mathlib documentation

topology.connected

Connected subsets of topological spaces #

In this file we define connected subsets of a topological spaces and various other properties and classes related to connectivity.

Main definitions #

We define the following properties for sets in a topological space:

For each of these definitions, we also have a class stating that the whole space satisfies that property: connected_space, totally_disconnected_space, totally_separated_space.

On the definition of connected sets/spaces #

In informal mathematics, connected spaces are assumed to be nonempty. We formalise the predicate without that assumption as is_preconnected. In other words, the only difference is whether the empty space counts as connected. There are good reasons to consider the empty space to be “too simple to be simple” See also https://ncatlab.org/nlab/show/too+simple+to+be+simple, and in particular https://ncatlab.org/nlab/show/too+simple+to+be+simple#relationship_to_biased_definitions.

def is_preconnected {α : Type u} [topological_space α] (s : set α) :
Prop

A preconnected set is one where there is no non-trivial open partition.

Equations
def is_connected {α : Type u} [topological_space α] (s : set α) :
Prop

A connected set is one that is nonempty and where there is no non-trivial open partition.

Equations
theorem is_connected.nonempty {α : Type u} [topological_space α] {s : set α} (h : is_connected s) :
theorem is_connected_singleton {α : Type u} [topological_space α] {x : α} :
theorem is_preconnected_of_forall {α : Type u} [topological_space α] {s : set α} (x : α) (H : (y : α), y s ( (t : set α) (H : t s), x t y t is_preconnected t)) :

If any point of a set is joined to a fixed point by a preconnected subset, then the original set is preconnected as well.

theorem is_preconnected_of_forall_pair {α : Type u} [topological_space α] {s : set α} (H : (x : α), x s (y : α), y s ( (t : set α) (H : t s), x t y t is_preconnected t)) :

If any two points of a set are contained in a preconnected subset, then the original set is preconnected as well.

theorem is_preconnected_sUnion {α : Type u} [topological_space α] (x : α) (c : set (set α)) (H1 : (s : set α), s c x s) (H2 : (s : set α), s c is_preconnected s) :

A union of a family of preconnected sets with a common point is preconnected as well.

theorem is_preconnected_Union {α : Type u} [topological_space α] {ι : Sort u_1} {s : ι set α} (h₁ : ( (i : ι), s i).nonempty) (h₂ : (i : ι), is_preconnected (s i)) :
is_preconnected ( (i : ι), s i)
theorem is_preconnected.union {α : Type u} [topological_space α] (x : α) {s t : set α} (H1 : x s) (H2 : x t) (H3 : is_preconnected s) (H4 : is_preconnected t) :
theorem is_preconnected.union' {α : Type u} [topological_space α] {s t : set α} (H : (s t).nonempty) (hs : is_preconnected s) (ht : is_preconnected t) :
theorem is_connected.union {α : Type u} [topological_space α] {s t : set α} (H : (s t).nonempty) (Hs : is_connected s) (Ht : is_connected t) :

The directed sUnion of a set S of preconnected subsets is preconnected.

theorem is_preconnected.bUnion_of_refl_trans_gen {α : Type u} [topological_space α] {ι : Type u_1} {t : set ι} {s : ι set α} (H : (i : ι), i t is_preconnected (s i)) (K : (i : ι), i t (j : ι), j t relation.refl_trans_gen (λ (i j : ι), (s i s j).nonempty i t) i j) :
is_preconnected ( (n : ι) (H : n t), s n)

The bUnion of a family of preconnected sets is preconnected if the graph determined by whether two sets intersect is preconnected.

theorem is_connected.bUnion_of_refl_trans_gen {α : Type u} [topological_space α] {ι : Type u_1} {t : set ι} {s : ι set α} (ht : t.nonempty) (H : (i : ι), i t is_connected (s i)) (K : (i : ι), i t (j : ι), j t relation.refl_trans_gen (λ (i j : ι), (s i s j).nonempty i t) i j) :
is_connected ( (n : ι) (H : n t), s n)

The bUnion of a family of preconnected sets is preconnected if the graph determined by whether two sets intersect is preconnected.

