statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
comap_coe_nhds_infty : comap (coe : X → alexandroff X) (𝓝 ∞) = coclosed_compact X | by simp [nhds_infty_eq, comap_sup, comap_map coe_injective] | lemma | alexandroff.comap_coe_nhds_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_infty {f : filter (alexandroff X)} :
f ≤ 𝓝 ∞ ↔ ∀ s : set X, is_closed s → is_compact s → coe '' sᶜ ∪ {∞} ∈ f | by simp only [has_basis_nhds_infty.ge_iff, and_imp] | lemma | alexandroff.le_nhds_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"and_imp",
"filter",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter_le_nhds_infty {f : ultrafilter (alexandroff X)} :
(f : filter (alexandroff X)) ≤ 𝓝 ∞ ↔ ∀ s : set X, is_closed s → is_compact s → coe '' s ∉ f | by simp only [le_nhds_infty, ← compl_image_coe, ultrafilter.mem_coe,
ultrafilter.compl_mem_iff_not_mem] | lemma | alexandroff.ultrafilter_le_nhds_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"filter",
"is_closed",
"is_compact",
"ultrafilter",
"ultrafilter.compl_mem_iff_not_mem",
"ultrafilter.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_infty' {α : Type*} {f : alexandroff X → α} {l : filter α} :
tendsto f (𝓝 ∞) l ↔ tendsto f (pure ∞) l ∧ tendsto (f ∘ coe) (coclosed_compact X) l | by simp [nhds_infty_eq, and_comm] | lemma | alexandroff.tendsto_nhds_infty' | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_infty {α : Type*} {f : alexandroff X → α} {l : filter α} :
tendsto f (𝓝 ∞) l ↔
∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : set X, is_closed t ∧ is_compact t ∧ maps_to (f ∘ coe) tᶜ s | tendsto_nhds_infty'.trans $ by simp only [tendsto_pure_left,
has_basis_coclosed_compact.tendsto_left_iff, forall_and_distrib, and_assoc, exists_prop] | lemma | alexandroff.tendsto_nhds_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"exists_prop",
"filter",
"forall_and_distrib",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_infty' {Y : Type*} [topological_space Y] {f : alexandroff X → Y} :
continuous_at f ∞ ↔ tendsto (f ∘ coe) (coclosed_compact X) (𝓝 (f ∞)) | tendsto_nhds_infty'.trans $ and_iff_right (tendsto_pure_nhds _ _) | lemma | alexandroff.continuous_at_infty' | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"continuous_at",
"tendsto_pure_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_infty {Y : Type*} [topological_space Y] {f : alexandroff X → Y} :
continuous_at f ∞ ↔
∀ s ∈ 𝓝 (f ∞), ∃ t : set X, is_closed t ∧ is_compact t ∧ maps_to (f ∘ coe) tᶜ s | continuous_at_infty'.trans $
by simp only [has_basis_coclosed_compact.tendsto_left_iff, exists_prop, and_assoc] | lemma | alexandroff.continuous_at_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"continuous_at",
"exists_prop",
"is_closed",
"is_compact",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_coe {Y : Type*} [topological_space Y] {f : alexandroff X → Y} {x : X} :
continuous_at f x ↔ continuous_at (f ∘ coe) x | by rw [continuous_at, nhds_coe_eq, tendsto_map'_iff, continuous_at] | lemma | alexandroff.continuous_at_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"continuous_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_coe [noncompact_space X] :
dense_range (coe : X → alexandroff X) | begin
rw [dense_range, ← compl_infty],
exact dense_compl_singleton _
end | lemma | alexandroff.dense_range_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"dense_compl_singleton",
"dense_range",
"noncompact_space"
] | If `X` is not a compact space, then the natural embedding `X → alexandroff X` has dense range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_embedding_coe [noncompact_space X] :
