mathlib documentation

topology.opens

Open sets #

Summary #

We define the subtype of open sets in a topological space.

Main Definitions #

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def topological_space.opens (α : Type u_1) [topological_space α] :
Type u_1

The type of open subsets of a topological space.

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@[protected, instance]
def topological_space.opens.set.has_coe {α : Type u_1} [topological_space α] :
has_coe (topological_space.opens α) (set α)
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theorem topological_space.opens.val_eq_coe {α : Type u_1} [topological_space α] (U : topological_space.opens α) :
U.val = ↑U
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theorem topological_space.opens.coe_mk {α : Type u_1} [topological_space α] {U : set α} {hU : is_open U} :
↑⟨U, hU⟩ = U

the coercion opens α → set α applied to a pair is the same as taking the first component

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@[protected, instance]
def topological_space.opens.has_subset {α : Type u_1} [topological_space α] :
has_subset (topological_space.opens α)
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@[protected, instance]
def topological_space.opens.has_mem {α : Type u_1} [topological_space α] :
has_mem α (topological_space.opens α)
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@[simp]
theorem topological_space.opens.subset_coe {α : Type u_1} [topological_space α] {U V : topological_space.opens α} :
↑U ⊆ ↑V = (U ⊆ V)
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@[simp]
theorem topological_space.opens.mem_coe {α : Type u_1} [topological_space α] {x : α} {U : topological_space.opens α} :
x ∈ ↑U = (x ∈ U)
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@[ext]
theorem topological_space.opens.ext {α : Type u_1} [topological_space α] {U V : topological_space.opens α} (h : ↑U = ↑V) :
U = V
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@[ext]
theorem topological_space.opens.ext_iff {α : Type u_1} [topological_space α] {U V : topological_space.opens α} :
↑U = ↑V ↔ U = V
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@[protected, instance]
def topological_space.opens.partial_order {α : Type u_1} [topological_space α] :
partial_order (topological_space.opens α)
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def topological_space.opens.interior {α : Type u_1} [topological_space α] (s : set α) :
topological_space.opens α

The interior of a set, as an element of opens.

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theorem topological_space.opens.gc {α : Type u_1} [topological_space α] :
galois_connection coe topological_space.opens.interior
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def topological_space.opens.gi {α : Type u_1} [topological_space α] :
galois_insertion (⇑order_dual.to_dual ∘ topological_space.opens.interior ∘ ⇑order_dual.of_dual) (⇑order_dual.to_dual ∘ subtype.val ∘ ⇑order_dual.of_dual)

The galois insertion between sets and opens, but ordered by reverse inclusion.

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@[simp]
theorem topological_space.opens.gi_choice_val {α : Type u_1} [topological_space α] {s : order_dual (set α)} {hs : (⇑order_dual.to_dual ∘ subtype.val ∘ ⇑order_dual.of_dual) ((⇑order_dual.to_dual ∘ topological_space.opens.interior ∘ ⇑order_dual.of_dual) s) ≤ s} :
(topological_space.opens.gi.choice s hs).val = s
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@[protected, instance]
def topological_space.opens.complete_lattice {α : Type u_1} [topological_space α] :
complete_lattice (topological_space.opens α)
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theorem topological_space.opens.le_def {α : Type u_1} [topological_space α] {U V : topological_space.opens α} :
U ≤ V ↔ ↑U ≤ ↑V
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@[simp]
theorem topological_space.opens.mk_inf_mk {α : Type u_1} [topological_space α] {U V : set α} {hU : is_open U} {hV : is_open V} :
⟨U, hU⟩ ⊓ ⟨V, hV⟩ = ⟨U ⊓ V, _⟩
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@[simp, norm_cast]
theorem topological_space.opens.coe_inf {α : Type u_1} [topological_space α] {U V : topological_space.opens α} :
↑(U ⊓ V) = ↑U ⊓ ↑V
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@[simp]
theorem topological_space.opens.coe_bot {α : Type u_1} [topological_space α] :
↑⊥ = ∅
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@[simp]
theorem topological_space.opens.coe_top {α : Type u_1} [topological_space α] :
↑⊤ = set.univ
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@[protected, instance]
def topological_space.opens.has_inter {α : Type u_1} [topological_space α] :
has_inter (topological_space.opens α)
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@[protected, instance]
def topological_space.opens.has_union {α : Type u_1} [topological_space α] :
has_union (topological_space.opens α)
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@[protected, instance]
def topological_space.opens.has_emptyc {α : Type u_1} [topological_space α] :
has_emptyc (topological_space.opens α)
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@[protected, instance]
def topological_space.opens.inhabited {α : Type u_1} [topological_space α] :
inhabited (topological_space.opens α)
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@[simp]
theorem topological_space.opens.inter_eq {α : Type u_1} [topological_space α] (U V : topological_space.opens α) :
U ∩ V = U ⊓ V
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@[simp]
theorem topological_space.opens.union_eq {α : Type u_1} [topological_space α] (U V : topological_space.opens α) :
U ∪ V = U ⊔ V
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@[simp]
theorem topological_space.opens.empty_eq {α : Type u_1} [topological_space α] :
∅ = ⊥
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@[simp]
theorem topological_space.opens.Sup_s {α : Type u_1} [topological_space α] {Us : set (topological_space.opens α)} :
↑(Sup Us) = ⋃₀(coe '' Us)
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theorem topological_space.opens.supr_def {α : Type u_1} [topological_space α] {ι : Sort u_2} (s : ι → topological_space.opens α) :
(⨆ (i : ι), s i) = ⟨⋃ (i : ι), ↑(s i), _⟩
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@[simp]
theorem topological_space.opens.supr_mk {α : Type u_1} [topological_space α] {ι : Sort u_2} (s : ι → set α) (h : ∀ (i : ι), is_open (s i)) :
(⨆ (i : ι), ⟨s i, _⟩) = ⟨⋃ (i : ι), s i, _⟩
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@[simp]
theorem topological_space.opens.supr_s {α : Type u_1} [topological_space α] {ι : Sort u_2} (s : ι → topological_space.opens α) :
(↑⨆ (i : ι), s i) = ⋃ (i : ι), ↑(s i)
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@[simp]
theorem topological_space.opens.mem_supr {α : Type u_1} [topological_space α] {ι : Sort u_2} {x : α} {s : ι → topological_space.opens α} :
x ∈ supr s ↔ ∃ (i : ι), x ∈ s i
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@[simp]
theorem topological_space.opens.mem_Sup {α : Type u_1} [topological_space α] {Us : set (topological_space.opens α)} {x : α} :
x ∈ Sup Us ↔ ∃ (u : topological_space.opens α) (H : u ∈ Us), x ∈ u
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theorem topological_space.opens.open_embedding_of_le {α : Type u_1} [topological_space α] {U V : topological_space.opens α} (i : U ≤ V) :
open_embedding (set.inclusion i)
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theorem topological_space.opens.not_nonempty_iff_eq_bot {α : Type u_1} [topological_space α] (U : topological_space.opens α) :
¬↑U.nonempty ↔ U = ⊥
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theorem topological_space.opens.ne_bot_iff_nonempty {α : Type u_1} [topological_space α] (U : topological_space.opens α) :
U ≠ ⊥ ↔ ↑U.nonempty
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def topological_space.opens.is_basis {α : Type u_1} [topological_space α] (B : set (topological_space.opens α)) :
Prop