theorem is_preconnected.Union_of_refl_trans_gen {α : Type u} [topological_space α] {ι : Type u_1} {s : ι set α} (H : (i : ι), is_preconnected (s i)) (K : (i j : ι), relation.refl_trans_gen (λ (i j : ι), (s i s j).nonempty) i j) :
is_preconnected ( (n : ι), s n)

Preconnectedness of the Union of a family of preconnected sets indexed by the vertices of a preconnected graph, where two vertices are joined when the corresponding sets intersect.

theorem is_connected.Union_of_refl_trans_gen {α : Type u} [topological_space α] {ι : Type u_1} [nonempty ι] {s : ι set α} (H : (i : ι), is_connected (s i)) (K : (i j : ι), relation.refl_trans_gen (λ (i j : ι), (s i s j).nonempty) i j) :
is_connected ( (n : ι), s n)
theorem is_preconnected.Union_of_chain {α : Type u} {β : Type v} [topological_space α] [linear_order β] [succ_order β] [is_succ_archimedean β] {s : β set α} (H : (n : β), is_preconnected (s n)) (K : (n : β), (s n s (order.succ n)).nonempty) :
is_preconnected ( (n : β), s n)

The Union of connected sets indexed by a type with an archimedean successor (like or ) such that any two neighboring sets meet is preconnected.

theorem is_connected.Union_of_chain {α : Type u} {β : Type v} [topological_space α] [linear_order β] [succ_order β] [is_succ_archimedean β] [nonempty β] {s : β set α} (H : (n : β), is_connected (s n)) (K : (n : β), (s n s (order.succ n)).nonempty) :
is_connected ( (n : β), s n)

The Union of connected sets indexed by a type with an archimedean successor (like or ) such that any two neighboring sets meet is connected.

theorem is_preconnected.bUnion_of_chain {α : Type u} {β : Type v} [topological_space α] [linear_order β] [succ_order β] [is_succ_archimedean β] {s : β set α} {t : set β} (ht : t.ord_connected) (H : (n : β), n t is_preconnected (s n)) (K : (n : β), n t order.succ n t (s n s (order.succ n)).nonempty) :
is_preconnected ( (n : β) (H : n t), s n)

The Union of preconnected sets indexed by a subset of a type with an archimedean successor (like or ) such that any two neighboring sets meet is preconnected.

theorem is_connected.bUnion_of_chain {α : Type u} {β : Type v} [topological_space α] [linear_order β] [succ_order β] [is_succ_archimedean β] {s : β set α} {t : set β} (hnt : t.nonempty) (ht : t.ord_connected) (H : (n : β), n t is_connected (s n)) (K : (n : β), n t order.succ n t (s n s (order.succ n)).nonempty) :
is_connected ( (n : β) (H : n t), s n)

The Union of connected sets indexed by a subset of a type with an archimedean successor (like or ) such that any two neighboring sets meet is preconnected.

theorem is_preconnected.subset_closure {α : Type u} [topological_space α] {s t : set α} (H : is_preconnected s) (Kst : s t) (Ktcs : t closure s) :

Theorem of bark and tree : if a set is within a (pre)connected set and its closure, then it is (pre)connected as well.

theorem is_connected.subset_closure {α : Type u} [topological_space α] {s t : set α} (H : is_connected s) (Kst : s t) (Ktcs : t closure s) :

The closure of a (pre)connected set is (pre)connected as well.

theorem is_connected.closure {α : Type u} [topological_space α] {s : set α} (H : is_connected s) :
theorem is_preconnected.image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} (H : is_preconnected s) (f : α β) (hf : continuous_on f s) :

The image of a (pre)connected set is (pre)connected as well.