dense_embedding (coe : X → alexandroff X) | { dense := dense_range_coe, .. open_embedding_coe } | lemma | alexandroff.dense_embedding_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"dense",
"dense_embedding",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_coe {x y : X} : (x : alexandroff X) ⤳ y ↔ x ⤳ y | open_embedding_coe.to_inducing.specializes_iff | lemma | alexandroff.specializes_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_coe {x y : X} : inseparable (x : alexandroff X) y ↔ inseparable x y | open_embedding_coe.to_inducing.inseparable_iff | lemma | alexandroff.inseparable_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_specializes_infty_coe {x : X} : ¬specializes ∞ (x : alexandroff X) | is_closed_infty.not_specializes rfl (coe_ne_infty x) | lemma | alexandroff.not_specializes_infty_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_inseparable_infty_coe {x : X} : ¬inseparable ∞ (x : alexandroff X) | λ h, not_specializes_infty_coe h.specializes | lemma | alexandroff.not_inseparable_infty_coe | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_inseparable_coe_infty {x : X} : ¬inseparable (x : alexandroff X) ∞ | λ h, not_specializes_infty_coe h.specializes' | lemma | alexandroff.not_inseparable_coe_infty | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff {x y : alexandroff X} :
inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ inseparable x' y' | by induction x using alexandroff.rec; induction y using alexandroff.rec;
simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe] | lemma | alexandroff.inseparable_iff | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"alexandroff.rec",
"inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_continuous_cofinite_topology_of_symm [infinite X] [discrete_topology X] :
¬(continuous (@cofinite_topology.of (alexandroff X)).symm) | begin
inhabit X,
simp only [continuous_iff_continuous_at, continuous_at, not_forall],
use [cofinite_topology.of ↑(default : X)],
simpa [nhds_coe_eq, nhds_discrete, cofinite_topology.nhds_eq]
using (finite_singleton ((default : X) : alexandroff X)).infinite_compl
end | lemma | alexandroff.not_continuous_cofinite_topology_of_symm | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"cofinite_topology.nhds_eq",
"cofinite_topology.of",
"continuous",
"continuous_at",
"continuous_iff_continuous_at",
"discrete_topology",
"infinite",
"nhds_discrete",
"not_forall"
] | If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`cofinite_topology (alexandroff X)` to `alexandroff X` is not continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.homeo_of_equiv_compact_to_t2.t1_counterexample :
∃ (α β : Type) (Iα : topological_space α) (Iβ : topological_space β), by exactI
compact_space α ∧ t1_space β ∧ ∃ f : α ≃ β, continuous f ∧ ¬ continuous f.symm | ⟨alexandroff ℕ, cofinite_topology (alexandroff ℕ), infer_instance, infer_instance,
infer_instance, infer_instance, cofinite_topology.of, cofinite_topology.continuous_of,
alexandroff.not_continuous_cofinite_topology_of_symm⟩ | lemma | continuous.homeo_of_equiv_compact_to_t2.t1_counterexample | topology | src/topology/alexandroff.lean | [
"data.fintype.option",
"topology.separation",
"topology.sets.opens"
] | [
"alexandroff",
"cofinite_topology",
"cofinite_topology.continuous_of",
"cofinite_topology.of",
"compact_space",
"continuous",
"t1_space",
"topological_space"
] | A concrete counterexample shows that `continuous.homeo_of_equiv_compact_to_t2`
cannot be generalized from `t2_space` to `t1_space`.
