nndist_comm",
"nndist_le_pi_nndist"
] | The (sup metric) nonnegative distance between `Pi.single i a` and `Pi.single j b` for
`i ≠ j` is `max (nndist a 0) (nndist b 0)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_single_single {Y : Type*} [PseudoMetricSpace Y] [Zero Y] [DecidableEq β]
(i j : β) (a b : Y) (h : i ≠ j) :
dist (Pi.single i a : β → Y) (Pi.single j b) = max (dist a 0) (dist b 0) | by
simp only [dist_nndist, nndist_single_single i j a b h, NNReal.coe_max] | lemma | dist_single_single | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Pi.lean | [] | [
"NNReal.coe_max",
"PseudoMetricSpace",
"dist_nndist",
"nndist_single_single"
] | The (sup metric) distance between `Pi.single i a` and `Pi.single j b` for
`i ≠ j` is `max (dist a 0) (dist b 0)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z | by
simpa only [dist_comm x] using! abs_sub_left_of_mem_uIcc h | lemma | Real.dist_left_le_of_mem_uIcc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Real.lean | [] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z | by
simpa only [dist_comm _ z] using! abs_sub_right_of_mem_uIcc h | lemma | Real.dist_right_le_of_mem_uIcc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Real.lean | [] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
dist x y ≤ dist x' y' | abs_sub_le_of_uIcc_subset_uIcc <| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm]) | lemma | Real.dist_le_of_mem_uIcc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Real.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
dist x y ≤ y' - x' | by
simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using
Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy) | lemma | Real.dist_le_of_mem_Icc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Real.lean | [] | [
"Real.dist_eq",
"Real.dist_le_of_mem_uIcc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
dist x y ≤ 1 | by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy | lemma | Real.dist_le_of_mem_Icc_01 | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Real.lean | [] | [
"Real.dist_le_of_mem_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_of_mem_pi_Icc (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') : dist x y ≤ dist x' y' | by
refine (dist_pi_le_iff dist_nonneg).2 fun b =>
(Real.dist_le_of_mem_uIcc ?_ ?_).trans (dist_le_pi_dist x' y' b) <;> refine Icc_subset_uIcc ?_
exacts [⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩] | lemma | Real.dist_le_of_mem_pi_Icc | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Real.lean | [] | [
"Real.dist_le_of_mem_uIcc",
"dist_le_pi_dist",
"dist_nonneg",
"dist_pi_le_iff",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUltrametricDist (X : Type*) [Dist X] : Prop where
dist_triangle_max : ∀ x y z : X, dist x z ≤ max (dist x y) (dist y z) | class | IsUltrametricDist | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"Dist",
"dist_triangle_max"
] | The `dist : X → X → ℝ` respects the ultrametric inequality
of `dist(x, z) ≤ max (dist(x,y)) (dist(y,z))`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_triangle_max : dist x z ≤ max (dist x y) (dist y z) | IsUltrametricDist.dist_triangle_max x y z | lemma | dist_triangle_max | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_eq_max_of_dist_ne_dist (h : dist x y ≠ dist y z) :
dist x z = max (dist x y) (dist y z) | by
apply le_antisymm (dist_triangle_max x y z)
rcases h.lt_or_gt with h | h
· rw [max_eq_right h.le]
apply (le_max_iff.mp <| dist_triangle_max y x z).resolve_left
simpa only [not_le, dist_comm x y] using h
· rw [max_eq_left h.le, dist_comm x y, dist_comm x z]
apply (le_max_iff.mp <| dist_triangle_ma... | lemma | IsUltrametricDist.dist_eq_max_of_dist_ne_dist | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"dist_comm",
"dist_triangle_max",
"le_antisymm",
"not_le"
] | All triangles are isosceles in an ultrametric space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
subtype (p : X → Prop) : IsUltrametricDist (Subtype p) | ⟨fun _ _ _ ↦ by simpa [Subtype.dist_eq] using dist_triangle_max _ _ _⟩ | instance | IsUltrametricDist.subtype | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsUltrametricDist",
"Subtype.dist_eq",
"dist_triangle_max"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_of_mem {x y : X} {r : ℝ} (h : y ∈ ball x r) : ball x r = ball y r | by
ext a
simp_rw [mem_ball] at h ⊢
constructor <;> intro h' <;>
exact (dist_triangle_max _ _ _).trans_lt (max_lt h' (dist_comm x _ ▸ h)) | lemma | IsUltrametricDist.ball_eq_of_mem | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"ball_eq_of_mem",
"dist_comm",
"dist_triangle_max"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_subset_trichotomy :
ball x r ⊆ ball y s ∨ ball y s ⊆ ball x r ∨ Disjoint (ball x r) (ball y s) | by
wlog! hrs : r ≤ s generalizing x y r s
· rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
exact this y x s r hrs.le
· refine Set.disjoint_or_nonempty_inter (ball x r) (ball y s) |>.symm.imp (fun h ↦ ?_) (Or.inr ·)
obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
have hx :=... | lemma | IsUltrametricDist.ball_subset_trichotomy | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"Disjoint",
"Set.disjoint_or_nonempty_inter",
"Set.mem_inter_iff",
"ball_eq_of_mem",
"disjoint_comm",
"symm",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_or_disjoint :
ball x r = ball y r ∨ Disjoint (ball x r) (ball y r) | by
refine Set.disjoint_or_nonempty_inter (ball x r) (ball y r) |>.symm.imp (fun h ↦ ?_) id