A set of opens α is a basis if the set of corresponding sets is a topological basis.

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theorem topological_space.opens.is_basis_iff_nbhd {α : Type u_1} [topological_space α] {B : set (topological_space.opens α)} :
topological_space.opens.is_basis B ↔ ∀ {U : topological_space.opens α} {x : α}, x ∈ U → (∃ (U' : topological_space.opens α) (H : U' ∈ B), x ∈ U' ∧ U' ⊆ U)
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theorem topological_space.opens.is_basis_iff_cover {α : Type u_1} [topological_space α] {B : set (topological_space.opens α)} :
topological_space.opens.is_basis B ↔ ∀ (U : topological_space.opens α), ∃ (Us : set (topological_space.opens α)) (H : Us ⊆ B), U = Sup Us
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def topological_space.opens.comap {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : C(α, β)) :
topological_space.opens β →o topological_space.opens α

The preimage of an open set, as an open set.

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@[simp]
theorem topological_space.opens.comap_id {α : Type u_1} [topological_space α] :
topological_space.opens.comap continuous_map.id = order_hom.id
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theorem topological_space.opens.comap_mono {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : C(α, β)) {V W : topological_space.opens β} (hVW : V ⊆ W) :
⇑(topological_space.opens.comap f) V ⊆ ⇑(topological_space.opens.comap f) W
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@[simp]
theorem topological_space.opens.coe_comap {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : C(α, β)) (U : topological_space.opens β) :
↑(⇑(topological_space.opens.comap f) U) = ⇑f ⁻¹' ↑U
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@[simp]
theorem topological_space.opens.comap_val {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : C(α, β)) (U : topological_space.opens β) :
(⇑(topological_space.opens.comap f) U).val = ⇑f ⁻¹' ↑U
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@[protected]
theorem topological_space.opens.comap_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (g : C(β, γ)) (f : C(α, β)) :
topological_space.opens.comap (g.comp f) = (topological_space.opens.comap f).comp (topological_space.opens.comap g)
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@[protected]
theorem topological_space.opens.comap_comap {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (g : C(β, γ)) (f : C(α, β)) (U : topological_space.opens γ) :
⇑(topological_space.opens.comap f) (⇑(topological_space.opens.comap g) U) = ⇑(topological_space.opens.comap (g.comp f)) U
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@[protected, simp]
def topological_space.opens.equiv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ₜ β) :
topological_space.opens α ≃ topological_space.opens β

A homeomorphism induces an equivalence on open sets, by taking comaps.

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@[protected, simp]
def topological_space.opens.order_iso {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ₜ β) :
topological_space.opens α ≃o topological_space.opens β

A homeomorphism induces an order isomorphism on open sets, by taking comaps.

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def topological_space.open_nhds_of {α : Type u_1} [topological_space α] (x : α) :
Type u_1

The open neighborhoods of a point. See also opens or nhds.

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@[protected, instance]
def topological_space.open_nhds_of.inhabited {α : Type u_1} [topological_space α] (x : α) :
inhabited (topological_space.open_nhds_of x)
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