statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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joined_in.mem (h : joined_in F x y) : x ∈ F ∧ y ∈ F | begin
rcases h with ⟨γ, γ_in⟩,
have : γ 0 ∈ F ∧ γ 1 ∈ F, by { split; apply γ_in },
simpa using this
end | lemma | joined_in.mem | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.source_mem (h : joined_in F x y) : x ∈ F | h.mem.1 | lemma | joined_in.source_mem | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.target_mem (h : joined_in F x y) : y ∈ F | h.mem.2 | lemma | joined_in.target_mem | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.some_path (h : joined_in F x y) : path x y | classical.some h | def | joined_in.some_path | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in",
"path"
] | When `x` and `y` are joined in `F`, choose a path from `x` to `y` inside `F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
joined_in.some_path_mem (h : joined_in F x y) (t : I) : h.some_path t ∈ F | classical.some_spec h t | lemma | joined_in.some_path_mem | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.joined_subtype (h : joined_in F x y) :
joined (⟨x, h.source_mem⟩ : F) (⟨y, h.target_mem⟩ : F) | ⟨{ to_fun := λ t, ⟨h.some_path t, h.some_path_mem t⟩,
continuous_to_fun := by continuity,
source' := by simp,
target' := by simp }⟩ | lemma | joined_in.joined_subtype | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuity",
"joined",
"joined_in"
] | If `x` and `y` are joined in the set `F`, then they are joined in the subtype `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
joined_in.of_line {f : ℝ → X} (hf : continuous_on f I) (h₀ : f 0 = x) (h₁ : f 1 = y)
(hF : f '' I ⊆ F) : joined_in F x y | ⟨path.of_line hf h₀ h₁, λ t, hF $ path.of_line_mem hf h₀ h₁ t⟩ | lemma | joined_in.of_line | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_on",
"joined_in",
"path.of_line_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.joined (h : joined_in F x y) : joined x y | ⟨h.some_path⟩ | lemma | joined_in.joined | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined",
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in_iff_joined (x_in : x ∈ F) (y_in : y ∈ F) :
joined_in F x y ↔ joined (⟨x, x_in⟩ : F) (⟨y, y_in⟩ : F) | ⟨λ h, h.joined_subtype, λ h, ⟨h.some_path.map continuous_subtype_coe, by simp⟩⟩ | lemma | joined_in_iff_joined | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_subtype_coe",
"joined",
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in_univ : joined_in univ x y ↔ joined x y | by simp [joined_in, joined, exists_true_iff_nonempty] | lemma | joined_in_univ | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"exists_true_iff_nonempty",
"joined",
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.mono {U V : set X} (h : joined_in U x y) (hUV : U ⊆ V) : joined_in V x y | ⟨h.some_path, λ t, hUV (h.some_path_mem t)⟩ | lemma | joined_in.mono | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.refl (h : x ∈ F) : joined_in F x x | ⟨path.refl x, λ t, h⟩ | lemma | joined_in.refl | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.symm (h : joined_in F x y) : joined_in F y x | begin
cases h.mem with hx hy,
simp [joined_in_iff_joined, *] at *,
exact h.symm
end | lemma | joined_in.symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in",
"joined_in_iff_joined"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined_in.trans (hxy : joined_in F x y) (hyz : joined_in F y z) : joined_in F x z | begin
cases hxy.mem with hx hy,
cases hyz.mem with hx hy,
simp [joined_in_iff_joined, *] at *,
exact hxy.trans hyz
end | lemma | joined_in.trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in",
"joined_in_iff_joined"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_component (x : X) | {y | joined x y} | def | path_component | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined"
] | The path component of `x` is the set of points that can be joined to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_path_component_self (x : X) : x ∈ path_component x | joined.refl x | lemma | mem_path_component_self | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined.refl",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_component.nonempty (x : X) : (path_component x).nonempty | ⟨x, mem_path_component_self x⟩ | lemma | path_component.nonempty | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"mem_path_component_self",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_path_component_of_mem (h : x ∈ path_component y) : y ∈ path_component x | joined.symm h | lemma | mem_path_component_of_mem | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined.symm",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_component_symm : x ∈ path_component y ↔ y ∈ path_component x | ⟨λ h, mem_path_component_of_mem h, λ h, mem_path_component_of_mem h⟩ | lemma | path_component_symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"mem_path_component_of_mem",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_component_congr (h : x ∈ path_component y) : path_component x = path_component y | begin
ext z,
split,
{ intro h',
rw path_component_symm,
exact (h.trans h').symm },
{ intro h',
rw path_component_symm at h' ⊢,
exact h'.trans h },
end | lemma | path_component_congr | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path_component",