have h₁ := ball_eq_of_mem <| Set.inter_subset_left h.some_mem
have h₂ := ball_eq_of_mem <| Set.inter_subset_right h.some_mem
exact h₁.trans h₂.symm | lemma | IsUltrametricDist.ball_eq_or_disjoint | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"Disjoint",
"Set.disjoint_or_nonempty_inter",
"Set.inter_subset_left",
"Set.inter_subset_right",
"ball_eq_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_eq_of_mem {x y : X} {r : ℝ} (h : y ∈ closedBall x r) :
closedBall x r = closedBall y r | by
ext
simp_rw [mem_closedBall] at h ⊢
constructor <;> intro h' <;>
exact (dist_triangle_max _ _ _).trans (max_le h' (dist_comm x _ ▸ h)) | lemma | IsUltrametricDist.closedBall_eq_of_mem | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"dist_comm",
"dist_triangle_max",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_subset_trichotomy :
closedBall x r ⊆ closedBall y s ∨ closedBall y s ⊆ closedBall x r ∨
Disjoint (closedBall x r) (closedBall y s) | by
wlog! hrs : r ≤ s generalizing x y r s
· rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
exact this y x s r hrs.le
· refine Set.disjoint_or_nonempty_inter (closedBall x r) (closedBall y s) |>.symm.imp
(fun h ↦ ?_) (Or.inr ·)
obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_... | lemma | IsUltrametricDist.closedBall_subset_trichotomy | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"Disjoint",
"Set.disjoint_or_nonempty_inter",
"Set.mem_inter_iff",
"disjoint_comm",
"symm",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_ball (x : X) (r : ℝ) : IsClosed (ball x r) | by
cases le_or_gt r 0 with
| inl hr =>
simp [ball_eq_empty.mpr hr]
| inr h =>
rw [← isOpen_compl_iff, isOpen_iff]
push _ ∈ _
intro y hy
cases ball_eq_or_disjoint x y r with
| inl hd =>
rw [hd] at hy
simp [h.not_ge] at hy
| inr hd =>
use r
simp [h, ← Set.le_iff_s... | lemma | IsUltrametricDist.isClosed_ball | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsClosed",
"Set.le_iff_subset",
"isOpen_compl_iff",
"le_compl_iff_disjoint_left",
"le_or_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_ball : IsClopen (ball x r) | ⟨isClosed_ball x r, isOpen_ball⟩ | lemma | IsUltrametricDist.isClopen_ball | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsClopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_ball_eq_empty : frontier (ball x r) = ∅ | isClopen_iff_frontier_eq_empty.mp (isClopen_ball x r) | lemma | IsUltrametricDist.frontier_ball_eq_empty | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_eq_or_disjoint :
closedBall x r = closedBall y r ∨ Disjoint (closedBall x r) (closedBall y r) | by
refine Set.disjoint_or_nonempty_inter (closedBall x r) (closedBall y r) |>.symm.imp
(fun h ↦ ?_) id
have h₁ := closedBall_eq_of_mem <| Set.inter_subset_left h.some_mem
have h₂ := closedBall_eq_of_mem <| Set.inter_subset_right h.some_mem
exact h₁.trans h₂.symm | lemma | IsUltrametricDist.closedBall_eq_or_disjoint | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"Disjoint",
"Set.disjoint_or_nonempty_inter",
"Set.inter_subset_left",
"Set.inter_subset_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_closedBall {r : ℝ} (hr : r ≠ 0) : IsOpen (closedBall x r) | by
cases lt_or_gt_of_ne hr with
| inl h =>
simp [closedBall_eq_empty.mpr h]
| inr h =>
rw [isOpen_iff]
simp only [gt_iff_lt]
intro y hy
cases closedBall_eq_or_disjoint x y r with
| inl hd =>
use r
simp [h, hd, ball_subset_closedBall]
| inr hd =>
simp [closedBall_eq_of... | lemma | IsUltrametricDist.isOpen_closedBall | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsOpen",
"lt_or_gt_of_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_closedBall {r : ℝ} (hr : r ≠ 0) : IsClopen (closedBall x r) | ⟨Metric.isClosed_closedBall, isOpen_closedBall x hr⟩ | lemma | IsUltrametricDist.isClopen_closedBall | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsClopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_closedBall_eq_empty {r : ℝ} (hr : r ≠ 0) : frontier (closedBall x r) = ∅ | isClopen_iff_frontier_eq_empty.mp (isClopen_closedBall x hr) | lemma | IsUltrametricDist.frontier_closedBall_eq_empty | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_sphere {r : ℝ} (hr : r ≠ 0) : IsOpen (sphere x r) | by
rw [← closedBall_diff_ball, sdiff_eq]
exact (isOpen_closedBall x hr).inter (isClosed_ball x r).isOpen_compl | lemma | IsUltrametricDist.isOpen_sphere | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsOpen",
"sdiff_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_sphere {r : ℝ} (hr : r ≠ 0) : IsClopen (sphere x r) | ⟨Metric.isClosed_sphere, isOpen_sphere x hr⟩ | lemma | IsUltrametricDist.isClopen_sphere | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | [] | [
"IsClopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousMap.isUltrametricDist {X Y : Type*}
[TopologicalSpace X] [CompactSpace X] [MetricSpace Y] [IsUltrametricDist Y] :
IsUltrametricDist C(X, Y) | by
constructor
intro f g h
rw [ContinuousMap.dist_le (by positivity)]
refine fun x ↦ (dist_triangle_max (f x) (g x) (h x)).trans (max_le_max ?_ ?_) <;>
exact ContinuousMap.dist_apply_le_dist x | instance | ContinuousMap.isUltrametricDist | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/ContinuousMaps.lean | [] | [
"CompactSpace",
"ContinuousMap.dist_apply_le_dist",
"ContinuousMap.dist_le",
"IsUltrametricDist",
"MetricSpace",
"TopologicalSpace",
"dist_triangle_max",
"max_le_max",
"trans"
