mathlib documentation

topology.basic

Basic theory of topological spaces.

The main definition is the type class topological space α which endows a type α with a topology. Then set α gets predicates is_open, is_closed and functions interior, closure and frontier. Each point x of α gets a neighborhood filter 𝓝 x. A filter F on α has x as a cluster point if is_cluster_pt x F : 𝓝 x ⊓ F ≠ ⊥. A map f : ι → α clusters at x along F : filter ι if map_cluster_pt x F f : cluster_pt x (map f F). In particular the notion of cluster point of a sequence u is map_cluster_pt x at_top u.

This file also defines locally finite families of subsets of α.

For topological spaces α and β, a function f : α → β and a point a : α, continuous_at f a means f is continuous at a, and global continuity is continuous f. There is also a version of continuity pcontinuous for partially defined functions.

Notation

Implementation notes

Topology in mathlib heavily uses filters (even more than in Bourbaki). See explanations in https://leanprover-community.github.io/theories/topology.html.

References

Tags

topological space, interior, closure, frontier, neighborhood, continuity, continuous function

 Topological spaces

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

A topology on α.

Instances
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def topological_space.of_closed {α : Type u} (T : set (set α)) (empty_mem : ∅ ∈ T) (sInter_mem : ∀ (A : set (set α)), A ⊆ T → ⋂₀A ∈ T) (union_mem : ∀ (A B : set α), A ∈ T → B ∈ T → A ∪ B ∈ T) :
topological_space α

A constructor for topologies by specifying the closed sets, and showing that they satisfy the appropriate conditions.

Equations
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@[ext]
theorem topological_space_eq {α : Type u} {f g : topological_space α} (a : f.is_open = g.is_open) :
f = g

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

is_open s means that s is open in the ambient topological space on α

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@[simp]
theorem is_open_univ {α : Type u} [t : topological_space α] :
is_open set.univ

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theorem is_open_inter {α : Type u} {s₁ s₂ : set α} [t : topological_space α] (h₁ : is_open s₁) (h₂ : is_open s₂) :
is_open (s₁ ∩ s₂)

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theorem is_open_sUnion {α : Type u} [t : topological_space α] {s : set (set α)} (h : ∀ (t_1 : set α), t_1 ∈ s → is_open t_1) :
is_open (⋃₀s)

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theorem is_open_fold {α : Type u} {s : set α} {t : topological_space α} :
t.is_open s = is_open s

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theorem is_open_Union {α : Type u} {ι : Sort w} [topological_space α] {f : ι → set α} (h : ∀ (i : ι), is_open (f i)) :
is_open (⋃ (i : ι), f i)

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theorem is_open_bUnion {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α} (h : ∀ (i : β), i ∈ s → is_open (f i)) :
is_open (⋃ (i : β) (H : i ∈ s), f i)

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theorem is_open_union {α : Type u} {s₁ s₂ : set α} [topological_space α] (h₁ : is_open s₁) (h₂ : is_open s₂) :
is_open (s₁ ∪ s₂)

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@[simp]
theorem is_open_empty {α : Type u} [topological_space α] :
is_open ∅

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theorem is_open_sInter {α : Type u} [topological_space α] {s : set (set α)} (hs : s.finite) (a : ∀ (t : set α), t ∈ s → is_open t) :
is_open (⋂₀s)

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theorem is_open_bInter {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α} (hs : s.finite) (a : ∀ (i : β), i ∈ s → is_open (f i)) :
is_open (⋂ (i : β) (H : i ∈ s), f i)

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theorem is_open_Inter {α : Type u} {β : Type v} [topological_space α] [fintype β] {s : β → set α} (h : ∀ (i : β), is_open (s i)) :
is_open (⋂ (i : β), s i)

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theorem is_open_Inter_prop {α : Type u} [topological_space α] {p : Prop} {s : p → set α} (h : ∀ (h : p), is_open (s h)) :
is_open (set.Inter s)

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theorem is_open_const {α : Type u} [topological_space α] {p : Prop} :
is_open {a : α | p}

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theorem is_open_and {α : Type u} {p₁ p₂ : α → Prop} [topological_space α] (a : is_open {a : α | p₁ a}) (a_1 : is_open {a : α | p₂ a}) :
is_open {a : α | p₁ a ∧ p₂ a}

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

A set is closed if its complement is open

Equations
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@[simp]
theorem is_closed_empty {α : Type u} [topological_space α] :
is_closed ∅

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@[simp]
theorem is_closed_univ {α : Type u} [topological_space α] :
is_closed set.univ

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theorem is_closed_union {α : Type u} {s₁ s₂ : set α} [topological_space α] (a : is_closed s₁) (a_1 : is_closed s₂) :
is_closed (s₁ ∪ s₂)