"path_component_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_component_subset_component (x : X) : path_component x ⊆ connected_component x | λ y h, (is_connected_range h.some_path.continuous).subset_connected_component
⟨0, by simp⟩ ⟨1, by simp⟩ | lemma | path_component_subset_component | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"connected_component",
"is_connected_range",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_component_in (x : X) (F : set X) | {y | joined_in F x y} | def | path_component_in | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | The path component of `x` in `F` is the set of points that can be joined to `x` in `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
path_component_in_univ (x : X) : path_component_in x univ = path_component x | by simp [path_component_in, path_component, joined_in, joined, exists_true_iff_nonempty] | lemma | path_component_in_univ | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"exists_true_iff_nonempty",
"joined",
"joined_in",
"path_component",
"path_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined.mem_path_component (hyz : joined y z) (hxy : y ∈ path_component x) :
z ∈ path_component x | hxy.trans hyz | lemma | joined.mem_path_component | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected (F : set X) : Prop | ∃ x ∈ F, ∀ {y}, y ∈ F → joined_in F x y | def | is_path_connected | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined_in"
] | A set `F` is path connected if it contains a point that can be joined to all other in `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_path_connected_iff_eq : is_path_connected F ↔ ∃ x ∈ F, path_component_in x F = F | begin
split ; rintros ⟨x, x_in, h⟩ ; use [x, x_in],
{ ext y,
exact ⟨λ hy, hy.mem.2, h⟩ },
{ intros y y_in,
rwa ← h at y_in },
end | lemma | is_path_connected_iff_eq | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"path_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.joined_in (h : is_path_connected F) : ∀ x y ∈ F, joined_in F x y | λ x x_in x y_in, let ⟨b, b_in, hb⟩ := h in (hb x_in).symm.trans (hb y_in) | lemma | is_path_connected.joined_in | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected_iff : is_path_connected F ↔ F.nonempty ∧ ∀ x y ∈ F, joined_in F x y | ⟨λ h, ⟨let ⟨b, b_in, hb⟩ := h in ⟨b, b_in⟩, h.joined_in⟩,
λ ⟨⟨b, b_in⟩, h⟩, ⟨b, b_in, λ x x_in, h b b_in x x_in⟩⟩ | lemma | is_path_connected_iff | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"joined_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.image {Y : Type*} [topological_space Y] (hF : is_path_connected F)
{f : X → Y} (hf : continuous f) : is_path_connected (f '' F) | begin
rcases hF with ⟨x, x_in, hx⟩,
use [f x, mem_image_of_mem f x_in],
rintros _ ⟨y, y_in, rfl⟩,
exact ⟨(hx y_in).some_path.map hf, λ t, ⟨_, (hx y_in).some_path_mem t, rfl⟩⟩,
end | lemma | is_path_connected.image | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"is_path_connected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.mem_path_component (h : is_path_connected F) (x_in : x ∈ F) (y_in : y ∈ F) :
y ∈ path_component x | (h.joined_in x x_in y y_in).joined | lemma | is_path_connected.mem_path_component | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"joined",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.subset_path_component (h : is_path_connected F) (x_in : x ∈ F) :
F ⊆ path_component x | λ y y_in, h.mem_path_component x_in y_in | lemma | is_path_connected.subset_path_component | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"path_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.union {U V : set X} (hU : is_path_connected U) (hV : is_path_connected V)
(hUV : (U ∩ V).nonempty) : is_path_connected (U ∪ V) | begin
rcases hUV with ⟨x, xU, xV⟩,
use [x, or.inl xU],
rintros y (yU | yV),
{ exact (hU.joined_in x xU y yU).mono (subset_union_left U V) },
{ exact (hV.joined_in x xV y yV).mono (subset_union_right U V) },
end | lemma | is_path_connected.union | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.preimage_coe {U W : set X} (hW : is_path_connected W) (hWU : W ⊆ U) :
is_path_connected ((coe : U → X) ⁻¹' W) | begin
rcases hW with ⟨x, x_in, hx⟩,
use [⟨x, hWU x_in⟩, by simp [x_in]],
rintros ⟨y, hyU⟩ hyW,
exact ⟨(hx hyW).joined_subtype.some_path.map (continuous_inclusion hWU), by simp⟩
end | lemma | is_path_connected.preimage_coe | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_inclusion",
"is_path_connected"
] | If a set `W` is path-connected, then it is also path-connected when seen as a set in a smaller
ambient type `U` (when `U` contains `W`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_path_connected.exists_path_through_family
{X : Type*} [topological_space X] {n : ℕ} {s : set X} (h : is_path_connected s)
(p : fin (n+1) → X) (hp : ∀ i, p i ∈ s) :
∃ γ : path (p 0) (p n), (range γ ⊆ s) ∧ (∀ i, p i ∈ range γ) | begin
let p' : ℕ → X := λ k, if h : k < n+1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩,
obtain ⟨γ, hγ⟩ : ∃ (γ : path (p' 0) (p' n)), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s,
{ have hp' : ∀ i ≤ n, p' i ∈ s,
{ intros i hi,
simp [p', nat.lt_succ_of_le hi, hp] },
clear_value p',
clear hp p,