] | Continuous maps from a compact space to an ultrametric space are an ultrametric space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Pi.instIsUltrametricDist {ι : Type*} {X : ι → Type*} [Fintype ι]
[(i : ι) → PseudoMetricSpace (X i)] [(i : ι) → IsUltrametricDist (X i)] :
IsUltrametricDist ((i : ι) → X i) | by
constructor
intro f g h
simp only [dist_pi_def, ← NNReal.coe_max, NNReal.coe_le_coe, ← Finset.sup_sup]
exact Finset.sup_mono_fun fun i _ ↦ IsUltrametricDist.dist_triangle_max (f i) (g i) (h i) | instance | Pi.instIsUltrametricDist | Topology.MetricSpace.Ultra | Mathlib/Topology/MetricSpace/Ultra/Pi.lean | [] | [
"Finset.sup_mono_fun",
"Finset.sup_sup",
"Fintype",
"IsUltrametricDist",
"NNReal.coe_le_coe",
"NNReal.coe_max",
"PseudoMetricSpace",
"dist_pi_def"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
exists_countably_generated :
∃ u : UniformSpace X, u.toTopologicalSpace = t ∧ (uniformity X).IsCountablyGenerated | class | TopologicalSpace.PseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"TopologicalSpace",
"UniformSpace",
"uniformity"
] | A topological space is *pseudometrizable* if there exists a pseudometric space structure
compatible with the topology. To minimize imports, we implement this class in terms of the
existence of a countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compat... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pseudoMetrizableSpaceUniformity (X : Type*) [TopologicalSpace X]
[h : PseudoMetrizableSpace X] : UniformSpace X | h.exists_countably_generated.choose.replaceTopology
h.exists_countably_generated.choose_spec.1.symm | abbrev | TopologicalSpace.pseudoMetrizableSpaceUniformity | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"TopologicalSpace",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pseudoMetrizableSpaceUniformity_countably_generated
(X : Type*) [TopologicalSpace X] [h : PseudoMetrizableSpace X] :
𝓤[pseudoMetrizableSpaceUniformity X].IsCountablyGenerated | h.exists_countably_generated.choose_spec.2 | theorem | TopologicalSpace.pseudoMetrizableSpaceUniformity_countably_generated | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"TopologicalSpace"
] | The uniformity coming from `pseudoMetrizableSpaceUniformity` is countably generated.. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pseudoMetrizableSpace_prod [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y] :
PseudoMetrizableSpace (X × Y) | let : UniformSpace X := pseudoMetrizableSpaceUniformity X
have : (uniformity X).IsCountablyGenerated :=
pseudoMetrizableSpaceUniformity_countably_generated X
let : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
have : (uniformity Y).IsCountablyGenerated :=
pseudoMetrizableSpaceUniformity_countably_ge... | instance | TopologicalSpace.pseudoMetrizableSpace_prod | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"UniformSpace",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.IsInducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
(hf : IsInducing f) : PseudoMetrizableSpace X | let u : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
have : (uniformity Y).IsCountablyGenerated :=
pseudoMetrizableSpaceUniformity_countably_generated Y
⟨⟨u.comap f, u.toTopologicalSpace_comap.trans hf.eq_induced.symm,
Filter.comap.isCountablyGenerated (uniformity Y) (Prod.map f f)⟩⟩ | theorem | Topology.IsInducing.pseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"Filter.comap.isCountablyGenerated",
"UniformSpace",
"uniformity"
] | Given an inducing map of a topological space into a pseudometrizable space, the source space
is also pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetrizableSpace.subtype [PseudoMetrizableSpace X] (s : Set X) :
PseudoMetrizableSpace s | IsInducing.subtypeVal.pseudoMetrizableSpace | instance | TopologicalSpace.PseudoMetrizableSpace.subtype | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pseudoMetrizableSpace_pi [∀ i, PseudoMetrizableSpace (A i)] :
PseudoMetrizableSpace (∀ i, A i) | let := fun i => pseudoMetrizableSpaceUniformity (A i)
have := fun i => pseudoMetrizableSpaceUniformity_countably_generated (A i)
inferInstance | instance | TopologicalSpace.pseudoMetrizableSpace_pi | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoMetrizableSpace.regularSpace [PseudoMetrizableSpace X] : RegularSpace X | let := pseudoMetrizableSpaceUniformity X
inferInstance | instance | TopologicalSpace.PseudoMetrizableSpace.regularSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"RegularSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop extends
PseudoMetrizableSpace X, T0Space X | class | TopologicalSpace.MetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"T0Space",
"TopologicalSpace"
] | A topological space is metrizable if there exists a metric space structure compatible with the
topology. To minimize imports, we implement this class in terms of the existence of a
countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformit... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
metrizableSpace_prod [MetrizableSpace X] [MetrizableSpace Y] :
MetrizableSpace (X × Y) | where | instance | TopologicalSpace.metrizableSpace_prod | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Topology.IsEmbedding.metrizableSpace [MetrizableSpace Y] {f : X → Y}
(hf : IsEmbedding f) : MetrizableSpace X | where
toPseudoMetrizableSpace := hf.toIsInducing.pseudoMetrizableSpace
toT0Space := hf.t0Space | theorem | Topology.IsEmbedding.metrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [] | Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