Let `α = alexandroff ℕ` be the one-point compactification of `ℕ`, and let `β` be the same space
`alexandroff ℕ` with the cofinite topology. Then `α` is compact, `β` is T1, and the ident... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis (s : set (set α)) : Prop | (exists_subset_inter : ∀t₁∈s, ∀t₂∈s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃∈s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂)
(sUnion_eq : (⋃₀ s) = univ)
(eq_generate_from : t = generate_from s) | structure | topological_space.is_topological_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | 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). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.insert_empty {s : set (set α)} (h : is_topological_basis s) :
is_topological_basis (insert ∅ s) | begin
refine ⟨_, by rw [sUnion_insert, empty_union, h.sUnion_eq], _⟩,
{ rintro t₁ (rfl|h₁) t₂ (rfl|h₂) x ⟨hx₁, hx₂⟩, {cases hx₁}, {cases hx₁}, {cases hx₂},
obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x ⟨hx₁, hx₂⟩,
exact ⟨t₃, or.inr h₃, hs⟩ },
{ rw h.eq_generate_from,
refine le_antisymm (le_... | lemma | topological_space.is_topological_basis.insert_empty | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open_empty",
"le_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.diff_empty {s : set (set α)} (h : is_topological_basis s) :
is_topological_basis (s \ {∅}) | begin
refine ⟨_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], _⟩,
{ rintro t₁ ⟨h₁, -⟩ t₂ ⟨h₂, -⟩ x hx,
obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x hx,
exact ⟨t₃, ⟨h₃, nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩ },
{ rw h.eq_generate_from,
refine le_antisymm (generate_from_anti $ diff_subset s ... | lemma | topological_space.is_topological_basis.diff_empty | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"eq_or_ne",
"is_open_empty",
"le_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_of_subbasis {s : set (set α)} (hs : t = generate_from s) :
is_topological_basis ((λ f, ⋂₀ f) '' {f : set (set α) | f.finite ∧ f ⊆ s}) | begin
refine ⟨_, _, hs.trans (le_antisymm (le_generate_from _) $ generate_from_anti $ λ t ht, _)⟩,
{ rintro _ ⟨t₁, ⟨hft₁, ht₁b⟩, rfl⟩ _ ⟨t₂, ⟨hft₂, ht₂b⟩, rfl⟩ x h,
exact ⟨_, ⟨_, ⟨hft₁.union hft₂, union_subset ht₁b ht₂b⟩, sInter_union t₁ t₂⟩, h, subset.rfl⟩ },
{ rw [sUnion_image, Union₂_eq_univ_iff],
exac... | lemma | topological_space.is_topological_basis_of_subbasis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open_sInter",
"le_generate_from"
] | If a family of sets `s` generates the topology, then intersections of finite
subcollections of `s` form a topological basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis_of_open_of_nhds {s : set (set α)}
(h_open : ∀ u ∈ s, is_open u)
(h_nhds : ∀(a:α) (u : set α), a ∈ u → is_open u → ∃v ∈ s, a ∈ v ∧ v ⊆ u) :
is_topological_basis s | begin
refine ⟨λ t₁ ht₁ t₂ ht₂ x hx, h_nhds _ _ hx (is_open.inter (h_open _ ht₁) (h_open _ ht₂)), _, _⟩,
{ refine sUnion_eq_univ_iff.2 (λ a, _),
rcases h_nhds a univ trivial is_open_univ with ⟨u, h₁, h₂, -⟩,
exact ⟨u, h₁, h₂⟩ },
{ refine (le_generate_from h_open).antisymm (λ u hu, _),
refine (@is_open_... | lemma | topological_space.is_topological_basis_of_open_of_nhds | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"infi₂_le_of_le",
"is_open",
"is_open.inter",
"is_open_iff_nhds",
"is_open_univ",
"le_generate_from"
] | If a family of open sets `s` is such that every open neighbourhood contains some
member of `s`, then `s` is a topological basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.mem_nhds_iff {a : α} {s : set α} {b : set (set α)}
(hb : is_topological_basis b) : s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s | begin
change s ∈ (𝓝 a).sets ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s,
rw [hb.eq_generate_from, nhds_generate_from, binfi_sets_eq],
{ simp [and_assoc, and.left_comm] },
{ exact assume s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩,
have a ∈ s ∩ t, from ⟨hs₁, ht₁⟩,
let ⟨u, hu₁, hu₂, hu₃⟩ := hb.1 _ hs₂ _ ht₂ _ this in
⟨u, ⟨hu₂, hu₁... | lemma | topological_space.is_topological_basis.mem_nhds_iff | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | A set `s` is in the neighbourhood of `a` iff there is some basis set `t`, which