induct... | lemma | is_path_connected.exists_path_through_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"fin.coe_coe_of_lt",
"is_path_connected",
"le_rfl",
"nat.succ_le_iff",
"path",
"path.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.exists_path_through_family'
{X : Type*} [topological_space X] {n : ℕ} {s : set X} (h : is_path_connected s)
(p : fin (n+1) → X) (hp : ∀ i, p i ∈ s) :
∃ (γ : path (p 0) (p n)) (t : fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i | begin
rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩,
rcases hγ with ⟨h₁, h₂⟩,
simp only [range, mem_set_of_eq] at h₂,
rw range_subset_iff at h₁,
choose! t ht using h₂,
exact ⟨γ, t, h₁, ht⟩
end | lemma | is_path_connected.exists_path_through_family' | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_connected_space (X : Type*) [topological_space X] : Prop | (nonempty : nonempty X)
(joined : ∀ x y : X, joined x y) | class | path_connected_space | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined",
"topological_space"
] | A topological space is path-connected if it is non-empty and every two points can be
joined by a continuous path. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
path_connected_space_iff_zeroth_homotopy :
path_connected_space X ↔ nonempty (zeroth_homotopy X) ∧ subsingleton (zeroth_homotopy X) | begin
letI := path_setoid X,
split,
{ introI h,
refine ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩,
rintros ⟨x⟩ ⟨y⟩,
exact quotient.sound (path_connected_space.joined x y) },
{ unfold zeroth_homotopy,
rintros ⟨h, h'⟩,
resetI,
exact ⟨(nonempty_quotient_iff _).mp h, λ x y, quotient.exact $ su... | lemma | path_connected_space_iff_zeroth_homotopy | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"nonempty_quotient_iff",
"path_connected_space",
"path_setoid",
"zeroth_homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_path (x y : X) : path x y | nonempty.some (joined x y) | def | path_connected_space.some_path | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined",
"nonempty.some",
"path"
] | Use path-connectedness to build a path between two points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_path_connected_iff_path_connected_space : is_path_connected F ↔ path_connected_space F | begin
rw is_path_connected_iff,
split,
{ rintro ⟨⟨x, x_in⟩, h⟩,
refine ⟨⟨⟨x, x_in⟩⟩, _⟩,
rintros ⟨y, y_in⟩ ⟨z, z_in⟩,
have H := h y y_in z z_in,
rwa joined_in_iff_joined y_in z_in at H },
{ rintros ⟨⟨x, x_in⟩, H⟩,
refine ⟨⟨x, x_in⟩, λ y y_in z z_in, _⟩,
rw joined_in_iff_joined y_in z_in,... | lemma | is_path_connected_iff_path_connected_space | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"is_path_connected_iff",
"joined_in_iff_joined",
"path_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_connected_space_iff_univ : path_connected_space X ↔ is_path_connected (univ : set X) | begin
split,
{ introI h,
haveI := @path_connected_space.nonempty X _ _,
inhabit X,
refine ⟨default, mem_univ _, _⟩,
simpa using path_connected_space.joined default },
{ intro h,
have h' := h.joined_in,
cases h with x h,
exact ⟨⟨x⟩, by simpa using h'⟩ },
end | lemma | path_connected_space_iff_univ | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"path_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_connected_space_iff_eq : path_connected_space X ↔ ∃ x : X, path_component x = univ | by simp [path_connected_space_iff_univ, is_path_connected_iff_eq] | lemma | path_connected_space_iff_eq | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected_iff_eq",
"path_component",
"path_connected_space",
"path_connected_space_iff_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_connected_space.connected_space [path_connected_space X] : connected_space X | begin
rw connected_space_iff_connected_component,
rcases is_path_connected_iff_eq.mp (path_connected_space_iff_univ.mp ‹_›) with ⟨x, x_in, hx⟩,
use x,
rw ← univ_subset_iff,
exact (by simpa using hx : path_component x = univ) ▸ path_component_subset_component x
end | instance | path_connected_space.connected_space | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"connected_space",
"connected_space_iff_connected_component",
"path_component",
"path_component_subset_component",
"path_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_path_connected.is_connected (hF : is_path_connected F) : is_connected F | begin
rw is_connected_iff_connected_space,
rw is_path_connected_iff_path_connected_space at hF,
exact @path_connected_space.connected_space _ _ hF
end | lemma | is_path_connected.is_connected | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_connected",
"is_connected_iff_connected_space",
"is_path_connected",
"is_path_connected_iff_path_connected_space",
"path_connected_space.connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_path_through_family {n : ℕ} (p : fin (n+1) → X) :
∃ γ : path (p 0) (p n), (∀ i, p i ∈ range γ) | begin
have : is_path_connected (univ : set X) := path_connected_space_iff_univ.mp (by apply_instance),
rcases this.exists_path_through_family p (λ i, true.intro) with ⟨γ, -, h⟩,
exact ⟨γ, h⟩
end | lemma | path_connected_space.exists_path_through_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_path_through_family' {n : ℕ} (p : fin (n+1) → X) :
∃ (γ : path (p 0) (p n)) (t : fin (n + 1) → I), ∀ i, γ (t i) = p i | begin