MetrizableSpace.subtype [MetrizableSpace X] (s : Set X) : MetrizableSpace s | IsEmbedding.subtypeVal.metrizableSpace | instance | TopologicalSpace.MetrizableSpace.subtype | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
metrizableSpace_pi [∀ i, MetrizableSpace (A i)] : MetrizableSpace (∀ i, A i) | where | instance | TopologicalSpace.metrizableSpace_pi | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
(hs : IsSeparable s) : SecondCountableTopology s | let ⟨u, hu, hs⟩ := hs
have := hu.to_subtype
have : SeparableSpace (closure u) :=
⟨Set.range (u.inclusion subset_closure), Set.countable_range (u.inclusion subset_closure),
Subtype.dense_iff.2 <| by rw [← Set.range_comp, Set.val_comp_inclusion, Subtype.range_coe]⟩
let := pseudoMetrizableSpaceUniformity (... | theorem | TopologicalSpace.IsSeparable.secondCountableTopology | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"IsSeparable",
"SecondCountableTopology",
"Set",
"Set.countable_range",
"Set.range_comp",
"Set.val_comp_inclusion",
"Subtype.range_coe",
"Topology.IsEmbedding.inclusion",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparable.exists_countable_dense_subset [PseudoMetrizableSpace X]
{s : Set X} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t | by
let := pseudoMetrizableSpaceUniformity X
have := pseudoMetrizableSpaceUniformity_countably_generated X
apply subset_countable_closure_of_almost_dense_set
intro U hU
obtain ⟨t, htc, hst⟩ := hs
refine ⟨t, htc, fun x hx => ?_⟩
obtain ⟨y, hyx, hyt⟩ := mem_closure_iff_ball.1 (hst hx) (symmetrize_mem_uniform... | theorem | TopologicalSpace.IsSeparable.exists_countable_dense_subset | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"IsSeparable",
"Set",
"SetRel.symmetrize_subset_inv",
"closure",
"symmetrize_mem_uniformity"
] | If a set `s` is separable in a pseudo metrizable space, then it admits a countable dense
subset. This is not obvious, as the countable set whose closure covers `s` given by the definition
of separability does not need in general to be contained in `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSeparable.separableSpace [PseudoMetrizableSpace X] {s : Set X} (hs : IsSeparable s) :
SeparableSpace s | by
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
lift t to Set s using hts
refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall] | theorem | TopologicalSpace.IsSeparable.separableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Basic.lean | [] | [
"IsSeparable",
"Set"
] | If a set `s` is separable, then the corresponding subtype is separable in a
pseudo metrizable space.
This is not obvious, as the countable set whose closure covers `s` does not need in
general to be contained in `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompletelyPseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
complete : ∃ m : PseudoMetricSpace X, m.toUniformSpace.toTopologicalSpace = t ∧
@CompleteSpace X m.toUniformSpace | class | TopologicalSpace.IsCompletelyPseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"PseudoMetricSpace",
"TopologicalSpace"
] | A topological space is completely pseudometrizable if there exists a pseudometric space
structure compatible with the topology which makes the space complete.
To endow such a space with a compatible distance, use
`letI := upgradeIsCompletelyPseudoMetrizable X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpgradedIsCompletelyPseudoMetrizableSpace (X : Type*) extends
PseudoMetricSpace X, CompleteSpace X | class | TopologicalSpace.UpgradedIsCompletelyPseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"PseudoMetricSpace"
] | A convenience class, for a completely pseudometrizable space endowed with a complete
pseudometric. No instance of this class should be registered: It should be used as
`letI := upgradeIsCompletelyPseudoMetrizable X` to endow a completely pseudometrizable
space with a complete pseudometric. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyPseudoMetrizableMetric (X : Type*) [TopologicalSpace X]
[h : IsCompletelyPseudoMetrizableSpace X] : PseudoMetricSpace X | h.complete.choose.replaceTopology h.complete.choose_spec.1.symm | def | TopologicalSpace.completelyPseudoMetrizableMetric | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"PseudoMetricSpace",
"TopologicalSpace"
] | Construct on a completely pseudometrizable space a pseudometric (compatible with the topology)
which is complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
complete_completelyPseudoMetrizableMetric (X : Type*) [ht : TopologicalSpace X]
[h : IsCompletelyPseudoMetrizableSpace X] :
@CompleteSpace X (completelyPseudoMetrizableMetric X).toUniformSpace | by
convert! h.complete.choose_spec.2
exact PseudoMetricSpace.replaceTopology_eq _ _ | theorem | TopologicalSpace.complete_completelyPseudoMetrizableMetric | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"PseudoMetricSpace.replaceTopology_eq",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upgradeIsCompletelyPseudoMetrizable (X : Type*) [TopologicalSpace X]
[IsCompletelyPseudoMetrizableSpace X] :
UpgradedIsCompletelyPseudoMetrizableSpace X | letI := completelyPseudoMetrizableMetric X
{ complete_completelyPseudoMetrizableMetric X with } | def | TopologicalSpace.upgradeIsCompletelyPseudoMetrizable | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | This definition endows a completely pseudometrizable space with a complete pseudometric.