theorem is_connected.image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} (H : is_connected s) (f : α β) (hf : continuous_on f s) :
theorem is_preconnected_closed_iff {α : Type u} [topological_space α] {s : set α} :
is_preconnected s (t t' : set α), is_closed t is_closed t' s t t' (s t).nonempty (s t').nonempty (s (t t')).nonempty
theorem inducing.is_preconnected_image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {f : α β} (hf : inducing f) :
theorem is_preconnected.preimage_of_open_map {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set β} (hs : is_preconnected s) {f : α β} (hinj : function.injective f) (hf : is_open_map f) (hsf : s set.range f) :
theorem is_preconnected.preimage_of_closed_map {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set β} (hs : is_preconnected s) {f : α β} (hinj : function.injective f) (hf : is_closed_map f) (hsf : s set.range f) :
theorem is_connected.preimage_of_open_map {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set β} (hs : is_connected s) {f : α β} (hinj : function.injective f) (hf : is_open_map f) (hsf : s set.range f) :
theorem is_connected.preimage_of_closed_map {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set β} (hs : is_connected s) {f : α β} (hinj : function.injective f) (hf : is_closed_map f) (hsf : s set.range f) :
theorem is_preconnected.subset_or_subset {α : Type u} [topological_space α] {s u v : set α} (hu : is_open u) (hv : is_open v) (huv : disjoint u v) (hsuv : s u v) (hs : is_preconnected s) :
s u s v
theorem is_preconnected.subset_left_of_subset_union {α : Type u} [topological_space α] {s u v : set α} (hu : is_open u) (hv : is_open v) (huv : disjoint u v) (hsuv : s u v) (hsu : (s u).nonempty) (hs : is_preconnected s) :
s u
theorem is_preconnected.subset_right_of_subset_union {α : Type u} [topological_space α] {s u v : set α} (hu : is_open u) (hv : is_open v) (huv : disjoint u v) (hsuv : s u v) (hsv : (s v).nonempty) (hs : is_preconnected s) :
s v
theorem is_preconnected.subset_of_closure_inter_subset {α : Type u} [topological_space α] {s u : set α} (hs : is_preconnected s) (hu : is_open u) (h'u : (s u).nonempty) (h : closure u s u) :
s u

If a preconnected set s intersects an open set u, and limit points of u inside s are contained in u, then the whole set s is contained in u.

theorem is_preconnected.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} (hs : is_preconnected s) (ht : is_preconnected t) :
theorem is_connected.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} (hs : is_connected s) (ht : is_connected t) :
theorem is_preconnected_univ_pi {ι : Type u_1} {π : ι Type u_2} [Π (i : ι), topological_space (π i)] {s : Π (i : ι), set (π i)} (hs : (i : ι), is_preconnected (s i)) :
@[simp]
theorem is_connected_univ_pi {ι : Type u_1} {π : ι Type u_2} [Π (i : ι), topological_space (π i)] {s : Π (i : ι), set (π i)} :
theorem sigma.is_connected_iff {ι : Type u_1} {π : ι Type u_2} [Π (i : ι), topological_space (π i)] {s : set (Σ (i : ι), π i)} :
is_connected s (i : ι) (t : set (π i)), is_connected t s = sigma.mk i '' t
theorem sigma.is_preconnected_iff {ι : Type u_1} {π : ι Type u_2} [hι : nonempty ι] [Π (i : ι), topological_space (π i)] {s : set (Σ (i : ι), π i)} :
is_preconnected s (i : ι) (t : set (π i)), is_preconnected t s = sigma.mk i '' t
theorem sum.is_connected_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set β)} :
theorem sum.is_preconnected_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set β)} :
def connected_component {α : Type u} [topological_space α] (x : α) :
set α

The connected component of a point is the maximal connected set that contains this point.

Equations
def connected_component_in {α : Type u} [topological_space α] (F : set α) (x : α) :
set α

Given a set F in a topological space α and a point x : α, the connected component of x in F is the connected component of x in the subtype F seen as a set in α. This definition does not make sense if x is not in F so we return the empty set in this case.

Equations
theorem connected_component_in_eq_image {α : Type u} [topological_space α] {F : set α} {x : α} (h : x F) :
theorem connected_component_in_eq_empty {α : Type u} [topological_space α] {F : set α} {x : α} (h : x F) :
theorem mem_connected_component_in {α : Type u} [topological_space α] {x : α} {F : set α} (hx : x F) :
theorem connected_component_in_subset {α : Type u} [topological_space α] (F : set α) (x : α) :
theorem is_preconnected.subset_connected_component {α : Type u} [topological_space α] {x : α} {s : set α} (H1 : is_preconnected s) (H2 : x s) :
theorem is_preconnected.subset_connected_component_in {α : Type u} [topological_space α] {s : set α} {x : α} {F : set α} (hs : is_preconnected s) (hxs : x s) (hsF : s F) :
theorem is_connected.subset_connected_component {α : Type u} [topological_space α] {x : α} {s : set α} (H1 : is_connected s) (H2 : x s) :
theorem is_preconnected.connected_component_in {α : Type u} [topological_space α] {x : α} {F : set α} (h : is_preconnected F) (hx : x F) :
theorem connected_component_in_mono {α : Type u} [topological_space α] (x : α) {F G : set α} (h : F G) :
@[class]
structure preconnected_space (α : Type u) [topological_space α] :
Prop

A preconnected space is one where there is no non-trivial open partition.