contains `a` and is itself contained in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.is_open_iff {s : set α} {b : set (set α)} (hb : is_topological_basis b) :
is_open s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s | by simp [is_open_iff_mem_nhds, hb.mem_nhds_iff] | lemma | topological_space.is_topological_basis.is_open_iff | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"is_open_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.nhds_has_basis {b : set (set α)} (hb : is_topological_basis b) {a : α} :
(𝓝 a).has_basis (λ t : set α, t ∈ b ∧ a ∈ t) (λ t, t) | ⟨λ s, hb.mem_nhds_iff.trans $ by simp only [exists_prop, and_assoc]⟩ | lemma | topological_space.is_topological_basis.nhds_has_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.is_open {s : set α} {b : set (set α)}
(hb : is_topological_basis b) (hs : s ∈ b) : is_open s | by { rw hb.eq_generate_from, exact generate_open.basic s hs } | lemma | topological_space.is_topological_basis.is_open | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.mem_nhds {a : α} {s : set α} {b : set (set α)}
(hb : is_topological_basis b) (hs : s ∈ b) (ha : a ∈ s) : s ∈ 𝓝 a | (hb.is_open hs).mem_nhds ha | lemma | topological_space.is_topological_basis.mem_nhds | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.exists_subset_of_mem_open {b : set (set α)}
(hb : is_topological_basis b) {a:α} {u : set α} (au : a ∈ u)
(ou : is_open u) : ∃v ∈ b, a ∈ v ∧ v ⊆ u | hb.mem_nhds_iff.1 $ is_open.mem_nhds ou au | lemma | topological_space.is_topological_basis.exists_subset_of_mem_open | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.open_eq_sUnion' {B : set (set α)}
(hB : is_topological_basis B) {u : set α} (ou : is_open u) :
u = ⋃₀ {s ∈ B | s ⊆ u} | ext $ λ a,
⟨λ ha, let ⟨b, hb, ab, bu⟩ := hB.exists_subset_of_mem_open ha ou in ⟨b, ⟨hb, bu⟩, ab⟩,
λ ⟨b, ⟨hb, bu⟩, ab⟩, bu ab⟩ | lemma | topological_space.is_topological_basis.open_eq_sUnion' | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | Any open set is the union of the basis sets contained in it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.open_eq_sUnion {B : set (set α)}
(hB : is_topological_basis B) {u : set α} (ou : is_open u) :
∃ S ⊆ B, u = ⋃₀ S | ⟨{s ∈ B | s ⊆ u}, λ s h, h.1, hB.open_eq_sUnion' ou⟩ | lemma | topological_space.is_topological_basis.open_eq_sUnion | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.open_iff_eq_sUnion {B : set (set α)}
(hB : is_topological_basis B) {u : set α} :
is_open u ↔ ∃ S ⊆ B, u = ⋃₀ S | ⟨hB.open_eq_sUnion, λ ⟨S, hSB, hu⟩, hu.symm ▸ is_open_sUnion (λ s hs, hB.is_open (hSB hs))⟩ | lemma | topological_space.is_topological_basis.open_iff_eq_sUnion | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"is_open_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.open_eq_Union {B : set (set α)}
(hB : is_topological_basis B) {u : set α} (ou : is_open u) :
∃ (β : Type u) (f : β → set α), u = (⋃ i, f i) ∧ ∀ i, f i ∈ B | ⟨↥{s ∈ B | s ⊆ u}, coe, by { rw ← sUnion_eq_Union, apply hB.open_eq_sUnion' ou }, λ s, and.left s.2⟩ | lemma | topological_space.is_topological_basis.open_eq_Union | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.mem_closure_iff {b : set (set α)} (hb : is_topological_basis b)
{s : set α} {a : α} :
a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).nonempty | (mem_closure_iff_nhds_basis' hb.nhds_has_basis).trans $ by simp only [and_imp] | lemma | topological_space.is_topological_basis.mem_closure_iff | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"and_imp",
"closure",
"mem_closure_iff_nhds_basis'"
] | A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.dense_iff {b : set (set α)} (hb : is_topological_basis b) {s : set α} :
dense s ↔ ∀ o ∈ b, set.nonempty o → (o ∩ s).nonempty | begin
simp only [dense, hb.mem_closure_iff],
exact ⟨λ h o hb ⟨a, ha⟩, h a o hb ha, λ h a o hb ha, h o hb ⟨a, ha⟩⟩
end | lemma | topological_space.is_topological_basis.dense_iff | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense",
"set.nonempty"
] | A set is dense iff it has non-trivial intersection with all basis sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.is_open_map_iff {β} [topological_space β] {B : set (set α)}
(hB : is_topological_basis B) {f : α → β} :