Use it as: `letI := upgradeIsCompletelyPseudoMetrizable X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pi_countable {ι : Type*} [Countable ι] {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, IsCompletelyPseudoMetrizableSpace (X i)] :
IsCompletelyPseudoMetrizableSpace (Π i, X i) | by
letI := fun i ↦ upgradeIsCompletelyPseudoMetrizable (X i)
infer_instance | instance | TopologicalSpace.IsCompletelyPseudoMetrizableSpace.pi_countable | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"Countable",
"TopologicalSpace"
] | A countable product of completely pseudometrizable spaces is completely pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prod [TopologicalSpace X] [IsCompletelyPseudoMetrizableSpace X] [TopologicalSpace Y]
[IsCompletelyPseudoMetrizableSpace Y] : IsCompletelyPseudoMetrizableSpace (X × Y) | letI := upgradeIsCompletelyPseudoMetrizable X
letI := upgradeIsCompletelyPseudoMetrizable Y
inferInstance | instance | TopologicalSpace.IsCompletelyPseudoMetrizableSpace.prod | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | The product of two completely pseudometrizable spaces is completely pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
sum [TopologicalSpace X] [IsCompletelyPseudoMetrizableSpace X] [TopologicalSpace Y]
[IsCompletelyPseudoMetrizableSpace Y] : IsCompletelyPseudoMetrizableSpace (X ⊕ Y) | letI := upgradeIsCompletelyPseudoMetrizable X
letI := upgradeIsCompletelyPseudoMetrizable Y
inferInstance | instance | TopologicalSpace.IsCompletelyPseudoMetrizableSpace.sum | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | The disjoint union of two completely pseudometrizable spaces is completely pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Topology.IsClosedEmbedding.IsCompletelyPseudoMetrizableSpace [TopologicalSpace X]
[TopologicalSpace Y] [IsCompletelyPseudoMetrizableSpace Y] {f : X → Y}
(hf : IsClosedEmbedding f) :
IsCompletelyPseudoMetrizableSpace X | by
letI := upgradeIsCompletelyPseudoMetrizable Y
letI : PseudoMetricSpace X := hf.isEmbedding.comapPseudoMetricSpace
have : CompleteSpace X := by
rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
exact hf.isClosed_range.isComplete
infer_instance | theorem | Topology.IsClosedEmbedding.IsCompletelyPseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"PseudoMetricSpace",
"TopologicalSpace",
"completeSpace_iff_isComplete_range"
] | Given a closed embedding into a completely pseudometrizable space,
the source space is also completely pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.IsClosed.isCompletelyPseudoMetrizableSpace
[TopologicalSpace X] [IsCompletelyPseudoMetrizableSpace X]
{s : Set X} (hs : IsClosed s) : IsCompletelyPseudoMetrizableSpace s | hs.isClosedEmbedding_subtypeVal.IsCompletelyPseudoMetrizableSpace | theorem | IsClosed.isCompletelyPseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"IsClosed",
"Set",
"TopologicalSpace"
] | A closed subset of a completely pseudometrizable space is also completely pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompletelyMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
complete : ∃ m : MetricSpace X, m.toUniformSpace.toTopologicalSpace = t ∧
@CompleteSpace X m.toUniformSpace | class | TopologicalSpace.IsCompletelyMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"MetricSpace",
"TopologicalSpace"
] | A topological space is completely metrizable if there exists a metric space structure
compatible with the topology which makes the space complete.
To endow such a space with a compatible distance, use
`letI := upgradeIsCompletelyMetrizable X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompletelyMetrizableSpace.toIsCompletelyPseudoMetrizableSpace [TopologicalSpace X]
[IsCompletelyMetrizableSpace X] : IsCompletelyPseudoMetrizableSpace X | by
obtain ⟨m, _⟩ := ‹_›
use m.toPseudoMetricSpace | instance | TopologicalSpace.IsCompletelyMetrizableSpace.toIsCompletelyPseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | A completely metrizable space is completely pseudometrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompletelyMetrizableSpace_of_isCompletelyPseudoMetrizableSpace [TopologicalSpace X]
[IsCompletelyPseudoMetrizableSpace X] [T0Space X] :
IsCompletelyMetrizableSpace X | by
letI := upgradeIsCompletelyPseudoMetrizable X
use MetricSpace.ofT0PseudoMetricSpace X
exact ⟨rfl, by infer_instance⟩ | lemma | TopologicalSpace.IsCompletelyMetrizableSpace_of_isCompletelyPseudoMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"MetricSpace.ofT0PseudoMetricSpace",
"T0Space",
"TopologicalSpace"
] | A completely pseudometrizable T0 space is completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpgradedIsCompletelyMetrizableSpace (X : Type*) extends MetricSpace X, CompleteSpace X | class | TopologicalSpace.UpgradedIsCompletelyMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"MetricSpace"
] | A convenience class, for a completely metrizable space endowed with a complete metric.