have : is_path_connected (univ : set X) := path_connected_space_iff_univ.mp (by apply_instance),
rcases this.exists_path_through_family' p (λ i, true.intro) with ⟨γ, t, -, h⟩,
exact ⟨γ, t, h⟩
end | lemma | path_connected_space.exists_path_through_family' | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
loc_path_connected_space (X : Type*) [topological_space X] : Prop | (path_connected_basis : ∀ x : X, (𝓝 x).has_basis (λ s : set X, s ∈ 𝓝 x ∧ is_path_connected s) id) | class | loc_path_connected_space | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"topological_space"
] | A topological space is locally path connected, at every point, path connected
neighborhoods form a neighborhood basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
loc_path_connected_of_bases {p : ι → Prop} {s : X → ι → set X}
(h : ∀ x, (𝓝 x).has_basis p (s x)) (h' : ∀ x i, p i → is_path_connected (s x i)) :
loc_path_connected_space X | begin
constructor,
intro x,
apply (h x).to_has_basis,
{ intros i pi,
exact ⟨s x i, ⟨(h x).mem_of_mem pi, h' x i pi⟩, by refl⟩ },
{ rintros U ⟨U_in, hU⟩,
rcases (h x).mem_iff.mp U_in with ⟨i, pi, hi⟩,
tauto }
end | lemma | loc_path_connected_of_bases | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_path_connected",
"loc_path_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_connected_space_iff_connected_space [loc_path_connected_space X] :
path_connected_space X ↔ connected_space X | begin
split,
{ introI h,
apply_instance },
{ introI hX,
rw path_connected_space_iff_eq,
use (classical.arbitrary X),
refine is_clopen.eq_univ ⟨_, _⟩ (by simp),
{ rw is_open_iff_mem_nhds,
intros y y_in,
rcases (path_connected_basis y).ex_mem with ⟨U, ⟨U_in, hU⟩⟩,
apply mem_of_... | lemma | path_connected_space_iff_connected_space | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"classical.arbitrary",
"connected_space",
"is_clopen.eq_univ",
"is_closed_iff_nhds",
"is_open_iff_mem_nhds",
"joined.mem_path_component",
"loc_path_connected_space",
"mem_of_mem_nhds",
"path_component_congr",
"path_connected_space",
"path_connected_space_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_connected_subset_basis [loc_path_connected_space X] {U : set X} (h : is_open U)
(hx : x ∈ U) : (𝓝 x).has_basis (λ s : set X, s ∈ 𝓝 x ∧ is_path_connected s ∧ s ⊆ U) id | (path_connected_basis x).has_basis_self_subset (is_open.mem_nhds h hx) | lemma | path_connected_subset_basis | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_open",
"is_open.mem_nhds",
"is_path_connected",
"loc_path_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
loc_path_connected_of_is_open [loc_path_connected_space X] {U : set X} (h : is_open U) :
loc_path_connected_space U | ⟨begin
rintros ⟨x, x_in⟩,
rw nhds_subtype_eq_comap,
constructor,
intros V,
rw (has_basis.comap (coe : U → X) (path_connected_subset_basis h x_in)).mem_iff,
split,
{ rintros ⟨W, ⟨W_in, hW, hWU⟩, hWV⟩,
exact ⟨coe ⁻¹' W, ⟨⟨preimage_mem_comap W_in, hW.preimage_coe hWU⟩, hWV⟩⟩ },
{ rintros ⟨W, ⟨W_in, hW⟩... | lemma | loc_path_connected_of_is_open | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_subtype_coe",
"is_open",
"is_open.mem_nhds",
"loc_path_connected_space",
"nhds_subtype_eq_comap",
"path_connected_subset_basis",
"subtype.coe_image_subset",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.is_connected_iff_is_path_connected
[loc_path_connected_space X] {U : set X} (U_op : is_open U) :
is_path_connected U ↔ is_connected U | begin
rw [is_connected_iff_connected_space, is_path_connected_iff_path_connected_space],
haveI := loc_path_connected_of_is_open U_op,
exact path_connected_space_iff_connected_space
end | lemma | is_open.is_connected_iff_is_path_connected | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"is_connected",
"is_connected_iff_connected_space",
"is_open",
"is_path_connected",
"is_path_connected_iff_path_connected_space",
"loc_path_connected_of_is_open",
"loc_path_connected_space",
"path_connected_space_iff_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc_pt.nhds_inter {x : α} {U : set α} (h_acc : acc_pt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
acc_pt x (𝓟 (U ∩ C)) | begin
have : 𝓝[≠] x ≤ 𝓟 U,
{ rw le_principal_iff,
exact mem_nhds_within_of_mem_nhds hU, },
rw [acc_pt, ← inf_principal, ← inf_assoc, inf_of_le_left this],
exact h_acc,
end | theorem | acc_pt.nhds_inter | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"acc_pt",
"inf_assoc",
"mem_nhds_within_of_mem_nhds"
] | If `x` is an accumulation point of a set `C` and `U` is a neighborhood of `x`,
then `x` is an accumulation point of `U ∩ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preperfect (C : set α) : Prop | ∀ x ∈ C, acc_pt x (𝓟 C) | def | preperfect | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"acc_pt"
] | A set `C` is preperfect if all of its points are accumulation points of itself.
If `C` is nonempty and `α` is a T1 space, this is equivalent to the closure of `C` being perfect.
See `preperfect_iff_perfect_closure`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
perfect (C : set α) : Prop | (closed : is_closed C)
(acc : preperfect C) | structure | perfect | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"is_closed",
"preperfect"
] | A set `C` is called perfect if it is closed and all of its
points are accumulation points of itself.