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theorem is_closed_sInter {α : Type u} [topological_space α] {s : set (set α)} (a : ∀ (t : set α), t ∈ s → is_closed t) :
is_closed (⋂₀s)

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theorem is_closed_Inter {α : Type u} {ι : Sort w} [topological_space α] {f : ι → set α} (h : ∀ (i : ι), is_closed (f i)) :
is_closed (⋂ (i : ι), f i)

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@[simp]
theorem is_open_compl_iff {α : Type u} [topological_space α] {s : set α} :
is_open sᶜ ↔ is_closed s

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@[simp]
theorem is_closed_compl_iff {α : Type u} [topological_space α] {s : set α} :
is_closed sᶜ ↔ is_open s

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theorem is_open_diff {α : Type u} [topological_space α] {s t : set α} (h₁ : is_open s) (h₂ : is_closed t) :
is_open (s \ t)

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theorem is_closed_inter {α : Type u} {s₁ s₂ : set α} [topological_space α] (h₁ : is_closed s₁) (h₂ : is_closed s₂) :
is_closed (s₁ ∩ s₂)

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theorem is_closed_bUnion {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α} (hs : s.finite) (a : ∀ (i : β), i ∈ s → is_closed (f i)) :
is_closed (⋃ (i : β) (H : i ∈ s), f i)

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theorem is_closed_Union {α : Type u} {β : Type v} [topological_space α] [fintype β] {s : β → set α} (h : ∀ (i : β), is_closed (s i)) :
is_closed (set.Union s)

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theorem is_closed_Union_prop {α : Type u} [topological_space α] {p : Prop} {s : p → set α} (h : ∀ (h : p), is_closed (s h)) :
is_closed (set.Union s)

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theorem is_closed_imp {α : Type u} [topological_space α] {p q : α → Prop} (hp : is_open {x : α | p x}) (hq : is_closed {x : α | q x}) :
is_closed {x : α | p x → q x}

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theorem is_open_neg {α : Type u} {p : α → Prop} [topological_space α] (a : is_closed {a : α | p a}) :
is_open {a : α | ¬p a}

Interior of a set

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

The interior of a set s is the largest open subset of s.

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theorem mem_interior {α : Type u} [topological_space α] {s : set α} {x : α} :
x ∈ interior s ↔ ∃ (t : set α) (H : t ⊆ s), is_open t ∧ x ∈ t

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@[simp]
theorem is_open_interior {α : Type u} [topological_space α] {s : set α} :
is_open (interior s)

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theorem interior_subset {α : Type u} [topological_space α] {s : set α} :
interior s ⊆ s

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theorem interior_maximal {α : Type u} [topological_space α] {s t : set α} (h₁ : t ⊆ s) (h₂ : is_open t) :
t ⊆ interior s

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theorem is_open.interior_eq {α : Type u} [topological_space α] {s : set α} (h : is_open s) :
interior s = s

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theorem interior_eq_iff_open {α : Type u} [topological_space α] {s : set α} :
interior s = s ↔ is_open s

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theorem subset_interior_iff_open {α : Type u} [topological_space α] {s : set α} :
s ⊆ interior s ↔ is_open s

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theorem subset_interior_iff_subset_of_open {α : Type u} [topological_space α] {s t : set α} (h₁ : is_open s) :
s ⊆ interior t ↔ s ⊆ t

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theorem interior_mono {α : Type u} [topological_space α] {s t : set α} (h : s ⊆ t) :
interior s ⊆ interior t

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@[simp]
theorem interior_empty {α : Type u} [topological_space α] :
interior ∅ = ∅

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@[simp]
theorem interior_univ {α : Type u} [topological_space α] :
interior set.univ = set.univ

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@[simp]
theorem interior_interior {α : Type u} [topological_space α] {s : set α} :
interior (interior s) = interior s

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@[simp]
theorem interior_inter {α : Type u} [topological_space α] {s t : set α} :
interior (s ∩ t) = interior s ∩ interior t

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theorem interior_union_is_closed_of_interior_empty {α : Type u} [topological_space α] {s t : set α} (h₁ : is_closed s) (h₂ : interior t = ∅) :
interior (s ∪ t) = interior s

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theorem is_open_iff_forall_mem_open {α : Type u} {s : set α} [topological_space α] :
is_open s ↔ ∀ (x : α), x ∈ s → (∃ (t : set α) (H : t ⊆ s), is_open t ∧ x ∈ t)

 Closure of a set

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

The closure of s is the smallest closed set containing s.