Instances of this typeclass
@[class]
structure connected_space (α : Type u) [topological_space α] :
Prop

A connected space is a nonempty one where there is no non-trivial open partition.

Instances of this typeclass
theorem is_connected_range {α : Type u} {β : Type v} [topological_space α] [topological_space β] [connected_space α] {f : α β} (h : continuous f) :
@[protected, instance]
@[protected, instance]
@[protected, instance]
def pi.preconnected_space {ι : Type u_1} {π : ι Type u_2} [Π (i : ι), topological_space (π i)] [ (i : ι), preconnected_space (π i)] :
preconnected_space (Π (i : ι), π i)
@[protected, instance]
def pi.connected_space {ι : Type u_1} {π : ι Type u_2} [Π (i : ι), topological_space (π i)] [ (i : ι), connected_space (π i)] :
connected_space (Π (i : ι), π i)
theorem is_clopen.eq_univ {α : Type u} [topological_space α] [preconnected_space α] {s : set α} (h' : is_clopen s) (h : s.nonempty) :
theorem is_preconnected_iff_subset_of_disjoint {α : Type u} [topological_space α] {s : set α} :
is_preconnected s (u v : set α), is_open u is_open v s u v s (u v) = s u s v

A set s is preconnected if and only if for every cover by two open sets that are disjoint on s, it is contained in one of the two covering sets.

theorem is_connected_iff_sUnion_disjoint_open {α : Type u} [topological_space α] {s : set α} :
is_connected s (U : finset (set α)), ( (u v : set α), u U v U (s (u v)).nonempty u = v) ( (u : set α), u U is_open u) s ⋃₀ U ( (u : set α) (H : u U), s u)

A set s is connected if and only if for every cover by a finite collection of open sets that are pairwise disjoint on s, it is contained in one of the members of the collection.

theorem is_preconnected.subset_clopen {α : Type u} [topological_space α] {s t : set α} (hs : is_preconnected s) (ht : is_clopen t) (hne : (s t).nonempty) :
s t

Preconnected sets are either contained in or disjoint to any given clopen set.

theorem disjoint_or_subset_of_clopen {α : Type u} [topological_space α] {s t : set α} (hs : is_preconnected s) (ht : is_clopen t) :
disjoint s t s t

Preconnected sets are either contained in or disjoint to any given clopen set.

A set s is preconnected if and only if for every cover by two closed sets that are disjoint on s, it is contained in one of the two covering sets.

A closed set s is preconnected if and only if for every cover by two closed sets that are disjoint, it is contained in one of the two covering sets.

theorem is_clopen.connected_component_subset {α : Type u} [topological_space α] {s : set α} {x : α} (hs : is_clopen s) (hx : x s) :

The connected component of a point is always a subset of the intersection of all its clopen neighbourhoods.

theorem is_clopen.bUnion_connected_component_eq {α : Type u} [topological_space α] {Z : set α} (h : is_clopen Z) :
( (x : α) (H : x Z), connected_component x) = Z

A clopen set is the union of its connected components.

theorem preimage_connected_component_connected {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α β} (connected_fibers : (t : β), is_connected (f ⁻¹' {t})) (hcl : (T : set β), is_closed T is_closed (f ⁻¹' T)) (t : β) :

The preimage of a connected component is preconnected if the function has connected fibers and a subset is closed iff the preimage is.

theorem quotient_map.preimage_connected_component {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α β} (hf : quotient_map f) (h_fibers : (y : β), is_connected (f ⁻¹' {y})) (a : α) :
theorem quotient_map.image_connected_component {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α β} (hf : quotient_map f) (h_fibers : (y : β), is_connected (f ⁻¹' {y})) (a : α) :
@[class]
structure locally_connected_space (α : Type u_3) [topological_space α] :
Prop

A topological space is locally connected if each neighborhood filter admits a basis of connected open sets. Note that it is equivalent to each point having a basis of connected (non necessarily open) sets but in a non-trivial way, so we choose this definition and prove the equivalence later in locally_connected_space_iff_connected_basis.

Instances of this typeclass
theorem locally_connected_space_of_connected_bases {α : Type u} [topological_space α] {ι : Type u_1} (b : α ι set α) (p : α ι Prop) (hbasis : (x : α), (nhds x).has_basis (p x) (b x)) (hconnected : (x : α) (i : ι), p x i is_preconnected (b x i)) :
def is_totally_disconnected {α : Type u} [topological_space α] (s : set α) :
Prop

A set s is called totally disconnected if every subset t ⊆ s which is preconnected is a subsingleton, ie either empty or a singleton.

Equations
@[protected, instance]
def pi.totally_disconnected_space {α : Type u_1} {β : α Type u_2} [t₂ : Π (a : α), topological_space (β a)] [ (a : α), totally_disconnected_space (β a)] :
@[protected, instance]
def sigma.totally_disconnected_space {ι : Type u_1} {π : ι Type u_2} [Π (i : ι), topological_space (π i)] [ (i : ι), totally_disconnected_space (π i)] :
theorem is_totally_disconnected_of_clopen_set {X : Type u_1} [topological_space X] (hX : {x y : X}, x y ( (U : set X) (h_clopen : is_clopen U), x U y U)) :

Let X be a topological space, and suppose that for all distinct x,y ∈ X, there is some clopen set U such that x ∈ U and y ∉ U. Then X is totally disconnected.

A space is totally disconnected iff its connected components are subsingletons.

A space is totally disconnected iff its connected components are singletons.

@[simp]

The image of a connected component in a totally disconnected space is a singleton.

theorem embedding.is_totally_disconnected {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α β} (hf : embedding f) {s : set α} (h : is_totally_disconnected (f '' s)) :
def is_totally_separated {α : Type u} [topological_space α] (s : set α) :
Prop

A set s is called totally separated if any two points of this set can be separated by two disjoint open sets covering s.

Equations
@[class]
structure totally_separated_space (α : Type u) [topological_space α] :
Prop

A space is totally separated if any two points can be separated by two disjoint open sets covering the whole space.

Instances of this typeclass
theorem exists_clopen_of_totally_separated {α : Type u_1} [topological_space α] [totally_separated_space α] {x y : α} (hxy : x y) :
(U : set α) (hU : is_clopen U), x U y U

The setoid of connected components of a topological space

Equations

The lift to connected_components α of a continuous map from α to a totally disconnected space

Equations
theorem connected_components_lift_unique' {α : Type u} [topological_space α] {β : Sort u_1} {g₁ g₂ : connected_components α β} (hg : g₁ coe = g₂ coe) :
g₁ = g₂

The preimage of a singleton in connected_components is the connected component of an element in the equivalence class.

theorem connected_components_preimage_image {α : Type u} [topological_space α] (U : set α) :
coe ⁻¹' (coe '' U) = (x : α) (H : x U), connected_component x

The preimage of the image of a set under the quotient map to connected_components α is the union of the connected components of the elements in it.

theorem is_preconnected.constant {α : Type u} [topological_space α] {Y : Type u_1} [topological_space Y] [discrete_topology Y] {s : set α} (hs : is_preconnected s) {f : α Y} (hf : continuous_on f s) {x y : α} (hx : x s) (hy : y s) :
f x = f y

A preconnected set s has the property that every map to a discrete space that is continuous on s is constant on s

theorem is_preconnected_of_forall_constant {α : Type u} [topological_space α] {s : set α} (hs : (f : α bool), continuous_on f s (x : α), x s (y : α), y s f x = f y) :

If every map to bool (a discrete two-element space), that is continuous on a set s, is constant on s, then s is preconnected

theorem preconnected_space.constant {α : Type u} [topological_space α] {Y : Type u_1} [topological_space Y] [discrete_topology Y] (hp : preconnected_space α) {f : α Y} (hf : continuous f) {x y : α} :
f x = f y

A preconnected_space version of is_preconnected.constant