is_open_map f ↔ ∀ s ∈ B, is_open (f '' s) | begin
refine ⟨λ H o ho, H _ (hB.is_open ho), λ hf o ho, _⟩,
rw [hB.open_eq_sUnion' ho, sUnion_eq_Union, image_Union],
exact is_open_Union (λ s, hf s s.2.1)
end | lemma | topological_space.is_topological_basis.is_open_map_iff | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"is_open_Union",
"is_open_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.exists_nonempty_subset {B : set (set α)}
(hb : is_topological_basis B) {u : set α} (hu : u.nonempty) (ou : is_open u) :
∃ v ∈ B, set.nonempty v ∧ v ⊆ u | begin
cases hu with x hx,
rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx,
rcases hx with ⟨v, hv, hxv⟩,
exact ⟨v, hv.1, ⟨x, hxv⟩, hv.2⟩
end | lemma | topological_space.is_topological_basis.exists_nonempty_subset | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_opens : is_topological_basis { U : set α | is_open U } | is_topological_basis_of_open_of_nhds (by tauto) (by tauto) | lemma | topological_space.is_topological_basis_opens | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.prod {β} [topological_space β] {B₁ : set (set α)}
{B₂ : set (set β)} (h₁ : is_topological_basis B₁) (h₂ : is_topological_basis B₂) :
is_topological_basis (image2 (×ˢ) B₁ B₂) | begin
refine is_topological_basis_of_open_of_nhds _ _,
{ rintro _ ⟨u₁, u₂, hu₁, hu₂, rfl⟩,
exact (h₁.is_open hu₁).prod (h₂.is_open hu₂) },
{ rintro ⟨a, b⟩ u hu uo,
rcases (h₁.nhds_has_basis.prod_nhds h₂.nhds_has_basis).mem_iff.1 (is_open.mem_nhds uo hu)
with ⟨⟨s, t⟩, ⟨⟨hs, ha⟩, ht, hb⟩, hu⟩,
exa... | lemma | topological_space.is_topological_basis.prod | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open.mem_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.inducing {β} [topological_space β]
{f : α → β} {T : set (set β)} (hf : inducing f) (h : is_topological_basis T) :
is_topological_basis (image (preimage f) T) | begin
refine is_topological_basis_of_open_of_nhds _ _,
{ rintros _ ⟨V, hV, rfl⟩,
rwa hf.is_open_iff,
refine ⟨V, h.is_open hV, rfl⟩ },
{ intros a U ha hU,
rw hf.is_open_iff at hU,
obtain ⟨V, hV, rfl⟩ := hU,
obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV,
obtain ⟨W, hW, ha⟩ := ha,
refine ⟨f ... | lemma | topological_space.is_topological_basis.inducing | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"inducing",
"set.preimage_mono",
"set.subset_sUnion_of_mem",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_of_cover {ι} {U : ι → set α} (Uo : ∀ i, is_open (U i))
(Uc : (⋃ i, U i) = univ) {b : Π i, set (set (U i))} (hb : ∀ i, is_topological_basis (b i)) :
is_topological_basis (⋃ i : ι, image (coe : U i → α) '' (b i)) | begin
refine is_topological_basis_of_open_of_nhds (λ u hu, _) _,
{ simp only [mem_Union, mem_image] at hu,
rcases hu with ⟨i, s, sb, rfl⟩,
exact (Uo i).is_open_map_subtype_coe _ ((hb i).is_open sb) },
{ intros a u ha uo,
rcases Union_eq_univ_iff.1 Uc a with ⟨i, hi⟩,
lift a to ↥(U i) using hi,
... | lemma | topological_space.is_topological_basis_of_cover | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"continuous_subtype_coe",
"is_open",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.continuous {β : Type*} [topological_space β]
{B : set (set β)} (hB : is_topological_basis B) (f : α → β) (hf : ∀ s ∈ B, is_open (f ⁻¹' s)) :
continuous f | begin rw hB.eq_generate_from, exact continuous_generated_from hf end | lemma | topological_space.is_topological_basis.continuous | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"continuous",
"continuous_generated_from",
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separable_space : Prop | (exists_countable_dense : ∃s:set α, s.countable ∧ dense s) | class | topological_space.separable_space | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense"
] | 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 count... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_countable_dense [separable_space α] :
∃ s : set α, s.countable ∧ dense s | separable_space.exists_countable_dense | lemma | topological_space.exists_countable_dense | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_dense_seq [separable_space α] [nonempty α] : ∃ u : ℕ → α, dense_range u | begin
obtain ⟨s : set α, hs, s_dense⟩ := exists_countable_dense α,