Equations
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@[simp]
theorem is_closed_closure {α : Type u} [topological_space α] {s : set α} :
is_closed (closure s)

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theorem subset_closure {α : Type u} [topological_space α] {s : set α} :
s ⊆ closure s

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theorem closure_minimal {α : Type u} [topological_space α] {s t : set α} (h₁ : s ⊆ t) (h₂ : is_closed t) :
closure s ⊆ t

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theorem is_closed.closure_eq {α : Type u} [topological_space α] {s : set α} (h : is_closed s) :
closure s = s

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theorem is_closed.closure_subset {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) :
closure s ⊆ s

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theorem is_closed.closure_subset_iff {α : Type u} [topological_space α] {s t : set α} (h₁ : is_closed t) :
closure s ⊆ t ↔ s ⊆ t

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theorem closure_mono {α : Type u} [topological_space α] {s t : set α} (h : s ⊆ t) :
closure s ⊆ closure t

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theorem monotone_closure (α : Type u_1) [topological_space α] :
monotone closure

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theorem closure_inter_subset_inter_closure {α : Type u} [topological_space α] (s t : set α) :
closure (s ∩ t) ⊆ closure s ∩ closure t

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theorem is_closed_of_closure_subset {α : Type u} [topological_space α] {s : set α} (h : closure s ⊆ s) :
is_closed s

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theorem closure_eq_iff_is_closed {α : Type u} [topological_space α] {s : set α} :
closure s = s ↔ is_closed s

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theorem closure_subset_iff_is_closed {α : Type u} [topological_space α] {s : set α} :
closure s ⊆ s ↔ is_closed s

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@[simp]
theorem closure_empty {α : Type u} [topological_space α] :
closure ∅ = ∅

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@[simp]
theorem closure_empty_iff {α : Type u} [topological_space α] (s : set α) :
closure s = ∅ ↔ s = ∅

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theorem set.nonempty.closure {α : Type u} [topological_space α] {s : set α} (h : s.nonempty) :
(closure s).nonempty

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@[simp]
theorem closure_univ {α : Type u} [topological_space α] :
closure set.univ = set.univ

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@[simp]
theorem closure_closure {α : Type u} [topological_space α] {s : set α} :
closure (closure s) = closure s

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@[simp]
theorem closure_union {α : Type u} [topological_space α] {s t : set α} :
closure (s ∪ t) = closure s ∪ closure t

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theorem interior_subset_closure {α : Type u} [topological_space α] {s : set α} :
interior s ⊆ closure s

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theorem closure_eq_compl_interior_compl {α : Type u} [topological_space α] {s : set α} :
closure s = (interior sᶜ)ᶜ

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@[simp]
theorem interior_compl {α : Type u} [topological_space α] {s : set α} :
interior sᶜ = (closure s)ᶜ

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@[simp]
theorem closure_compl {α : Type u} [topological_space α] {s : set α} :
closure sᶜ = (interior s)ᶜ

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theorem mem_closure_iff {α : Type u} [topological_space α] {s : set α} {a : α} :
a ∈ closure s ↔ ∀ (o : set α), is_open o → a ∈ o → (o ∩ s).nonempty

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theorem dense_iff_inter_open {α : Type u} [topological_space α] {s : set α} :
closure s = set.univ ↔ ∀ (U : set α), is_open U → U.nonempty → (U ∩ s).nonempty

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theorem dense_of_subset_dense {α : Type u} [topological_space α] {s₁ s₂ : set α} (h : s₁ ⊆ s₂) (hd : closure s₁ = set.univ) :
closure s₂ = set.univ

Frontier of a set

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

The frontier of a set is the set of points between the closure and interior.

Equations
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theorem frontier_eq_closure_inter_closure {α : Type u} [topological_space α] {s : set α} :
frontier s = closure s ∩ closure sᶜ

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@[simp]
theorem frontier_compl {α : Type u} [topological_space α] (s : set α) :
frontier sᶜ = frontier s

The complement of a set has the same frontier as the original set.

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theorem frontier_inter_subset {α : Type u} [topological_space α] (s t : set α) :
frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t

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theorem frontier_union_subset {α : Type u} [topological_space α] (s t : set α) :
frontier (s ∪ t) ⊆ frontier s ∩ closure tᶜ ∪ closure sᶜ ∩ frontier t

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theorem is_closed.frontier_eq {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) :
frontier s = s \ interior s

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theorem is_open.frontier_eq {α : Type u} [topological_space α] {s : set α} (hs : is_open s) :
frontier s = closure s \ s

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theorem is_closed_frontier {α : Type u} [topological_space α] {s : set α} :
is_closed (frontier s)

The frontier of a set is closed.

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theorem interior_frontier {α : Type u} [topological_space α] {s : set α} (h : is_closed s) :
interior (frontier s) = ∅

The frontier of a closed set has no interior point.

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theorem closure_eq_interior_union_frontier {α : Type u} [topological_space α] (s : set α) :
closure s = interior s ∪ frontier s

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theorem closure_eq_self_union_frontier {α : Type u} [topological_space α] (s : set α) :
closure s = s ∪ frontier s

 Neighborhoods

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def nhds {α : Type u} [topological_space α] (a : α) :
filter α

A set is called a neighborhood of a if it contains an open set around a. The set of all neighborhoods of a forms a filter, the neighborhood filter at a, is here defined as the infimum over the principal filters of all open sets containing a.