No instance of this class should be registered: It should be used as
`letI := upgradeIsCompletelyMetrizable X` to endow a completely metrizable
space with a complete metric. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completelyMetrizableMetric (X : Type*) [TopologicalSpace X]
[h : IsCompletelyMetrizableSpace X] : MetricSpace X | h.complete.choose.replaceTopology h.complete.choose_spec.1.symm | def | TopologicalSpace.completelyMetrizableMetric | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"MetricSpace",
"TopologicalSpace"
] | Construct on a completely metrizable space a metric (compatible with the topology)
which is complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
complete_completelyMetrizableMetric (X : Type*) [ht : TopologicalSpace X]
[h : IsCompletelyMetrizableSpace X] :
@CompleteSpace X (completelyMetrizableMetric X).toUniformSpace | by
convert! h.complete.choose_spec.2
exact MetricSpace.replaceTopology_eq _ _ | theorem | TopologicalSpace.complete_completelyMetrizableMetric | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"MetricSpace.replaceTopology_eq",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upgradeIsCompletelyMetrizable (X : Type*) [TopologicalSpace X] [IsCompletelyMetrizableSpace X] :
UpgradedIsCompletelyMetrizableSpace X | letI := completelyMetrizableMetric X
{ complete_completelyMetrizableMetric X with } | def | TopologicalSpace.upgradeIsCompletelyMetrizable | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | This definition endows a completely metrizable space with a complete metric. Use it as:
`letI := upgradeIsCompletelyMetrizable X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pi_countable {ι : Type*} [Countable ι] {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, IsCompletelyMetrizableSpace (X i)] : IsCompletelyMetrizableSpace (Π i, X i) | by
letI := fun i ↦ upgradeIsCompletelyMetrizable (X i)
infer_instance | instance | TopologicalSpace.IsCompletelyMetrizableSpace.pi_countable | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"Countable",
"TopologicalSpace"
] | A countable product of completely metrizable spaces is completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
sigma {ι : Type*} {X : ι → Type*} [∀ n, TopologicalSpace (X n)]
[∀ n, IsCompletelyMetrizableSpace (X n)] : IsCompletelyMetrizableSpace (Σ n, X n) | letI := fun n ↦ upgradeIsCompletelyMetrizable (X n)
letI : MetricSpace (Σ n, X n) := Metric.Sigma.metricSpace
haveI : CompleteSpace (Σ n, X n) := Metric.Sigma.completeSpace
inferInstance | instance | TopologicalSpace.IsCompletelyMetrizableSpace.sigma | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"Metric.Sigma.completeSpace",
"Metric.Sigma.metricSpace",
"MetricSpace",
"TopologicalSpace"
] | A disjoint union of completely metrizable spaces is completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prod [TopologicalSpace X] [IsCompletelyMetrizableSpace X] [TopologicalSpace Y]
[IsCompletelyMetrizableSpace Y] : IsCompletelyMetrizableSpace (X × Y) | letI := upgradeIsCompletelyMetrizable X
letI := upgradeIsCompletelyMetrizable Y
inferInstance | instance | TopologicalSpace.IsCompletelyMetrizableSpace.prod | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | The product of two completely metrizable spaces is completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
sum [TopologicalSpace X] [IsCompletelyMetrizableSpace X] [TopologicalSpace Y]
[IsCompletelyMetrizableSpace Y] : IsCompletelyMetrizableSpace (X ⊕ Y) | letI := upgradeIsCompletelyMetrizable X
letI := upgradeIsCompletelyMetrizable Y
inferInstance | instance | TopologicalSpace.IsCompletelyMetrizableSpace.sum | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"TopologicalSpace"
] | The disjoint union of two completely metrizable spaces is completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Topology.IsClosedEmbedding.IsCompletelyMetrizableSpace [TopologicalSpace X]
[TopologicalSpace Y] [IsCompletelyMetrizableSpace Y] {f : X → Y} (hf : IsClosedEmbedding f) :
IsCompletelyMetrizableSpace X | by
letI := upgradeIsCompletelyMetrizable Y
letI : MetricSpace X := hf.isEmbedding.comapMetricSpace f
have : CompleteSpace X := by
rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing]
exact hf.isClosed_range.isComplete
