mathlib documentation

topology.path_connected

Path connectedness

Main definitions

In the file the unit interval [0, 1] in ℝ is denoted by I, and X is a topological space.

path (x y : X) is the type of paths from x to y, i.e., continuous maps from I to X mapping 0 to x and 1 to y.
path.map is the image of a path under a continuous map.
joined (x y : X) means there is a path between x and y.
joined.some_path (h : joined x y) selects some path between two points x and y.
path_component (x : X) is the set of points joined to x.
path_connected_space X is a predicate class asserting that X is non-empty and every two points of X are joined.

Then there are corresponding relative notions for F : set X.

joined_in F (x y : X) means there is a path γ joining x to y with values in F.
joined_in.some_path (h : joined_in F x y) selects a path from x to y inside F.
path_component_in F (x : X) is the set of points joined to x in F.
is_path_connected F asserts that F is non-empty and every two points of F are joined in F.
loc_path_connected_space X is a predicate class asserting that X is locally path-connected: each point has a basis of path-connected neighborhoods (we do not ask these to be open).

Main theorems

joined and joined_in F are transitive relations.

One can link the absolute and relative version in two directions, using (univ : set X) or the subtype ↥F.

path_connected_space_iff_univ : path_connected_space X ↔ is_path_connected (univ : set X)
is_path_connected_iff_path_connected_space : is_path_connected F ↔ path_connected_space ↥F

For locally path connected spaces, we have

path_connected_space_iff_connected_space : path_connected_space X ↔ connected_space X
is_connected_iff_is_path_connected (U_op : is_open U) : is_path_connected U ↔ is_connected U

Implementation notes

By default, all paths have I as their source and X as their target, but there is an operation I_extend that will extend any continuous map γ : I → X into a continuous map I_extend γ : ℝ → X that is constant before 0 and after 1.

This is used to define path.extend that turns γ : path x y into a continuous map γ.extend : ℝ → X whose restriction to I is the original γ, and is equal to x on (-∞, 0] and to y on [1, +∞).

The unit interval

theorem Icc_zero_one_symm {t : ℝ} :

t ∈ set.Icc 0 1 ↔ 1 - t ∈ set.Icc 0 1

@[instance]

def I_has_zero :

has_zero ↥(set.Icc 0 1)

Equations

I_has_zero = {zero := ⟨0, I_has_zero._proof_1⟩}

@[simp]

theorem coe_I_zero :

↑0 = 0

@[instance]

def I_has_one :

has_one ↥(set.Icc 0 1)

Equations

I_has_one = {one := ⟨1, I_has_one._proof_1⟩}

@[simp]

theorem coe_I_one :

↑1 = 1

def I_symm (a : ↥(set.Icc 0 1)) :

↥(set.Icc 0 1)

Unit interval central symmetry.

Equations

I_symm = λ (t : ↥(set.Icc 0 1)), ⟨1 - t.val, _⟩

@[simp]

theorem I_symm_zero :

@[simp]

theorem I_symm_one :

theorem continuous_I_symm :

continuous I_symm

def proj_I (a : ℝ) :

↥(set.Icc 0 1)

Projection of ℝ onto its unit interval.

Equations

proj_I = λ (t : ℝ), dite (t ≤ 0) (λ (h : t ≤ 0), ⟨0, proj_I._proof_1⟩) (λ (h : ¬t ≤ 0), dite (t ≤ 1) (λ (h' : t ≤ 1), ⟨t, _⟩) (λ (h' : ¬t ≤ 1), ⟨1, proj_I._proof_3⟩))

theorem proj_I_I {t : ℝ} (h : t ∈ set.Icc 0 1) :

proj_I t = ⟨t, h⟩

theorem proj_I_zero :

theorem proj_I_one :

theorem surjective_proj_I :

function.surjective proj_I

theorem range_proj_I :

set.range proj_I = set.univ

theorem continuous_proj_I :

continuous proj_I

def I_extend {β : Type u_1} (f : ↥(set.Icc 0 1) → β) (a : ℝ) :

β

Extension of a function defined on the unit interval to ℝ, by precomposing with the projection.