Note that we do not require `C` to be nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preperfect_iff_nhds : preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x | by simp only [preperfect, acc_pt_iff_nhds] | lemma | preperfect_iff_nhds | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"acc_pt_iff_nhds",
"preperfect"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preperfect.open_inter {U : set α} (hC : preperfect C) (hU : is_open U) :
preperfect (U ∩ C) | begin
rintros x ⟨xU, xC⟩,
apply (hC _ xC).nhds_inter,
exact hU.mem_nhds xU,
end | theorem | preperfect.open_inter | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"is_open",
"preperfect"
] | The intersection of a preperfect set and an open set is preperfect | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preperfect.perfect_closure (hC : preperfect C) : perfect (closure C) | begin
split, { exact is_closed_closure },
intros x hx,
by_cases h : x ∈ C; apply acc_pt.mono _ (principal_mono.mpr subset_closure),
{ exact hC _ h },
have : {x}ᶜ ∩ C = C := by simp [h],
rw [acc_pt, nhds_within, inf_assoc, inf_principal, this],
rw [closure_eq_cluster_pts] at hx,
exact hx,
end | theorem | preperfect.perfect_closure | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"acc_pt",
"acc_pt.mono",
"closure",
"closure_eq_cluster_pts",
"inf_assoc",
"is_closed_closure",
"nhds_within",
"perfect",
"preperfect",
"subset_closure"
] | The closure of a preperfect set is perfect.
For a converse, see `preperfect_iff_perfect_closure` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preperfect_iff_perfect_closure [t1_space α] :
preperfect C ↔ perfect (closure C) | begin
split; intro h, { exact h.perfect_closure },
intros x xC,
have H : acc_pt x (𝓟 (closure C)) := h.acc _ (subset_closure xC),
rw acc_pt_iff_frequently at *,
have : ∀ y , y ≠ x ∧ y ∈ closure C → ∃ᶠ z in 𝓝 y, z ≠ x ∧ z ∈ C,
{ rintros y ⟨hyx, yC⟩,
simp only [← mem_compl_singleton_iff, @and_comm _ (_ ... | theorem | preperfect_iff_perfect_closure | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"acc_pt",
"acc_pt_iff_frequently",
"closure",
"frequently_frequently_nhds",
"frequently_nhds_within_iff",
"mem_closure_iff_frequently",
"perfect",
"preperfect",
"subset_closure",
"t1_space"
] | In a T1 space, being preperfect is equivalent to having perfect closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
perfect.closure_nhds_inter {U : set α} (hC : perfect C) (x : α) (xC : x ∈ C) (xU : x ∈ U)
(Uop : is_open U) : perfect (closure (U ∩ C)) ∧ (closure (U ∩ C)).nonempty | begin
split,
{ apply preperfect.perfect_closure,
exact (hC.acc).open_inter Uop, },
apply nonempty.closure,
exact ⟨x, ⟨xU, xC⟩⟩,
end | theorem | perfect.closure_nhds_inter | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"closure",
"is_open",
"perfect",
"preperfect.perfect_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
perfect.splitting [t2_5_space α] (hC : perfect C) (hnonempty : C.nonempty) :
∃ C₀ C₁ : set α, (perfect C₀ ∧ C₀.nonempty ∧ C₀ ⊆ C) ∧
(perfect C₁ ∧ C₁.nonempty ∧ C₁ ⊆ C) ∧ disjoint C₀ C₁ | begin
cases hnonempty with y yC,
obtain ⟨x, xC, hxy⟩ : ∃ x ∈ C, x ≠ y,
{ have := hC.acc _ yC,
rw acc_pt_iff_nhds at this,
rcases this univ (univ_mem) with ⟨x, xC, hxy⟩,
exact ⟨x, xC.2, hxy⟩, },
obtain ⟨U, xU, Uop, V, yV, Vop, hUV⟩ := exists_open_nhds_disjoint_closure hxy,
use [closure (U ∩ C), clo... | lemma | perfect.splitting | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"acc_pt_iff_nhds",
"closure",
"closure_mono",
"disjoint",
"disjoint.mono",
"exists_open_nhds_disjoint_closure",
"perfect",
"t2_5_space"
] | Given a perfect nonempty set in a T2.5 space, we can find two disjoint perfect subsets
This is the main inductive step in the proof of the Cantor-Bendixson Theorem | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_countable_union_perfect_of_is_closed [second_countable_topology α]
(hclosed : is_closed C) :
∃ V D : set α, (V.countable) ∧ (perfect D) ∧ (C = V ∪ D) | begin
obtain ⟨b, bct, bnontrivial, bbasis⟩ := topological_space.exists_countable_basis α,
let v := {U ∈ b | (U ∩ C).countable},
let V := ⋃ U ∈ v, U,
let D := C \ V,
have Vct : (V ∩ C).countable,
{ simp only [Union_inter, mem_sep_iff],
apply countable.bUnion,
{ exact countable.mono (inter_subset_left... | theorem | exists_countable_union_perfect_of_is_closed | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"by_contradiction",
"countable",
"is_closed",
"is_open_bUnion",
"perfect",
"preperfect_iff_nhds",
"set.countable_singleton",
"topological_space.exists_countable_basis"