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theorem nhds_def {α : Type u} [topological_space α] (a : α) :
𝓝 a = ⨅ (s : set α) (H : s ∈ {s : set α | a ∈ s ∧ is_open s}), 𝓟 s

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theorem nhds_basis_opens {α : Type u} [topological_space α] (a : α) :
(𝓝 a).has_basis (λ (s : set α), a ∈ s ∧ is_open s) (λ (x : set α), x)

The open sets containing a are a basis for the neighborhood filter. See nhds_basis_opens' for a variant using open neighborhoods instead.

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theorem le_nhds_iff {α : Type u} [topological_space α] {f : filter α} {a : α} :
f ≤ 𝓝 a ↔ ∀ (s : set α), a ∈ s → is_open s → s ∈ f

A filter lies below the neighborhood filter at a iff it contains every open set around a.

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theorem nhds_le_of_le {α : Type u} [topological_space α] {f : filter α} {a : α} {s : set α} (h : a ∈ s) (o : is_open s) (sf : 𝓟 s ≤ f) :
𝓝 a ≤ f

To show a filter is above the neighborhood filter at a, it suffices to show that it is above the principal filter of some open set s containing a.

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theorem mem_nhds_sets_iff {α : Type u} [topological_space α] {a : α} {s : set α} :
s ∈ 𝓝 a ↔ ∃ (t : set α) (H : t ⊆ s), is_open t ∧ a ∈ t

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theorem eventually_nhds_iff {α : Type u} [topological_space α] {a : α} {p : α → Prop} :
(∀ᶠ (x : α) in 𝓝 a, p x) ↔ ∃ (t : set α), (∀ (x : α), x ∈ t → p x) ∧ is_open t ∧ a ∈ t

A predicate is true in a neighborhood of a iff it is true for all the points in an open set containing a.

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theorem map_nhds {α : Type u} {β : Type v} [topological_space α] {a : α} {f : α → β} :
filter.map f (𝓝 a) = ⨅ (s : set α) (H : s ∈ {s : set α | a ∈ s ∧ is_open s}), 𝓟 (f '' s)

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theorem mem_of_nhds {α : Type u} [topological_space α] {a : α} {s : set α} (a_1 : s ∈ 𝓝 a) :
a ∈ s

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theorem filter.eventually.self_of_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α} (h : ∀ᶠ (y : α) in 𝓝 a, p y) :
p a

If a predicate is true in a neighborhood of a, then it is true for a.

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theorem mem_nhds_sets {α : Type u} [topological_space α] {a : α} {s : set α} (hs : is_open s) (ha : a ∈ s) :
s ∈ 𝓝 a

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theorem nhds_basis_opens' {α : Type u} [topological_space α] (a : α) :
(𝓝 a).has_basis (λ (s : set α), s ∈ 𝓝 a ∧ is_open s) (λ (x : set α), x)

The open neighborhoods of a are a basis for the neighborhood filter. See nhds_basis_opens for a variant using open sets around a instead.

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theorem filter.eventually.eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α} (h : ∀ᶠ (y : α) in 𝓝 a, p y) :
∀ᶠ (y : α) in 𝓝 a, ∀ᶠ (x : α) in 𝓝 y, p x

If a predicate is true in a neighbourhood of a, then for y sufficiently close to a this predicate is true in a neighbourhood of y.

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@[simp]
theorem eventually_eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α} :
(∀ᶠ (y : α) in 𝓝 a, ∀ᶠ (x : α) in 𝓝 y, p x) ↔ ∀ᶠ (x : α) in 𝓝 a, p x

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@[simp]
theorem nhds_bind_nhds {α : Type u} {a : α} [topological_space α] :
(𝓝 a).bind 𝓝 = 𝓝 a

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@[simp]
theorem eventually_eventually_eq_nhds {α : Type u} {β : Type v} [topological_space α] {f g : α → β} {a : α} :
(∀ᶠ (y : α) in 𝓝 a, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 a] g

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theorem filter.eventually_eq.eq_of_nhds {α : Type u} {β : Type v} [topological_space α] {f g : α → β} {a : α} (h : f =ᶠ[𝓝 a] g) :
f a = g a

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@[simp]
theorem eventually_eventually_le_nhds {α : Type u} {β : Type v} [topological_space α] [has_le β] {f g : α → β} {a : α} :
(∀ᶠ (y : α) in 𝓝 a, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 a] g

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theorem filter.eventually_eq.eventually_eq_nhds {α : Type u} {β : Type v} [topological_space α] {f g : α → β} {a : α} (h : f =ᶠ[𝓝 a] g) :
∀ᶠ (y : α) in 𝓝 a, f =ᶠ[𝓝 y] g

If two functions are equal in a neighbourhood of a, then for y sufficiently close to a these functions are equal in a neighbourhood of y.