infer_instance | theorem | Topology.IsClosedEmbedding.IsCompletelyMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"CompleteSpace",
"MetricSpace",
"TopologicalSpace",
"completeSpace_iff_isComplete_range"
] | Given a closed embedding into a completely metrizable space,
the source space is also completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.IsClosed.isCompletelyMetrizableSpace
[TopologicalSpace X] [IsCompletelyMetrizableSpace X]
{s : Set X} (hs : IsClosed s) : IsCompletelyMetrizableSpace s | hs.isClosedEmbedding_subtypeVal.IsCompletelyMetrizableSpace | theorem | IsClosed.isCompletelyMetrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"IsClosed",
"Set",
"TopologicalSpace"
] | A closed subset of a completely metrizable space is also completely metrizable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
univ [TopologicalSpace X] [IsCompletelyMetrizableSpace X] :
IsCompletelyMetrizableSpace (univ : Set X) | isClosed_univ.isCompletelyMetrizableSpace | instance | TopologicalSpace.IsCompletelyMetrizableSpace.univ | Topology.Metrizable | Mathlib/Topology/Metrizable/CompletelyMetrizable.lean | [] | [
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x) : PseudoMetricSpace X | where
dist x y := ↑(⨅ l : List X, ((x::l).zipWith d (l ++ [y])).sum : ℝ≥0)
dist_self x := NNReal.coe_eq_zero.2 <|
nonpos_iff_eq_zero.1 <| (ciInf_le (OrderBot.bddBelow _) []).trans_eq <| by simp [dist_self]
dist_comm x y :=
NNReal.coe_inj.2 <| by
refine reverse_surjective.iInf_congr _ fun l ↦ ?_
... | def | PseudoMetricSpace.ofPreNNDist | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"NNReal.coe_add",
"NNReal.coe_le_coe",
"NNReal.le_iInf_add_iInf",
"PseudoMetricSpace",
"ciInf_le",
"dist_comm",
"dist_self",
"dist_triangle"
] | The maximal pseudometric space structure on `X` such that `dist x y ≤ d x y` for all `x y`,
where `d : X → X → ℝ≥0` is a function such that `d x x = 0` and `d x y = d y x` for all `x`, `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x) (x y : X) :
@dist X (@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x
y =
↑(⨅ l : List X, ((x::l).zipWith d (l ++ [y])).sum : ℝ≥0) | rfl | theorem | PseudoMetricSpace.dist_ofPreNNDist | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"PseudoMetricSpace.ofPreNNDist",
"dist_comm",
"dist_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_ofPreNNDist_le (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x) (x y : X) :
@dist X (@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x
y ≤
d x y | NNReal.coe_le_coe.2 <| (ciInf_le (OrderBot.bddBelow _) []).trans_eq <| by simp | theorem | PseudoMetricSpace.dist_ofPreNNDist_le | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"PseudoMetricSpace.ofPreNNDist",
"ciInf_le",
"dist_comm",
"dist_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x)
(hd : ∀ x₁ x₂ x₃ x₄, d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
↑(d x y) ≤ 2 * @dist X
(@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm... | by
/- We need to show that `d x y` is at most twice the sum `L` of `d xᵢ xᵢ₊₁` over a path
`x₀=x, ..., xₙ=y`. We prove it by induction on the length `n` of the sequence. Find an edge that
splits the path into two parts of almost equal length: both `d x₀ x₁ + ... + d xₖ₋₁ xₖ` and
`d xₖ₊₁ xₖ₊₂ + ... + d xₙ₋... | theorem | PseudoMetricSpace.le_two_mul_dist_ofPreNNDist | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"BddAbove",
"IsTrans",
"NNReal.coe_le_coe",
"NNReal.coe_mul",
"NNReal.coe_two",
"NNReal.mul_iInf",
"Nat.strong_induction_on",
"Option.coe_def",
"PseudoMetricSpace.ofPreNNDist",
"Set",
"csSup_le",
"dist_comm",
"dist_self",
"eq_or_ne",
"le_ciInf",
"le_csSup",
"le_rfl",
"min_eq_left",... | Consider a function `d : X → X → ℝ≥0` such that `d x x = 0` and `d x y = d y x` for all `x`,
`y`. Let `dist` be the largest pseudometric distance such that `dist x y ≤ d x y`, see
`PseudoMetricSpace.ofPreNNDist`. Suppose that `d` satisfies the following triangle-like
inequality: `d x₁ x₄ ≤ 2 * max (d x₁ x₂, d x₂ x₃, d ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X]
[IsCountablyGenerated (𝓤 X)] : ∃ I : PseudoMetricSpace X, I.toUniformSpace = ‹_› | by
classical
/- Choose a fast decreasing antitone basis `U : ℕ → SetRel X X` of the uniformity filter `𝓤 X`.