] | The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_perfect_nonempty_of_is_closed_of_not_countable [second_countable_topology α]
(hclosed : is_closed C) (hunc : ¬ C.countable) :
∃ D : set α, perfect D ∧ D.nonempty ∧ D ⊆ C | begin
rcases exists_countable_union_perfect_of_is_closed hclosed with ⟨V, D, Vct, Dperf, VD⟩,
refine ⟨D, ⟨Dperf, _⟩⟩,
split,
{ rw nonempty_iff_ne_empty,
by_contradiction,
rw [h, union_empty] at VD,
rw VD at hunc,
contradiction, },
rw VD,
exact subset_union_right _ _,
end | theorem | exists_perfect_nonempty_of_is_closed_of_not_countable | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"by_contradiction",
"exists_countable_union_perfect_of_is_closed",
"is_closed",
"perfect"
] | Any uncountable closed set in a second countable space contains a nonempty perfect subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
let D | closure (emetric.ball x (ε / 2) ∩ C) in
perfect D ∧ D.nonempty ∧ D ⊆ C ∧ emetric.diam D ≤ ε :=
begin
have : x ∈ (emetric.ball x (ε / 2)),
{ apply emetric.mem_ball_self,
rw ennreal.div_pos_iff,
exact ⟨ne_of_gt ε_pos, by norm_num⟩, },
have := hC.closure_nhds_inter x xC this emetric.is_open_ball,
refine ... | lemma | perfect.small_diam_aux | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"closure",
"emetric.ball",
"emetric.diam",
"emetric.diam_ball",
"emetric.diam_closure",
"emetric.diam_mono",
"emetric.is_open_ball",
"emetric.mem_ball_self",
"ennreal.div_mul_cancel",
"ennreal.div_pos_iff",
"is_closed.closure_subset_iff",
"mul_comm",
"perfect"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
perfect.small_diam_splitting (ε_pos : 0 < ε) : ∃ C₀ C₁ : set α,
(perfect C₀ ∧ C₀.nonempty ∧ C₀ ⊆ C ∧ emetric.diam C₀ ≤ ε) ∧
(perfect C₁ ∧ C₁.nonempty ∧ C₁ ⊆ C ∧ emetric.diam C₁ ≤ ε) ∧ disjoint C₀ C₁ | begin
rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩,
cases non0 with x₀ hx₀,
cases non1 with x₁ hx₁,
rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩,
rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩,
refine ⟨clos... | lemma | perfect.small_diam_splitting | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"closure",
"disjoint",
"disjoint.mono",
"emetric.ball",
"emetric.diam",
"perfect"
] | A refinement of `perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
perfect.exists_nat_bool_injection [complete_space α] :
∃ f : (ℕ → bool) → α, (range f) ⊆ C ∧ continuous f ∧ injective f | begin
obtain ⟨u, -, upos', hu⟩ := exists_seq_strict_anti_tendsto' (zero_lt_one' ℝ≥0∞),
have upos := λ n, (upos' n).1,
let P := subtype (λ E : set α, perfect E ∧ E.nonempty),
choose C0 C1 h0 h1 hdisj using λ {C : set α} (hC : perfect C) (hnonempty : C.nonempty)
{ε : ℝ≥0∞} (hε : 0 < ε), hC.small_diam_splittin... | theorem | perfect.exists_nat_bool_injection | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"cantor_scheme.disjoint",
"complete_space",
"continuity",
"continuous",
"exists_seq_strict_anti_tendsto'",
"ih",
"induced_map",
"nat.one_le_iff_ne_zero",
"perfect",
"pi_nat.res_length",
"subtype.val_inj",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'",
"zero_lt_one'"
] | Any nonempty perfect set in a complete metric space admits a continuous injection
from the cantor space, `ℕ → bool`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.exists_nat_bool_injection_of_not_countable {α : Type*}
[topological_space α] [polish_space α] {C : set α} (hC : is_closed C) (hunc : ¬ C.countable) :
∃ f : (ℕ → bool) → α, (range f) ⊆ C ∧ continuous f ∧ function.injective f | begin
letI := upgrade_polish_space α,
obtain ⟨D, hD, Dnonempty, hDC⟩ := exists_perfect_nonempty_of_is_closed_of_not_countable hC hunc,
obtain ⟨f, hfD, hf⟩ := hD.exists_nat_bool_injection Dnonempty,
exact ⟨f, hfD.trans hDC, hf⟩,
end | theorem | is_closed.exists_nat_bool_injection_of_not_countable | topology | src/topology/perfect.lean | [
"topology.metric_space.polish",
"topology.metric_space.cantor_scheme"
] | [
"continuous",
"exists_perfect_nonempty_of_is_closed_of_not_countable",
"is_closed",
"polish_space",
"topological_space",
"upgrade_polish_space"
] | Any closed uncountable subset of a Polish space admits a continuous injection
from the Cantor space `ℕ → bool`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_quasi_separated (s : set α) : Prop | ∀ (U V : set α), U ⊆ s → is_open U → is_compact U → V ⊆ s →
is_open V → is_compact V → is_compact (U ∩ V) | def | is_quasi_separated | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_compact",
"is_open"
] | A subset `s` of a topological space is quasi-separated if the intersections of any pairs of
compact open subsets of `s` are still compact.
