mathlib documentation

topology.path_connected

Path connectedness #

Main definitions #

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

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

Main theorems #

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

For locally path connected spaces, we have

Implementation notes #

By default, all paths have I as their source and X as their target, but there is an operation set.Icc_extend that will extend any continuous map γ : I → X into a continuous map Icc_extend zero_le_one 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, +∞).

Paths #

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

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} :
Equations
@[ext]
theorem path.ext {X : Type u_1} [topological_space X] {x y : X} {γ₁ γ₂ : path x y} :
γ₁ = γ₂γ₁ = γ₂
@[simp]
theorem path.coe_mk {X : Type u_1} [topological_space X] {x y : X} (f : unit_interval → X) (h₁ : continuous f) (h₂ : f 0 = x) (h₃ : f 1 = y) :
{to_fun := f, continuous' := h₁, source' := h₂, target' := h₃} = f
theorem path.continuous {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :
@[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)) × unit_interval) X

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

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

The constant path from a point to itself

Equations
@[simp]
theorem path.refl_range {X : Type u_1} [topological_space X] {a : X} :
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
@[simp]
theorem path.refl_symm {X : Type u_1} [topological_space X] {a : X} :
@[simp]
theorem path.symm_range {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :
def path.extend {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :
→ X

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

Equations
theorem path.continuous_extend {X : Type u_1} [topological_space X] {x y : X} (γ : path x y) :
@[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 : } (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) :
theorem path.extend_of_le_zero {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t : } (ht : t 0) :
γ.extend t = a
theorem path.extend_of_one_le {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t : } (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 : ), a
def path.of_line {X : Type u_1} [topological_space X] {x y : X} {f : → X} (hf : continuous_on f unit_interval) (h₀ : f 0 = x) (h₁ : f 1 = y) :
path x y

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

Equations
theorem path.of_line_mem {X : Type u_1} [topological_space X] {x y : X} {f : → X} (hf : continuous_on f unit_interval) (h₀ : f 0 = x) (h₁ : f 1 = y) (t : unit_interval) :
(path.of_line hf h₀ h₁) t f '' unit_interval
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
@[simp]
theorem path.refl_trans_refl {X : Type u_1} [topological_space X] {a : X} :
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
@[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
@[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 : ) :
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
def path.truncate_of_le {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t₀ t₁ : } (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
theorem path.truncate_range {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) {t₀ t₁ : } :
theorem path.truncate_continuous_family {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) :

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 : ) :
@[simp]
theorem path.truncate_self {X : Type u_1} [topological_space X] {a b : X} (γ : path a b) (t : ) :
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
theorem joined.refl {X : Type u_1} [topological_space X] (x : X) :
joined 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
theorem joined.symm {X : Type u_1} [topological_space X] {x y : X} (h : joined x y) :
joined y x
theorem joined.trans {X : Type u_1} [topological_space X] {x y z : X} (hxy : joined x y) (hyz : joined y z) :
joined x 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
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

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
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
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 : unit_interval) :
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 : → X} (hf : continuous_on f unit_interval) (h₀ : f 0 = x) (h₁ : f 1 = y) (hF : f '' unit_interval 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) :
joined 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} :
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) :
set X

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

Equations
@[simp]
theorem mem_path_component_self {X : Type u_1} [topological_space X] (x : X) :
@[simp]
theorem path_component.nonempty {X : Type u_1} [topological_space X] (x : X) :
theorem mem_path_component_of_mem {X : Type u_1} [topological_space X] {x y : X} (h : x path_component y) :
theorem path_component_symm {X : Type u_1} [topological_space X] {x y : X} :
theorem path_component_congr {X : Type u_1} [topological_space X] {x y : X} (h : x path_component y) :
def path_component_in {X : Type u_1} [topological_space X] (x : X) (F : set X) :
set X

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

Equations
@[simp]
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) :

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
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 Fy Fjoined_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) :
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) :
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) :
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) :
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) :

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 : } {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 : } {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)unit_interval), (∀ (t : unit_interval), γ t s) ∀ (i : fin (n + 1)), γ (t i) = p i

Path connected spaces #

@[class]
structure path_connected_space (X : Type u_3) [topological_space X] :
Prop

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

Instances
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
theorem path_connected_space.exists_path_through_family {X : Type u_1} [topological_space X] [path_connected_space X] {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 : } (p : fin (n + 1) → X) :
∃ (γ : path (p 0) (p n)) (t : fin (n + 1)unit_interval), ∀ (i : fin (n + 1)), γ (t i) = p i

Locally path connected spaces #

@[class]
structure loc_path_connected_space (X : Type u_3) [topological_space X] :
Prop

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

Instances
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 iis_path_connected (s x i)) :
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