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theorem filter.eventually_le.eventually_le_nhds {α : Type u} {β : Type v} [topological_space α] [has_le β] {f g : α → β} {a : α} (h : f ≤ᶠ[𝓝 a] g) :
∀ᶠ (y : α) in 𝓝 a, f ≤ᶠ[𝓝 y] g

If f x ≤ g x in a neighbourhood of a, then for y sufficiently close to a we have f x ≤ g x in a neighbourhood of y.

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theorem all_mem_nhds {α : Type u} [topological_space α] (x : α) (P : set α → Prop) (hP : ∀ (s t : set α), s ⊆ t → P s → P t) :
(∀ (s : set α), s ∈ 𝓝 x → P s) ↔ ∀ (s : set α), is_open s → x ∈ s → P s

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theorem all_mem_nhds_filter {α : Type u} {β : Type v} [topological_space α] (x : α) (f : set α → set β) (hf : ∀ (s t : set α), s ⊆ t → f s ⊆ f t) (l : filter β) :
(∀ (s : set α), s ∈ 𝓝 x → f s ∈ l) ↔ ∀ (s : set α), is_open s → x ∈ s → f s ∈ l

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theorem rtendsto_nhds {α : Type u} {β : Type v} [topological_space α] {r : rel β α} {l : filter β} {a : α} :
filter.rtendsto r l (𝓝 a) ↔ ∀ (s : set α), is_open s → a ∈ s → r.core s ∈ l

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theorem rtendsto'_nhds {α : Type u} {β : Type v} [topological_space α] {r : rel β α} {l : filter β} {a : α} :
filter.rtendsto' r l (𝓝 a) ↔ ∀ (s : set α), is_open s → a ∈ s → r.preimage s ∈ l

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theorem ptendsto_nhds {α : Type u} {β : Type v} [topological_space α] {f : β →. α} {l : filter β} {a : α} :
filter.ptendsto f l (𝓝 a) ↔ ∀ (s : set α), is_open s → a ∈ s → f.core s ∈ l

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theorem ptendsto'_nhds {α : Type u} {β : Type v} [topological_space α] {f : β →. α} {l : filter β} {a : α} :
filter.ptendsto' f l (𝓝 a) ↔ ∀ (s : set α), is_open s → a ∈ s → f.preimage s ∈ l

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theorem tendsto_nhds {α : Type u} {β : Type v} [topological_space α] {f : β → α} {l : filter β} {a : α} :
filter.tendsto f l (𝓝 a) ↔ ∀ (s : set α), is_open s → a ∈ s → f ⁻¹' s ∈ l

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theorem tendsto_const_nhds {α : Type u} {β : Type v} [topological_space α] {a : α} {f : filter β} :
filter.tendsto (λ (b : β), a) f (𝓝 a)

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theorem pure_le_nhds {α : Type u} [topological_space α] :
pure ≤ 𝓝

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theorem tendsto_pure_nhds {β : Type v} {α : Type u_1} [topological_space β] (f : α → β) (a : α) :
filter.tendsto f (pure a) (𝓝 (f a))

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theorem order_top.tendsto_at_top_nhds {β : Type v} {α : Type u_1} [order_top α] [topological_space β] (f : α → β) :
filter.tendsto f filter.at_top (𝓝 (f ⊤))

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@[simp, instance]
def nhds_ne_bot {α : Type u} [topological_space α] {a : α} :
(𝓝 a).ne_bot

Cluster points

In this section we define cluster points (also known as limit points and accumulation points) of a filter and of a sequence.

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def cluster_pt {α : Type u} [topological_space α] (x : α) (F : filter α) :
Prop

A point x is a cluster point of a filter F if 𝓝 x ⊓ F ≠ ⊥. Also known as an accumulation point or a limit point.

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theorem cluster_pt.ne_bot {α : Type u} [topological_space α] {x : α} {F : filter α} (h : cluster_pt x F) :
(𝓝 x ⊓ F).ne_bot

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theorem cluster_pt_iff {α : Type u} [topological_space α] {x : α} {F : filter α} :
cluster_pt x F ↔ ∀ {U V : set α}, U ∈ 𝓝 x → V ∈ F → (U ∩ V).nonempty

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theorem cluster_pt_principal_iff {α : Type u} [topological_space α] {x : α} {s : set α} :
cluster_pt x (𝓟 s) ↔ ∀ (U : set α), U ∈ 𝓝 x → (U ∩ s).nonempty

x is a cluster point of a set s if every neighbourhood of x meets s on a nonempty set.