Equations

I_extend f = f ∘ proj_I

theorem continuous.I_extend {X : Type u_1} [topological_space X] {f : ↥(set.Icc 0 1) → X} (hf : continuous f) :

continuous (I_extend f)

theorem I_extend_extends {β : Type u_3} (f : ↥(set.Icc 0 1) → β) {t : ℝ} (ht : t ∈ set.Icc 0 1) :

I_extend f t = f ⟨t, ht⟩

@[simp]

theorem I_extend_zero {β : Type u_3} (f : ↥(set.Icc 0 1) → β) :

I_extend f 0 = f 0

@[simp]

theorem I_extend_one {β : Type u_3} (f : ↥(set.Icc 0 1) → β) :

I_extend f 1 = f 1

@[simp]

theorem I_extend_range {β : Type u_3} (f : ↥(set.Icc 0 1) → β) :

set.range (I_extend f) = set.range f

@[instance]

def set.Icc.connected_space :

connected_space ↥(set.Icc 0 1)

@[instance]

def set.Icc.compact_space :

compact_space ↥(set.Icc 0 1)

Paths

@[nolint]

structure path {X : Type u_1} [topological_space X] (x y : X) :

Type u_1

to_fun : ↥(set.Icc 0 1) → X
continuous' : continuous c.to_fun
source' : c.to_fun 0 = x
target' : c.to_fun 1 = y

Continuous path connecting two points x and y in a topological space

@[instance]

def path.has_coe_to_fun {X : Type u_1} [topological_space X] {x y : X} :

has_coe_to_fun (path x y)

Equations

path.has_coe_to_fun = {F := λ (x : path x y), ↥(set.Icc 0 1) → X, coe := path.to_fun y}

@[ext]

theorem path.ext {X : Type u_1} [topological_space X] {x y : X} {γ₁ γ₂ : path x y} (a : ⇑γ₁ = ⇑γ₂) :

γ₁ = γ₂

theorem path.continuous {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

continuous ⇑γ

@[simp]

theorem path.source {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

⇑γ 0 = x

@[simp]

theorem path.target {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

⇑γ 1 = y

@[instance]

def path.has_uncurry_path {X : Type u_1} {α : Type u_2} [topological_space X] {x y : α → X} :

function.has_uncurry (Π (a : α), path (x a) (y a)) (α × ↥(set.Icc 0 1)) X

Any function φ : Π (a : α), path (x a) (y a) can be seen as a function α × I → X.

Equations

path.has_uncurry_path = {uncurry := λ (φ : Π (a : α), path (x a) (y a)) (p : α × ↥(set.Icc 0 1)), ⇑(φ p.fst) p.snd}

def path.refl {X : Type u_1} [topological_space X] (x : X) :

path x x

The constant path from a point to itself

Equations

path.refl x = {to_fun := λ (t : ↥(set.Icc 0 1)), x, continuous' := _, source' := _, target' := _}

@[simp]

theorem path.refl_range {X : Type u_1} [topological_space X] {a : X} :

set.range ⇑(path.refl a) = {a}

def path.symm {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

path y x

The reverse of a path from x to y, as a path from y to x

Equations

γ.symm = {to_fun := ⇑γ ∘ I_symm, continuous' := _, source' := _, target' := _}

@[simp]

theorem path.refl_symm {X : Type u_1} [topological_space X] {a : X} :

(path.refl a).symm = path.refl a

@[simp]

theorem path.symm_range {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

set.range ⇑(γ.symm) = set.range ⇑γ

def path.extend {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) (a : ℝ) :

X

A continuous map extending a path to ℝ, constant before 0 and after 1.

Equations

γ.extend = I_extend ⇑γ

theorem path.continuous_extend {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

continuous γ.extend

@[simp]

theorem path.extend_zero {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

γ.extend 0 = x

@[simp]

theorem path.extend_one {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :

γ.extend 1 = y

@[simp]

theorem path.extend_extends {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t : ℝ} (ht : t ∈ set.Icc 0 1) :

γ.extend t = ⇑γ ⟨t, ht⟩

@[simp]

theorem path.extend_extends' {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) (t : ↥(set.Icc 0 1)) :

γ.extend ↑t = ⇑γ t

@[simp]

theorem path.extend_range {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

set.range γ.extend = set.range ⇑γ

theorem path.extend_le_zero {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t : ℝ} (ht : t ≤ 0) :

γ.extend t = a

theorem path.extend_one_le {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t : ℝ} (ht : 1 ≤ t) :

γ.extend t = b

@[simp]

theorem path.refl_extend {X : Type u_1} [topological_space X] {a : X} :

(path.refl a).extend = λ (_x : ℝ), a

def path.of_line {X : Type u_1} [topological_space X] {x y : X} {f : ℝ → X} (hf : continuous_on f (set.Icc 0 1)) (h₀ : f 0 = x) (h₁ : f 1 = y) :

path x y

The path obtained from a map defined on ℝ by restriction to the unit interval.

Equations

path.of_line hf h₀ h₁ = {to_fun := f ∘ coe, continuous' := _, source' := h₀, target' := h₁}

theorem path.of_line_mem {X : Type u_1} [topological_space X] {x y : X} {f : ℝ → X} (hf : continuous_on f (set.Icc 0 1)) (h₀ : f 0 = x) (h₁ : f 1 = y) (t : ↥(set.Icc 0 1)) :

⇑(path.of_line hf h₀ h₁) t ∈ f '' set.Icc 0 1

def path.trans {X : Type u_1} [topological_space X] {x y z : X} (γ : path x y) (γ' : path y z) :

path x z

Concatenation of two paths from x to y and from y to z, putting the first path on [0, 1/2] and the second one on [1/2, 1].

Equations

γ.trans γ' = {to_fun := (λ (t : ℝ), ite (t ≤ 1 / 2) (γ.extend (2 * t)) (γ'.extend (2 * t - 1))) ∘ coe, continuous' := _, source' := _, target' := _}

@[simp]

theorem path.refl_trans_refl {X : Type u_1} [topological_space X] {a : X} :

(path.refl a).trans (path.refl a) = path.refl a

theorem path.trans_range {X : Type u_1} [topological_space X] {a b c : X} (γ₁ : path a b) (γ₂ : path b c) :

set.range ⇑(γ₁.trans γ₂) = set.range ⇑γ₁ ∪ set.range ⇑γ₂

def path.map {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) {Y : Type u_2} [topological_space Y] {f : X → Y} (h : continuous f) :

path (f x) (f y)

Image of a path from x to y by a continuous map

Equations

γ.map h = {to_fun := f ∘ ⇑γ, continuous' := _, source' := _, target' := _}

@[simp]

theorem path.map_coe {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) {Y : Type u_2} [topological_space Y] {f : X → Y} (h : continuous f) :

⇑(γ.map h) = f ∘ ⇑γ

def path.cast {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) {x' y' : X} (hx : x' = x) (hy : y' = y) :

path x' y'

Casting a path from x to y to a path from x' to y' when x' = x and y' = y

Equations

γ.cast hx hy = {to_fun := ⇑γ, continuous' := _, source' := _, target' := _}

@[simp]

theorem path.symm_cast {X : Type u_1} [topological_space X] {a₁ a₂ b₁ b₂ : X} (γ : path a₂ b₂) (ha : a₁ = a₂) (hb : b₁ = b₂) :

(γ.cast ha hb).symm = γ.symm.cast hb ha

@[simp]

theorem path.trans_cast {X : Type u_1} [topological_space X] {a₁ a₂ b₁ b₂ c₁ c₂ : X} (γ : path a₂ b₂) (γ' : path b₂ c₂) (ha : a₁ = a₂) (hb : b₁ = b₂) (hc : c₁ = c₂) :

(γ.cast ha hb).trans (γ'.cast hb hc) = (γ.trans γ').cast ha hc

@[simp]

theorem path.cast_coe {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) {x' y' : X} (hx : x' = x) (hy : y' = y) :

⇑(γ.cast hx hy) = ⇑γ

theorem path.symm_continuous_family {X : Type u_1} {ι : Type u_2} [topological_space X] [topological_space ι] {a b : ι → X} (γ : Π (t : ι), path (a t) (b t)) (h : continuous ↿γ) :

continuous ↿(λ (t : ι), (γ t).symm)

theorem path.continuous_uncurry_extend_of_continuous_family {X : Type u_1} {ι : Type u_2} [topological_space X] [topological_space ι] {a b : ι → X} (γ : Π (t : ι), path (a t) (b t)) (h : continuous ↿γ) :

continuous ↿(λ (t : ι), (γ t).extend)

theorem path.trans_continuous_family {X : Type u_1} {ι : Type u_2} [topological_space X] [topological_space ι] {a b c : ι → X} (γ₁ : Π (t : ι), path (a t) (b t)) (h₁ : continuous ↿γ₁) (γ₂ : Π (t : ι), path (b t) (c t)) (h₂ : continuous ↿γ₂) :

continuous ↿(λ (t : ι), (γ₁ t).trans (γ₂ t))

Truncating a path

def path.truncate {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) (t₀ t₁ : ℝ) :

path (γ.extend (min t₀ t₁)) (γ.extend t₁)

γ.truncate t₀ t₁ is the path which follows the path γ on the time interval [t₀, t₁] and stays still otherwise.

Equations

γ.truncate t₀ t₁ = {to_fun := λ (s : ↥(set.Icc 0 1)), γ.extend (min (max ↑s t₀) t₁), continuous' := _, source' := _, target' := _}

def path.truncate_of_le {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t₀ t₁ : ℝ} (h : t₀ ≤ t₁) :

path (γ.extend t₀) (γ.extend t₁)

γ.truncate_of_le t₀ t₁ h, where h : t₀ ≤ t₁ is γ.truncate t₀ t₁ casted as a path from γ.extend t₀ to γ.extend t₁.

Equations

γ.truncate_of_le h = (γ.truncate t₀ t₁).cast _ _

theorem path.truncate_range {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t₀ t₁ : ℝ} :

set.range ⇑(γ.truncate t₀ t₁) ⊆ set.range ⇑γ

theorem path.truncate_continuous_family {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

continuous (λ (x : ℝ × ℝ × ↥(set.Icc 0 1)), ⇑(γ.truncate x.fst x.snd.fst) x.snd.snd)

For a path γ, γ.truncate gives a "continuous family of paths", by which we mean the uncurried function which maps (t₀, t₁, s) to γ.truncate t₀ t₁ s is continuous.

theorem path.truncate_const_continuous_family {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) (t : ℝ) :

continuous ↿(γ.truncate t)

@[simp]

theorem path.truncate_self {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) (t : ℝ) :

γ.truncate t t = (path.refl (γ.extend t)).cast _ rfl

@[simp]

theorem path.truncate_zero_zero {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

γ.truncate 0 0 = (path.refl a).cast _ _

@[simp]

theorem path.truncate_one_one {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

γ.truncate 1 1 = (path.refl b).cast _ _

@[simp]

theorem path.truncate_zero_one {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

γ.truncate 0 1 = γ.cast _ _

Being joined by a path

def joined {X : Type u_1} [topological_space X] (x y : X) :

Prop

The relation "being joined by a path". This is an equivalence relation.

Equations

joined x y = nonempty (path x y)

theorem joined.refl {X : Type u_1} [topological_space X] (x : X) :

def joined.some_path {X : Type u_1} [topological_space X] {x y : X} (h : joined x y) :

path x y

When two points are joined, choose some path from x to y.

Equations

h.some_path = nonempty.some h

theorem joined.symm {X : Type u_1} [topological_space X] {x y : X} (h : joined x y) :

theorem joined.trans {X : Type u_1} [topological_space X] {x y z : X} (hxy : joined x y) (hyz : joined y z) :

def path_setoid (X : Type u_1) [topological_space X] :

The setoid corresponding the equivalence relation of being joined by a continuous path.

Equations

path_setoid X = {r := joined _inst_1, iseqv := _}

def zeroth_homotopy (X : Type u_1) [topological_space X] :

Type u_1

The quotient type of points of a topological space modulo being joined by a continuous path.

Equations

zeroth_homotopy X = quotient (path_setoid X)

@[instance]

def zeroth_homotopy.inhabited :

inhabited (zeroth_homotopy ℝ)

Equations

zeroth_homotopy.inhabited = {default := ⟦0⟧}

Being joined by a path inside a set

def joined_in {X : Type u_1} [topological_space X] (F : set X) (x y : X) :

Prop

The relation "being joined by a path in F". Not quite an equivalence relation since it's not reflexive for points that do not belong to F.

Equations

joined_in F x y = ∃ (γ : path x y), ∀ (t : ↥(set.Icc 0 1)), ⇑γ t ∈ F

theorem joined_in.mem {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

x ∈ F ∧ y ∈ F

theorem joined_in.source_mem {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

x ∈ F

theorem joined_in.target_mem {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

y ∈ F

def joined_in.some_path {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

path x y

When x and y are joined in F, choose a path from x to y inside F

Equations

h.some_path = classical.some h

theorem joined_in.some_path_mem {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) (t : ↥(set.Icc 0 1)) :

⇑(h.some_path) t ∈ F

theorem joined_in.joined_subtype {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

joined ⟨x, _⟩ ⟨y, _⟩

If x and y are joined in the set F, then they are joined in the subtype F.