Note that this is equivalent to `s` being a `quasi_separated_space` only when `s` is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_separated_space (α : Type*) [topological_space α] : Prop | (inter_is_compact : ∀ (U V : set α),
is_open U → is_compact U → is_open V → is_compact V → is_compact (U ∩ V)) | class | quasi_separated_space | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_compact",
"is_open",
"topological_space"
] | A topological space is quasi-separated if the intersections of any pairs of compact open
subsets are still compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_quasi_separated_univ_iff {α : Type*} [topological_space α] :
is_quasi_separated (set.univ : set α) ↔ quasi_separated_space α | begin
rw quasi_separated_space_iff,
simp [is_quasi_separated],
end | lemma | is_quasi_separated_univ_iff | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_quasi_separated",
"quasi_separated_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quasi_separated_univ {α : Type*} [topological_space α] [quasi_separated_space α] :
is_quasi_separated (set.univ : set α) | is_quasi_separated_univ_iff.mpr infer_instance | lemma | is_quasi_separated_univ | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_quasi_separated",
"quasi_separated_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quasi_separated.image_of_embedding {s : set α}
(H : is_quasi_separated s) (h : embedding f) : is_quasi_separated (f '' s) | begin
intros U V hU hU' hU'' hV hV' hV'',
convert (H (f ⁻¹' U) (f ⁻¹' V) _ (h.continuous.1 _ hU') _ _ (h.continuous.1 _ hV') _).image
h.continuous,
{ symmetry,
rw [← set.preimage_inter, set.image_preimage_eq_inter_range, set.inter_eq_left_iff_subset],
exact (set.inter_subset_left _ _).trans (hU.trans ... | lemma | is_quasi_separated.image_of_embedding | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"embedding",
"is_quasi_separated",
"set.image_preimage_eq_inter_range",
"set.image_subset_range",
"set.inter_eq_left_iff_subset",
"set.inter_subset_left",
"set.preimage_inter",
"set.subset_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.is_quasi_separated_iff (h : open_embedding f) {s : set α} :
is_quasi_separated s ↔ is_quasi_separated (f '' s) | begin
refine ⟨λ hs, hs.image_of_embedding h.to_embedding, _⟩,
intros H U V hU hU' hU'' hV hV' hV'',
rw [h.to_embedding.is_compact_iff_is_compact_image, set.image_inter h.inj],
exact H (f '' U) (f '' V)
(set.image_subset _ hU) (h.is_open_map _ hU') (hU''.image h.continuous)
(set.image_subset _ hV) (h.is_... | lemma | open_embedding.is_quasi_separated_iff | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_quasi_separated",
"open_embedding",
"set.image_inter",
"set.image_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quasi_separated_iff_quasi_separated_space (s : set α) (hs : is_open s) :
is_quasi_separated s ↔ quasi_separated_space s | begin
rw ← is_quasi_separated_univ_iff,
convert hs.open_embedding_subtype_coe.is_quasi_separated_iff.symm; simp
end | lemma | is_quasi_separated_iff_quasi_separated_space | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_open",
"is_quasi_separated",
"is_quasi_separated_univ_iff",
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quasi_separated.of_subset {s t : set α} (ht : is_quasi_separated t) (h : s ⊆ t) :
is_quasi_separated s | begin
intros U V hU hU' hU'' hV hV' hV'',
exact ht U V (hU.trans h) hU' hU'' (hV.trans h) hV' hV'',
end | lemma | is_quasi_separated.of_subset | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_quasi_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_space.to_quasi_separated_space [t2_space α] : quasi_separated_space α | ⟨λ U V hU hU' hV hV', hU'.inter hV'⟩ | instance | t2_space.to_quasi_separated_space | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"quasi_separated_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.to_quasi_separated_space [noetherian_space α] :
quasi_separated_space α | ⟨λ _ _ _ _ _ _, noetherian_space.is_compact _⟩ | instance | noetherian_space.to_quasi_separated_space | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quasi_separated.of_quasi_separated_space (s : set α) [quasi_separated_space α] :
is_quasi_separated s | is_quasi_separated_univ.of_subset (set.subset_univ _) | lemma | is_quasi_separated.of_quasi_separated_space | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_quasi_separated",
"quasi_separated_space",
"set.subset_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_space.of_open_embedding (h : open_embedding f) [quasi_separated_space β] :
quasi_separated_space α | is_quasi_separated_univ_iff.mp
(h.is_quasi_separated_iff.mpr $ is_quasi_separated.of_quasi_separated_space _) | lemma | quasi_separated_space.of_open_embedding | topology | src/topology/quasi_separated.lean | [
"topology.subset_properties",
"topology.separation",
"topology.noetherian_space"
] | [
"is_quasi_separated.of_quasi_separated_space",
"open_embedding",
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at (f : α → β) (s : set α) (x : α) | ∀ y < f x, ∀ᶠ x' in 𝓝[s] x, y < f x' | def | lower_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [] | A real function `f` is lower semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all
`x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general
preordered space, using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_on (f : α → β) (s : set α) | ∀ x ∈ s, lower_semicontinuous_within_at f s x | def | lower_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_within_at"
] | A real function `f` is lower semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`,
for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in
a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_at (f : α → β) (x : α) | ∀ y < f x, ∀ᶠ x' in 𝓝 x, y < f x' | def | lower_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [] | A real function `f` is lower semicontinuous at `x` if, for any `ε > 0`, for all `x'` close
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space,
using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous (f : α → β) | ∀ x, lower_semicontinuous_at f x | def | lower_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at"