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theorem cluster_pt_principal_iff_frequently {α : Type u} [topological_space α] {x : α} {s : set α} :
cluster_pt x (𝓟 s) ↔ ∃ᶠ (y : α) in 𝓝 x, y ∈ s

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theorem cluster_pt.of_le_nhds {α : Type u} [topological_space α] {x : α} {f : filter α} (H : f ≤ 𝓝 x) [f.ne_bot] :
cluster_pt x f

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theorem cluster_pt.of_le_nhds' {α : Type u} [topological_space α] {x : α} {f : filter α} (H : f ≤ 𝓝 x) (hf : f.ne_bot) :
cluster_pt x f

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theorem cluster_pt.of_nhds_le {α : Type u} [topological_space α] {x : α} {f : filter α} (H : 𝓝 x ≤ f) :
cluster_pt x f

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theorem cluster_pt.mono {α : Type u} [topological_space α] {x : α} {f g : filter α} (H : cluster_pt x f) (h : f ≤ g) :
cluster_pt x g

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theorem cluster_pt.of_inf_left {α : Type u} [topological_space α] {x : α} {f g : filter α} (H : cluster_pt x (f ⊓ g)) :
cluster_pt x f

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theorem cluster_pt.of_inf_right {α : Type u} [topological_space α] {x : α} {f g : filter α} (H : cluster_pt x (f ⊓ g)) :
cluster_pt x g

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def map_cluster_pt {α : Type u} [topological_space α] {ι : Type u_1} (x : α) (F : filter ι) (u : ι → α) :
Prop

A point x is a cluster point of a sequence u along a filter F if it is a cluster point of map u F.

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theorem map_cluster_pt_iff {α : Type u} [topological_space α] {ι : Type u_1} (x : α) (F : filter ι) (u : ι → α) :
map_cluster_pt x F u ↔ ∀ (s : set α), s ∈ 𝓝 x → (∃ᶠ (a : ι) in F, u a ∈ s)

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theorem map_cluster_pt_of_comp {α : Type u} [topological_space α] {ι : Type u_1} {δ : Type u_2} {F : filter ι} {φ : δ → ι} {p : filter δ} {x : α} {u : ι → α} [p.ne_bot] (h : filter.tendsto φ p F) (H : filter.tendsto (u ∘ φ) p (𝓝 x)) :
map_cluster_pt x F u

Interior, closure and frontier in terms of neighborhoods

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theorem interior_eq_nhds {α : Type u} [topological_space α] {s : set α} :
interior s = {a : α | 𝓝 a ≤ 𝓟 s}

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theorem mem_interior_iff_mem_nhds {α : Type u} [topological_space α] {s : set α} {a : α} :
a ∈ interior s ↔ s ∈ 𝓝 a

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theorem subset_interior_iff_nhds {α : Type u} [topological_space α] {s V : set α} :
s ⊆ interior V ↔ ∀ (x : α), x ∈ s → V ∈ 𝓝 x

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theorem is_open_iff_nhds {α : Type u} [topological_space α] {s : set α} :
is_open s ↔ ∀ (a : α), a ∈ s → 𝓝 a ≤ 𝓟 s

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theorem is_open_iff_mem_nhds {α : Type u} [topological_space α] {s : set α} :
is_open s ↔ ∀ (a : α), a ∈ s → s ∈ 𝓝 a

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theorem mem_closure_iff_frequently {α : Type u} [topological_space α] {s : set α} {a : α} :
a ∈ closure s ↔ ∃ᶠ (x : α) in 𝓝 a, x ∈ s

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theorem mem_closure_iff_cluster_pt {α : Type u} [topological_space α] {s : set α} {a : α} :
a ∈ closure s ↔ cluster_pt a (𝓟 s)

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theorem closure_eq_cluster_pts {α : Type u} [topological_space α] {s : set α} :
closure s = {a : α | cluster_pt a (𝓟 s)}

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theorem mem_closure_iff_nhds {α : Type u} [topological_space α] {s : set α} {a : α} :
a ∈ closure s ↔ ∀ (t : set α), t ∈ 𝓝 a → (t ∩ s).nonempty

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theorem mem_closure_iff_nhds' {α : Type u} [topological_space α] {s : set α} {a : α} :
a ∈ closure s ↔ ∀ (t : set α), t ∈ 𝓝 a → (∃ (y : ↥s), ↑y ∈ t)

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theorem mem_closure_iff_comap_ne_bot {α : Type u} [topological_space α] {A : set α} {x : α} :
x ∈ closure A ↔ (filter.comap coe (𝓝 x)).ne_bot

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theorem mem_closure_iff_nhds_basis {α : Type u} {β : Type v} [topological_space α] {a : α} {p : β → Prop} {s : β → set α} (h : (𝓝 a).has_basis p s) {t : set α} :
a ∈ closure t ↔ ∀ (i : β), p i → (∃ (y : α) (H : y ∈ t), y ∈ s i)

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theorem mem_closure_iff_ultrafilter {α : Type u} [topological_space α] {s : set α} {x : α} :
x ∈ closure s ↔ ∃ (u : filter.ultrafilter α), s ∈ u.val ∧ u.val ≤ 𝓝 x

x belongs to the closure of s if and only if some ultrafilter supported on s converges to x.

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theorem is_closed_iff_cluster_pt {α : Type u} [topological_space α] {s : set α} :
is_closed s ↔ ∀ (a : α), cluster_pt a (𝓟 s) → a ∈ s

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theorem is_closed_iff_nhds {α : Type u} [topological_space α] {s : set α} :
is_closed s ↔ ∀ (x : α), (∀ (U : set α), U ∈ 𝓝 x → (U ∩ s).nonempty) → x ∈ s

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theorem closure_inter_open {α : Type u} [topological_space α] {s t : set α} (h : is_open s) :
s ∩ closure t ⊆ closure (s ∩ t)

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theorem dense_inter_of_open_left {α : Type u} [topological_space α] {s t : set α} (hs : closure s = set.univ) (ht : closure t = set.univ) (hso : is_open s) :
closure (s ∩ t) = set.univ

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theorem dense_inter_of_open_right {α : Type u} [topological_space α] {s t : set α} (hs : closure s = set.univ) (ht : closure t = set.univ) (hto : is_open t) :
closure (s ∩ t) = set.univ

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theorem closure_diff {α : Type u} [topological_space α] {s t : set α} :
closure s \ closure t ⊆ closure (s \ t)

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theorem filter.frequently.mem_of_closed {α : Type u} [topological_space α] {a : α} {s : set α} (h : ∃ᶠ (x : α) in 𝓝 a, x ∈ s) (hs : is_closed s) :
a ∈ s

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theorem is_closed.mem_of_frequently_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α} {b : filter β} {a : α} {s : set α} (hs : is_closed s) (h : ∃ᶠ (x : β) in b, f x ∈ s) (hf : filter.tendsto f b (𝓝 a)) :
a ∈ s

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theorem is_closed.mem_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α} {b : filter β} {a : α} {s : set α} [b.ne_bot] (hs : is_closed s) (hf : filter.tendsto f b (𝓝 a)) (h : ∀ᶠ (x : β) in b, f x ∈ s) :
a ∈ s

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theorem mem_closure_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α} {b : filter β} {a : α} {s : set α} [b.ne_bot] (hf : filter.tendsto f b (𝓝 a)) (h : ∀ᶠ (x : β) in b, f x ∈ s) :
a ∈ closure s

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theorem tendsto_inf_principal_nhds_iff_of_forall_eq {α : Type u} {β : Type v} [topological_space α] {f : β → α} {l : filter β} {s : set β} {a : α} (h : ∀ (x : β), x ∉ s → f x = a) :
filter.tendsto f (l ⊓ 𝓟 s) (𝓝 a) ↔ filter.tendsto f l (𝓝 a)

Suppose that f sends the complement to s to a single point a, and l is some filter. Then f tends to a along l restricted to s if and only if it tends to a along l.

Limits of filters in topological spaces

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def Lim {α : Type u} [topological_space α] [nonempty α] (f : filter α) :
α

If f is a filter, then Lim f is a limit of the filter, if it exists.

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def lim {α : Type u} {β : Type v} [topological_space α] [nonempty α] (f : filter β) (g : β → α) :
α

If f is a filter in β and g : β → α is a function, then lim f is a limit of g at f, if it exists.

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theorem Lim_spec {α : Type u} [topological_space α] {f : filter α} (h : ∃ (a : α), f ≤ 𝓝 a) :
f ≤ 𝓝 (Lim f)

If a filter f is majorated by some 𝓝 a, then it is majorated by 𝓝 (Lim f). We formulate this lemma with a [nonempty α] argument of Lim derived from h to make it useful for types without a [nonempty α] instance. Because of the built-in proof irrelevance, Lean will unify this instance with any other instance.

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theorem lim_spec {α : Type u} {β : Type v} [topological_space α] {f : filter β} {g : β → α} (h : ∃ (a : α), filter.tendsto g f (𝓝 a)) :
filter.tendsto g f (𝓝 (lim f g))

If g tends to some 𝓝 a along f, then it tends to 𝓝 (lim f g). We formulate this lemma with a [nonempty α] argument of lim derived from h to make it useful for types without a [nonempty α] instance. Because of the built-in proof irrelevance, Lean will unify this instance with any other instance.

 Locally finite families

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def locally_finite {α : Type u} {β : Type v} [topological_space α] (f : β → set α) :
Prop

A family of sets in set α is locally finite if at every point x:α, there is a neighborhood of x which meets only finitely many sets in the family

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theorem locally_finite_of_finite {α : Type u} {β : Type v} [topological_space α] {f : β → set α} (h : set.univ.finite) :
locally_finite f

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theorem locally_finite_subset {α : Type u} {β : Type v} [topological_space α] {f₁ f₂ : β → set α} (hf₂ : locally_finite f₂) (hf : ∀ (b : β), f₁ b ⊆ f₂ b) :
locally_finite f₁

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theorem is_closed_Union_of_locally_finite {α : Type u} {β : Type v} [topological_space α] {f : β → set α} (h₁ : locally_finite f) (h₂ : ∀ (i : β), is_closed (f i)) :
is_closed (⋃ (i : β), f i)

Continuity

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def continuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α → β) :
Prop

A function between topological spaces is continuous if the preimage of every open set is open.

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theorem is_open.preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) {s : set β} (h : is_open s) :
is_open (f ⁻¹' s)

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def continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α → β) (x : α) :
Prop

A function between topological spaces is continuous at a point x₀ if f x tends to f x₀ when x tends to x₀.

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theorem continuous_at.tendsto {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} (h : continuous_at f x) :
f→_{x}f x

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theorem continuous_at.preimage_mem_nhds {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} {t : set β} (h : continuous_at f x) (ht : t ∈ 𝓝 (f x)) :
f ⁻¹' t ∈ 𝓝 x

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theorem preimage_interior_subset_interior_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {s : set β} (hf : continuous f) :
f ⁻¹' interior s ⊆ interior (f ⁻¹' s)

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theorem continuous_id {α : Type u_1} [topological_space α] :
continuous id

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theorem continuous.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] {g : β → γ} {f : α → β} (hg : continuous g) (hf : continuous f) :
continuous (g ∘ f)

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theorem continuous.iterate {α : Type u_1} [topological_space α] {f : α → α} (h : continuous f) (n : ℕ) :
continuous f^[n]

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theorem continuous_at.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] {g : β → γ} {f : α → β} {x : α} (hg : continuous_at g (f x)) (hf : continuous_at f x) :
continuous_at (g ∘ f) x

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theorem continuous.tendsto {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) (x : α) :
f→_{x}f x

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theorem continuous.continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} (h : continuous f) :
continuous_at f x

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theorem continuous_iff_continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} :
continuous f ↔ ∀ (x : α), continuous_at f x

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theorem continuous_const {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {b : β} :
continuous (λ (a : α), b)

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theorem continuous_at_const {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {x : α} {b : β} :
continuous_at (λ (a : α), b) x

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theorem continuous_at_id {α : Type u_1} [topological_space α] {x : α} :
continuous_at id x

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theorem continuous_at.iterate {α : Type u_1} [topological_space α] {f : α → α} {x : α} (hf : continuous_at f x) (hx : f x = x) (n : ℕ) :
continuous_at f^[n] x

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theorem continuous_iff_is_closed {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} :
continuous f ↔ ∀ (s : set β), is_closed s → is_closed (f ⁻¹' s)

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theorem is_closed.preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) {s : set β} (h : is_closed s) :
is_closed (f ⁻¹' s)

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theorem continuous_at_iff_ultrafilter {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (x : α) :
continuous_at f x ↔ ∀ (g : filter α), g.is_ultrafilter → g ≤ 𝓝 x → filter.map f g ≤ 𝓝 (f x)

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theorem continuous_iff_ultrafilter {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} :
continuous f ↔ ∀ (x : α) (g : filter α), g.is_ultrafilter → g ≤ 𝓝 x → filter.map f g ≤ 𝓝 (f x)

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theorem continuous_if {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {p : α → Prop} {f g : α → β} {h : Π (a : α), decidable (p a)} (hp : ∀ (a : α), a ∈ frontier {a : α | p a} → f a = g a) (hf : continuous f) (hg : continuous g) :
continuous (λ (a : α), ite (p a) (f a) (g a))

A piecewise defined function if p then f else g is continuous, if both f and g are continuous, and they coincide on the frontier (boundary) of the set {a | p a}.

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def pcontinuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α →. β) :
Prop

Continuity of a partial function

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theorem open_dom_of_pcontinuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α →. β} (h : pcontinuous f) :
is_open f.dom

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theorem pcontinuous_iff' {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α →. β} :
pcontinuous f ↔ ∀ {x : α} {y : β}, y ∈ f x → filter.ptendsto' f (𝓝 x) (𝓝 y)

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theorem image_closure_subset_closure_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {s : set α} (h : continuous f) :
f '' closure s ⊆ closure (f '' s)

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theorem mem_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set α} {t : set β} {f : α → β} {a : α} (hf : continuous f) (ha : a ∈ closure s) (ht : ∀ (a : α), a ∈ s → f a ∈ t) :
f a ∈ closure t