theorem joined_in.of_line {X : Type u_1} [topological_space X] {x y : X} {F : set X} {f : ℝ → X} (hf : continuous_on f (set.Icc 0 1)) (h₀ : f 0 = x) (h₁ : f 1 = y) (hF : f '' set.Icc 0 1 ⊆ F) :

joined_in F x y

theorem joined_in.joined {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

theorem joined_in_iff_joined {X : Type u_1} [topological_space X] {x y : X} {F : set X} (x_in : x ∈ F) (y_in : y ∈ F) :

joined_in F x y ↔ joined ⟨x, x_in⟩ ⟨y, y_in⟩

@[simp]

theorem joined_in_univ {X : Type u_1} [topological_space X] {x y : X} :

joined_in set.univ x y ↔ joined x y

theorem joined_in.mono {X : Type u_1} [topological_space X] {x y : X} {U V : set X} (h : joined_in U x y) (hUV : U ⊆ V) :

joined_in V x y

theorem joined_in.refl {X : Type u_1} [topological_space X] {x : X} {F : set X} (h : x ∈ F) :

joined_in F x x

theorem joined_in.symm {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : joined_in F x y) :

joined_in F y x

theorem joined_in.trans {X : Type u_1} [topological_space X] {x y z : X} {F : set X} (hxy : joined_in F x y) (hyz : joined_in F y z) :

joined_in F x z

Path component

def path_component {X : Type u_1} [topological_space X] (x : X) :

The path component of x is the set of points that can be joined to x.

Equations

path_component x = {y : X | joined x y}

@[simp]

theorem mem_path_component_self {X : Type u_1} [topological_space X] (x : X) :

x ∈ path_component x

@[simp]

theorem path_component.nonempty {X : Type u_1} [topological_space X] (x : X) :

(path_component x).nonempty

theorem mem_path_component_of_mem {X : Type u_1} [topological_space X] {x y : X} (h : x ∈ path_component y) :

y ∈ path_component x

theorem path_component_symm {X : Type u_1} [topological_space X] {x y : X} :

x ∈ path_component y ↔ y ∈ path_component x

theorem path_component_congr {X : Type u_1} [topological_space X] {x y : X} (h : x ∈ path_component y) :

path_component x = path_component y

theorem path_component_subset_component {X : Type u_1} [topological_space X] (x : X) :

path_component x ⊆ connected_component x

def path_component_in {X : Type u_1} [topological_space X] (x : X) (F : set X) :

The path component of x in F is the set of points that can be joined to x in F.

Equations

path_component_in x F = {y : X | joined_in F x y}

@[simp]

theorem path_component_in_univ {X : Type u_1} [topological_space X] (x : X) :

path_component_in x set.univ = path_component x

theorem joined.mem_path_component {X : Type u_1} [topological_space X] {x y z : X} (hyz : joined y z) (hxy : y ∈ path_component x) :

z ∈ path_component x

Path connected sets

def is_path_connected {X : Type u_1} [topological_space X] (F : set X) :

Prop

A set F is path connected if it contains a point that can be joined to all other in F.

Equations

is_path_connected F = ∃ (x : X) (H : x ∈ F), ∀ {y : X}, y ∈ F → joined_in F x y

theorem is_path_connected_iff_eq {X : Type u_1} [topological_space X] {F : set X} :

is_path_connected F ↔ ∃ (x : X) (H : x ∈ F), path_component_in x F = F

theorem is_path_connected.joined_in {X : Type u_1} [topological_space X] {F : set X} (h : is_path_connected F) (x y : X) (H : x ∈ F) (H_1 : y ∈ F) :

joined_in F x y

theorem is_path_connected_iff {X : Type u_1} [topological_space X] {F : set X} :

is_path_connected F ↔ F.nonempty ∧ ∀ (x y : X), x ∈ F → y ∈ F → joined_in F x y

theorem is_path_connected.image {X : Type u_1} [topological_space X] {F : set X} {Y : Type u_2} [topological_space Y] (hF : is_path_connected F) {f : X → Y} (hf : continuous f) :

is_path_connected (f '' F)

theorem is_path_connected.mem_path_component {X : Type u_1} [topological_space X] {x y : X} {F : set X} (h : is_path_connected F) (x_in : x ∈ F) (y_in : y ∈ F) :

y ∈ path_component x

theorem is_path_connected.subset_path_component {X : Type u_1} [topological_space X] {x : X} {F : set X} (h : is_path_connected F) (x_in : x ∈ F) :