cases set.countable_iff_exists_subset_range.mp hs with u hu,
exact ⟨u, s_dense.mono hu⟩,
end | lemma | topological_space.exists_dense_seq | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense_range"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_seq [separable_space α] [nonempty α] : ℕ → α | classical.some (exists_dense_seq α) | def | topological_space.dense_seq | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | A dense sequence 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range_dense_seq [separable_space α] [nonempty α] :
dense_range (dense_seq α) | classical.some_spec (exists_dense_seq α) | lemma | topological_space.dense_range_dense_seq | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense_range"
] | The sequence `dense_seq α` has dense range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable.to_separable_space [countable α] : separable_space α | { exists_countable_dense := ⟨set.univ, set.countable_univ, dense_univ⟩ } | instance | topological_space.countable.to_separable_space | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"countable",
"set.countable_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separable_space_of_dense_range {ι : Type*} [countable ι] (u : ι → α) (hu : dense_range u) :
separable_space α | ⟨⟨range u, countable_range u, hu⟩⟩ | lemma | topological_space.separable_space_of_dense_range | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"countable",
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.pairwise_disjoint.countable_of_is_open [separable_space α] {ι : Type*}
{s : ι → set α} {a : set ι} (h : a.pairwise_disjoint s) (ha : ∀ i ∈ a, is_open (s i))
(h'a : ∀ i ∈ a, (s i).nonempty) :
a.countable | begin
rcases exists_countable_dense α with ⟨u, ⟨u_encodable⟩, u_dense⟩,
have : ∀ i : a, ∃ y, y ∈ s i ∩ u :=
λ i, dense_iff_inter_open.1 u_dense (s i) (ha i i.2) (h'a i i.2),
choose f hfs hfu using this,
lift f to a → u using hfu,
have f_inj : injective f,
{ refine injective_iff_pairwise_ne.mpr
((h... | lemma | set.pairwise_disjoint.countable_of_is_open | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"encodable.of_inj",
"is_open",
"lift"
] | In a separable space, a family of nonempty disjoint open sets is countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.set.pairwise_disjoint.countable_of_nonempty_interior [separable_space α] {ι : Type*}
{s : ι → set α} {a : set ι} (h : a.pairwise_disjoint s)
(ha : ∀ i ∈ a, (interior (s i)).nonempty) :
a.countable | (h.mono $ λ i, interior_subset).countable_of_is_open (λ i hi, is_open_interior) ha | lemma | set.pairwise_disjoint.countable_of_nonempty_interior | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"interior",
"interior_subset",
"is_open_interior"
] | In a separable space, a family of disjoint sets with nonempty interiors is countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_separable (s : set α) | ∃ c : set α, c.countable ∧ s ⊆ closure c | def | topological_space.is_separable | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"closure",
"is_separable"
] | A set `s` in a topological space is separable if it is contained in the closure of a
countable set `c`. Beware that this definition does not require that `c` is contained in `s` (to
express the latter, use `separable_space s` or `is_separable (univ : set s))`. In metric spaces,
the two definitions are equivalent, see `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_separable.mono {s u : set α} (hs : is_separable s) (hu : u ⊆ s) :
is_separable u | begin
rcases hs with ⟨c, c_count, hs⟩,
exact ⟨c, c_count, hu.trans hs⟩
end | lemma | topological_space.is_separable.mono | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable.union {s u : set α} (hs : is_separable s) (hu : is_separable u) :
is_separable (s ∪ u) | begin
rcases hs with ⟨cs, cs_count, hcs⟩,
rcases hu with ⟨cu, cu_count, hcu⟩,
refine ⟨cs ∪ cu, cs_count.union cu_count, _⟩,
exact union_subset (hcs.trans (closure_mono (subset_union_left _ _)))
(hcu.trans (closure_mono (subset_union_right _ _)))
end | lemma | topological_space.is_separable.union | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"closure_mono",
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable.closure {s : set α} (hs : is_separable s) : is_separable (closure s) | begin