] | A real function `f` is lower semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space,
using an arbitrary `y < f x` instead of `f x - ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_within_at (f : α → β) (s : set α) (x : α) | ∀ y, f x < y → ∀ᶠ x' in 𝓝[s] x, f x' < y | def | upper_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [] | A real function `f` is upper semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all
`x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general
preordered space, using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_on (f : α → β) (s : set α) | ∀ x ∈ s, upper_semicontinuous_within_at f s x | def | upper_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_within_at"
] | A real function `f` is upper semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`,
for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a
general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous_at (f : α → β) (x : α) | ∀ y, f x < y → ∀ᶠ x' in 𝓝 x, f x' < y | def | upper_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [] | A real function `f` is upper semicontinuous at `x` if, for any `ε > 0`, for all `x'` close
enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space,
using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_semicontinuous (f : α → β) | ∀ x, upper_semicontinuous_at f x | def | upper_semicontinuous | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"upper_semicontinuous_at"
] | A real function `f` is upper semicontinuous if, for any `ε > 0`, for any `x`, for all `x'`
close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered
space, using an arbitrary `y > f x` instead of `f x + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_semicontinuous_within_at.mono (h : lower_semicontinuous_within_at f s x)
(hst : t ⊆ s) : lower_semicontinuous_within_at f t x | λ y hy, filter.eventually.filter_mono (nhds_within_mono _ hst) (h y hy) | lemma | lower_semicontinuous_within_at.mono | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually.filter_mono",
"lower_semicontinuous_within_at",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at_univ_iff :
lower_semicontinuous_within_at f univ x ↔ lower_semicontinuous_at f x | by simp [lower_semicontinuous_within_at, lower_semicontinuous_at, nhds_within_univ] | lemma | lower_semicontinuous_within_at_univ_iff | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_at",
"lower_semicontinuous_within_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at.lower_semicontinuous_within_at
(s : set α) (h : lower_semicontinuous_at f x) : lower_semicontinuous_within_at f s x | λ y hy, filter.eventually.filter_mono nhds_within_le_nhds (h y hy) | lemma | lower_semicontinuous_at.lower_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually.filter_mono",
"lower_semicontinuous_at",
"lower_semicontinuous_within_at",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on.lower_semicontinuous_within_at
(h : lower_semicontinuous_on f s) (hx : x ∈ s) :
lower_semicontinuous_within_at f s x | h x hx | lemma | lower_semicontinuous_on.lower_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on.mono (h : lower_semicontinuous_on f s) (hst : t ⊆ s) :
lower_semicontinuous_on f t | λ x hx, (h x (hst hx)).mono hst | lemma | lower_semicontinuous_on.mono | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_univ_iff :
lower_semicontinuous_on f univ ↔ lower_semicontinuous f | by simp [lower_semicontinuous_on, lower_semicontinuous, lower_semicontinuous_within_at_univ_iff] | lemma | lower_semicontinuous_on_univ_iff | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_on",
"lower_semicontinuous_within_at_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous.lower_semicontinuous_at
(h : lower_semicontinuous f) (x : α) : lower_semicontinuous_at f x | h x | lemma | lower_semicontinuous.lower_semicontinuous_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous.lower_semicontinuous_within_at
(h : lower_semicontinuous f) (s : set α) (x : α) : lower_semicontinuous_within_at f s x | (h x).lower_semicontinuous_within_at s | lemma | lower_semicontinuous.lower_semicontinuous_within_at | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous.lower_semicontinuous_on
(h : lower_semicontinuous f) (s : set α) : lower_semicontinuous_on f s | λ x hx, h.lower_semicontinuous_within_at s x | lemma | lower_semicontinuous.lower_semicontinuous_on | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_within_at_const :
lower_semicontinuous_within_at (λ x, z) s x | λ y hy, filter.eventually_of_forall (λ x, hy) | lemma | lower_semicontinuous_within_at_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually_of_forall",
"lower_semicontinuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_at_const :
lower_semicontinuous_at (λ x, z) x | λ y hy, filter.eventually_of_forall (λ x, hy) | lemma | lower_semicontinuous_at_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually_of_forall",
"lower_semicontinuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_on_const :
lower_semicontinuous_on (λ x, z) s | λ x hx, lower_semicontinuous_within_at_const | lemma | lower_semicontinuous_on_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous_on",
"lower_semicontinuous_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_semicontinuous_const :
lower_semicontinuous (λ (x : α), z) | λ x, lower_semicontinuous_at_const | lemma | lower_semicontinuous_const | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"lower_semicontinuous",
"lower_semicontinuous_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.lower_semicontinuous_indicator (hs : is_open s) (hy : 0 ≤ y) :
lower_semicontinuous (indicator s (λ x, y)) | begin
assume x z hz,
by_cases h : x ∈ s; simp [h] at hz,
{ filter_upwards [hs.mem_nhds h],
simp [hz] { contextual := tt} },
{ apply filter.eventually_of_forall (λ x', _),
by_cases h' : x' ∈ s;
simp [h', hz.trans_le hy, hz] }
end | lemma | is_open.lower_semicontinuous_indicator | topology | src/topology/semicontinuous.lean | [
"algebra.indicator_function",
"topology.continuous_on",
"topology.instances.ennreal"
] | [
"filter.eventually_of_forall",
"is_open",
"lower_semicontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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