F ⊆ path_component x

theorem is_path_connected.union {X : Type u_1} [topological_space X] {U V : set X} (hU : is_path_connected U) (hV : is_path_connected V) (hUV : (U ∩ V).nonempty) :

is_path_connected (U ∪ V)

theorem is_path_connected.preimage_coe {X : Type u_1} [topological_space X] {U W : set X} (hW : is_path_connected W) (hWU : W ⊆ U) :

is_path_connected (coe ⁻¹' W)

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).

theorem is_path_connected.exists_path_through_family {X : Type u_1} [topological_space X] {n : ℕ} {s : set X} (h : is_path_connected s) (p : fin (n + 1) → X) (hp : ∀ (i : fin (n + 1)), p i ∈ s) :

∃ (γ : path (p 0) (p ↑n)), set.range ⇑γ ⊆ s ∧ ∀ (i : fin (n + 1)), p i ∈ set.range ⇑γ

theorem is_path_connected.exists_path_through_family' {X : Type u_1} [topological_space X] {n : ℕ} {s : set X} (h : is_path_connected s) (p : fin (n + 1) → X) (hp : ∀ (i : fin (n + 1)), p i ∈ s) :

∃ (γ : path (p 0) (p ↑n)) (t : fin (n + 1) → ↥(set.Icc 0 1)), (∀ (t : ↥(set.Icc 0 1)), ⇑γ t ∈ s) ∧ ∀ (i : fin (n + 1)), ⇑γ (t i) = p i

Path connected spaces

@[class]

structure path_connected_space (X : Type u_4) [topological_space X] :

Prop

nonempty : nonempty X
joined : ∀ (x y : X), joined x y

A topological space is path-connected if it is non-empty and every two points can be joined by a continuous path.

Instances

normed_space.path_connected

theorem path_connected_space_iff_zeroth_homotopy {X : Type u_1} [topological_space X] :

path_connected_space X ↔ nonempty (zeroth_homotopy X) ∧ subsingleton (zeroth_homotopy X)

def path_connected_space.some_path {X : Type u_1} [topological_space X] [path_connected_space X] (x y : X) :

path x y

Use path-connectedness to build a path between two points.

Equations

path_connected_space.some_path x y = nonempty.some _

theorem is_path_connected_iff_path_connected_space {X : Type u_1} [topological_space X] {F : set X} :

is_path_connected F ↔ path_connected_space ↥F

theorem path_connected_space_iff_univ {X : Type u_1} [topological_space X] :

path_connected_space X ↔ is_path_connected set.univ

theorem path_connected_space_iff_eq {X : Type u_1} [topological_space X] :

path_connected_space X ↔ ∃ (x : X), path_component x = set.univ

@[instance]

def path_connected_space.connected_space {X : Type u_1} [topological_space X] [path_connected_space X] :

connected_space X

theorem path_connected_space.exists_path_through_family {X : Type u_1} [topological_space X] [path_connected_space X] {n : ℕ} (p : fin (n + 1) → X) :

∃ (γ : path (p 0) (p ↑n)), ∀ (i : fin (n + 1)), p i ∈ set.range ⇑γ

theorem path_connected_space.exists_path_through_family' {X : Type u_1} [topological_space X] [path_connected_space X] {n : ℕ} (p : fin (n + 1) → X) :

∃ (γ : path (p 0) (p ↑n)) (t : fin (n + 1) → ↥(set.Icc 0 1)), ∀ (i : fin (n + 1)), ⇑γ (t i) = p i

Locally path connected spaces

@[class]

structure loc_path_connected_space (X : Type u_4) [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

A topological space is locally path connected, at every point, path connected neighborhoods form a neighborhood basis.

Instances

normed_space.loc_path_connected

theorem loc_path_connected_of_bases {X : Type u_1} [topological_space X] {ι : Type u_2} {p : ι → Prop} {s : X → ι → set X} (h : ∀ (x : X), (𝓝 x).has_basis p (s x)) (h' : ∀ (x : X) (i : ι), p i → is_path_connected (s x i)) :

loc_path_connected_space X

theorem path_connected_space_iff_connected_space {X : Type u_1} [topological_space X] [loc_path_connected_space X] :

path_connected_space X ↔ connected_space X

theorem path_connected_subset_basis {X : Type u_1} [topological_space X] {x : X} [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

theorem loc_path_connected_of_is_open {X : Type u_1} [topological_space X] [loc_path_connected_space X] {U : set X} (h : is_open U) :

loc_path_connected_space ↥U

theorem is_open.is_connected_iff_is_path_connected {X : Type u_1} [topological_space X] [loc_path_connected_space X] {U : set X} (U_op : is_open U) :

is_path_connected U ↔ is_connected U