rcases hs with ⟨c, c_count, hs⟩,
exact ⟨c, c_count, by simpa using closure_mono hs⟩,
end | lemma | topological_space.is_separable.closure | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"closure",
"closure_mono",
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable_Union {ι : Type*} [countable ι] {s : ι → set α} (hs : ∀ i, is_separable (s i)) :
is_separable (⋃ i, s i) | begin
choose c hc h'c using hs,
refine ⟨⋃ i, c i, countable_Union hc, Union_subset_iff.2 (λ i, _)⟩,
exact (h'c i).trans (closure_mono (subset_Union _ i))
end | lemma | topological_space.is_separable_Union | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"closure_mono",
"countable",
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.countable.is_separable {s : set α} (hs : s.countable) : is_separable s | ⟨s, hs, subset_closure⟩ | lemma | set.countable.is_separable | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.finite.is_separable {s : set α} (hs : s.finite) : is_separable s | hs.countable.is_separable | lemma | set.finite.is_separable | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable_univ_iff :
is_separable (univ : set α) ↔ separable_space α | begin
split,
{ rintros ⟨c, c_count, hc⟩,
refine ⟨⟨c, c_count, by rwa [dense_iff_closure_eq, ← univ_subset_iff]⟩⟩ },
{ introsI h,
rcases exists_countable_dense α with ⟨c, c_count, hc⟩,
exact ⟨c, c_count, by rwa [univ_subset_iff, ← dense_iff_closure_eq]⟩ }
end | lemma | topological_space.is_separable_univ_iff | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense_iff_closure_eq",
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable_of_separable_space [h : separable_space α] (s : set α) : is_separable s | is_separable.mono (is_separable_univ_iff.2 h) (subset_univ _) | lemma | topological_space.is_separable_of_separable_space | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_separable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable.image {β : Type*} [topological_space β]
{s : set α} (hs : is_separable s) {f : α → β} (hf : continuous f) :
is_separable (f '' s) | begin
rcases hs with ⟨c, c_count, hc⟩,
refine ⟨f '' c, c_count.image _, _⟩,
rw image_subset_iff,
exact hc.trans (closure_subset_preimage_closure_image hf)
end | lemma | topological_space.is_separable.image | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"closure_subset_preimage_closure_image",
"continuous",
"is_separable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_separable_of_separable_space_subtype (s : set α) [separable_space s] : is_separable s | begin
have : is_separable ((coe : s → α) '' (univ : set s)) :=
(is_separable_of_separable_space _).image continuous_subtype_coe,
simpa only [image_univ, subtype.range_coe_subtype],
end | lemma | topological_space.is_separable_of_separable_space_subtype | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"continuous_subtype_coe",
"is_separable",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_pi {ι : Type*} {X : ι → Type*}
[∀ i, topological_space (X i)] {T : Π i, set (set (X i))}
(cond : ∀ i, is_topological_basis (T i)) :
is_topological_basis {S : set (Π i, X i) | ∃ (U : Π i, set (X i)) (F : finset ι),
(∀ i, i ∈ F → (U i) ∈ T i) ∧ S = (F : set ι).pi U } | begin
refine is_topological_basis_of_open_of_nhds _ _,
{ rintro _ ⟨U, F, h1, rfl⟩,
apply is_open_set_pi F.finite_to_set,
intros i hi,
exact (cond i).is_open (h1 i hi) },
{ intros a U ha hU,
obtain ⟨I, t, hta, htU⟩ :
∃ (I : finset ι) (t : Π (i : ι), set (X i)), (∀ i, t i ∈ 𝓝 (a i)) ∧ set.pi ... | lemma | is_topological_basis_pi | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"filter.mem_pi'",
"finset",
"is_open",
"is_open_set_pi",
"nhds_pi",
"set.pi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_infi {β : Type*} {ι : Type*} {X : ι → Type*}
[t : ∀ i, topological_space (X i)] {T : Π i, set (set (X i))}
(cond : ∀ i, is_topological_basis (T i)) (f : Π i, β → X i) :
@is_topological_basis β (⨅ i, induced (f i) (t i))
{ S | ∃ (U : Π i, set (X i)) (F : finset ι),
(∀ i, i ∈ F → U i ∈ T ... | begin
convert (is_topological_basis_pi cond).inducing (inducing_infi_to_pi _),
ext V,
split,
{ rintros ⟨U, F, h1, h2⟩,
have : (F : set ι).pi U = (⋂ (i : ι) (hi : i ∈ F),
(λ (z : Π j, X j), z i) ