mathlib documentation

topology.bases

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

A topological basis is one that satisfies the necessary conditions so that it suffices to take unions of the basis sets to get a topology (without taking finite intersections as well).

Equations
theorem topological_space.is_topological_basis_of_open_of_nhds {α : Type u} [t : topological_space α] {s : set (set α)} (h_open : ∀ (u : set α), u sis_open u) (h_nhds : ∀ (a : α) (u : set α), a uis_open u(∃ (v : set α) (H : v s), a v v u)) :

theorem topological_space.mem_nhds_of_is_topological_basis {α : Type u} [t : topological_space α] {a : α} {s : set α} {b : set (set α)} (hb : topological_space.is_topological_basis b) :
s 𝓝 a ∃ (t : set α) (H : t b), a t t s

theorem topological_space.is_topological_basis.nhds_has_basis {α : Type u} [t : topological_space α] {b : set (set α)} (hb : topological_space.is_topological_basis b) {a : α} :
(𝓝 a).has_basis (λ (t : set α), t b a t) (λ (t : set α), t)

theorem topological_space.is_open_of_is_topological_basis {α : Type u} [t : topological_space α] {s : set α} {b : set (set α)} (hb : topological_space.is_topological_basis b) (hs : s b) :

theorem topological_space.mem_basis_subset_of_mem_open {α : Type u} [t : topological_space α] {b : set (set α)} (hb : topological_space.is_topological_basis b) {a : α} {u : set α} (au : a u) (ou : is_open u) :
∃ (v : set α) (H : v b), a v v u

theorem topological_space.sUnion_basis_of_is_open {α : Type u} [t : topological_space α] {B : set (set α)} (hB : topological_space.is_topological_basis B) {u : set α} (ou : is_open u) :
∃ (S : set (set α)) (H : S B), u = ⋃₀S

theorem topological_space.Union_basis_of_is_open {α : Type u} [t : topological_space α] {B : set (set α)} (hB : topological_space.is_topological_basis B) {u : set α} (ou : is_open u) :
∃ (β : Type u) (f : β → set α), (u = ⋃ (i : β), f i) ∀ (i : β), f i B

@[class]
structure topological_space.separable_space (α : Type u) [t : topological_space α] :
Prop

A separable space is one with a countable dense subset, available through topological_space.exists_countable_dense. If α is also known to be nonempty, then topological_space.dense_seq provides a sequence ℕ → α with dense range, see topological_space.dense_range_dense_seq.

If α is a uniform space with countably generated uniformity filter (e.g., an emetric_space), then this condition is equivalent to topological_space.second_countable_topology α. In this case the latter should be used as a typeclass argument in theorems because Lean can automatically deduce separable_space from second_countable_topology but it can't deduce second_countable_topology and emetric_space.

Instances
theorem topological_space.exists_dense_seq (α : Type u) [t : topological_space α] [topological_space.separable_space α] [nonempty α] :
∃ (u : → α), dense_range u

A nonempty separable space admits a sequence with dense range. Instead of running cases on the conclusion of this lemma, you might want to use topological_space.dense_seq and topological_space.dense_range_dense_seq.

If α might be empty, then exists_countable_dense is the main way to use separability of α.

A sequence dense in a non-empty separable topological space.

If α might be empty, then exists_countable_dense is the main way to use separability of α.

Equations
@[simp]

The sequence dense_seq α has dense range.

theorem dense_range.separable_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space.separable_space α] [topological_space β] {f : α → β} (h : dense_range f) (h' : continuous f) :

If α is a separable space and f : α → β is a continuous map with dense range, then β is a separable space as well. E.g., the completion of a separable uniform space is separable.

@[class]
structure topological_space.first_countable_topology (α : Type u) [t : topological_space α] :
Prop

A first-countable space is one in which every point has a countable neighborhood basis.

Instances
@[instance]
def topological_space.second_countable_topology_fintype {ι : Type u_1} {π : ι → Type u_2} [fintype ι] [t : Π (a : ι), topological_space («π» a)] [sc : ∀ (a : ι), topological_space.second_countable_topology («π» a)] :

theorem topological_space.is_open_Union_countable {α : Type u} [t : topological_space α] [topological_space.second_countable_topology α] {ι : Type u_1} (s : ι → set α) (H : ∀ (i : ι), is_open (s i)) :
∃ (T : set ι), T.countable (⋃ (i : ι) (H : i T), s i) = ⋃ (i : ι), s i

theorem topological_space.is_open_sUnion_countable {α : Type u} [t : topological_space α] [topological_space.second_countable_topology α] (S : set (set α)) (H : ∀ (s : set α), s Sis_open s) :
∃ (T : set (set α)), T.countable T S ⋃₀T = ⋃₀S

theorem topological_space.countable_cover_nhds {α : Type u} [t : topological_space α] [topological_space.second_countable_topology α] {f : α → set α} (hf : ∀ (x : α), f x 𝓝 x) :
∃ (s : set α), s.countable (⋃ (x : α) (H : x s), f x) = set.univ

In a topological space with second countable topology, if f is a function that sends each point x to a neighborhood of x, then for some countable set s, the neighborhoods f x, x ∈ s, cover the whole space.