Define `d x y : ℝ≥0` to be `(1 / 2) ^ n`, where `n` is the minimal index of `U n` that
separates `x` and `y`: `(x, y) ∉ U n`, or `0` if `x` is not separated from `y`. This function
satisfies the... | theorem | UniformSpace.metrizable_uniformity | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"Inseparable",
"IsSymm",
"NNReal.coe_div",
"NNReal.coe_le_coe",
"NNReal.coe_lt_coe",
"NNReal.coe_pow",
"NNReal.coe_two",
"NNReal.half_lt_self",
"Nat.find",
"Nat.find_le_iff",
"Nat.find_spec",
"PseudoMetricSpace",
"PseudoMetricSpace.dist_ofPreNNDist_le",
"PseudoMetricSpace.le_two_mul_dist_o... | If `X` is a uniform space with countably generated uniformity filter, there exists a
`PseudoMetricSpace` structure compatible with the `UniformSpace` structure. Use
`UniformSpace.pseudoMetricSpace` or `UniformSpace.metricSpace` instead. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformSpace.pseudoMetricSpace (X : Type*) [UniformSpace X]
[IsCountablyGenerated (𝓤 X)] : PseudoMetricSpace X | (UniformSpace.metrizable_uniformity X).choose.replaceUniformity <|
congr_arg _ (UniformSpace.metrizable_uniformity X).choose_spec.symm | abbrev | UniformSpace.pseudoMetricSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"PseudoMetricSpace",
"UniformSpace",
"UniformSpace.metrizable_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.metricSpace (X : Type*) [UniformSpace X]
[IsCountablyGenerated (𝓤 X)] [T0Space X] : MetricSpace X | @MetricSpace.ofT0PseudoMetricSpace X (UniformSpace.pseudoMetricSpace X) _ | abbrev | UniformSpace.metricSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"MetricSpace",
"MetricSpace.ofT0PseudoMetricSpace",
"T0Space",
"UniformSpace",
"UniformSpace.pseudoMetricSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.metrizableSpace [UniformSpace X] [IsCountablyGenerated (𝓤 X)] [T0Space X] :
TopologicalSpace.MetrizableSpace X | inferInstance | theorem | UniformSpace.metrizableSpace | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"T0Space",
"TopologicalSpace.MetrizableSpace",
"UniformSpace"
] | A T₀ uniform space with countably generated `𝓤 X` is metrizable. This is not an instance to
avoid loops. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TopologicalSpace.pseudoMetrizableSpacePseudoMetric (X : Type*)
[TopologicalSpace X] [TopologicalSpace.PseudoMetrizableSpace X] : PseudoMetricSpace X | letI := TopologicalSpace.pseudoMetrizableSpaceUniformity X
haveI := TopologicalSpace.pseudoMetrizableSpaceUniformity_countably_generated X
UniformSpace.pseudoMetricSpace X | abbrev | TopologicalSpace.pseudoMetrizableSpacePseudoMetric | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"PseudoMetricSpace",
"TopologicalSpace",
"TopologicalSpace.PseudoMetrizableSpace",
"TopologicalSpace.pseudoMetrizableSpaceUniformity",
"TopologicalSpace.pseudoMetrizableSpaceUniformity_countably_generated",
"UniformSpace.pseudoMetricSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TopologicalSpace.metrizableSpaceMetric (X : Type*) [TopologicalSpace X]
[TopologicalSpace.MetrizableSpace X] : MetricSpace X | letI := pseudoMetrizableSpacePseudoMetric X
.ofT0PseudoMetricSpace X | abbrev | TopologicalSpace.metrizableSpaceMetric | Topology.Metrizable | Mathlib/Topology/Metrizable/Uniformity.lean | [] | [
"MetricSpace",
"TopologicalSpace",
"TopologicalSpace.MetrizableSpace"
] | Construct on a metrizable space a metric compatible with the topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_isInducing_l_infty : ∃ f : X → ℕ →ᵇ ℝ, IsInducing f | by
-- Choose a countable basis, and consider the set `s` of pairs of set `(U, V)` such that `U ∈ B`,
-- `V ∈ B`, and `closure U ⊆ V`.
rcases exists_countable_basis X with ⟨B, hBc, -, hB⟩
let s : Set (Set X × Set X) := { UV ∈ B ×ˢ B | closure UV.1 ⊆ UV.2 }
-- `s` is a countable set.
haveI : Encodable s := ((... | theorem | TopologicalSpace.exists_isInducing_l_infty | Topology.Metrizable | Mathlib/Topology/Metrizable/Urysohn.lean | [] | [
"BoundedContinuousFunction.dist_coe_le_dist",
"BoundedContinuousFunction.dist_le",
"BoundedContinuousFunction.isometry_extend",
"DiscreteTopology",
"Disjoint",
"Encodable",
"Encodable.encode'",
"Finite",
"Real.dist_le_of_mem_Icc",
"Real.dist_le_of_mem_Icc_01",
"Set",
"TopologicalSpace",
"clo... | For a regular topological space with second countable topology,
there exists an inducing map to `l^∞ = ℕ →ᵇ ℝ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_embedding_l_infty : ∃ f : X → ℕ →ᵇ ℝ, IsEmbedding f | let ⟨f, hf⟩ := exists_isInducing_l_infty X; ⟨f, hf.isEmbedding⟩ | theorem | TopologicalSpace.exists_embedding_l_infty | Topology.Metrizable | Mathlib/Topology/Metrizable/Urysohn.lean | [] | [] | A T₃ topological space with second countable topology can be embedded into `l^∞ = ℕ →ᵇ ℝ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
refl (X : Type*) [TopologicalSpace X] : OpenPartialHomeomorph X X | (Homeomorph.refl X).toOpenPartialHomeomorph | def | OpenPartialHomeomorph.refl | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"Homeomorph.refl",
"OpenPartialHomeomorph",
"TopologicalSpace",
"refl"
] | The identity on the whole space as an open partial homeomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
refl_partialEquiv : (OpenPartialHomeomorph.refl X).toPartialEquiv = PartialEquiv.refl X | rfl | theorem | OpenPartialHomeomorph.refl_partialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"OpenPartialHomeomorph.refl",
"PartialEquiv.refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl_symm : (OpenPartialHomeomorph.refl X).symm = OpenPartialHomeomorph.refl X | rfl | theorem | OpenPartialHomeomorph.refl_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [
"OpenPartialHomeomorph.refl",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
source_preimage_target : e.source ⊆ e ⁻¹' e.target | e.mapsTo | theorem | OpenPartialHomeomorph.source_preimage_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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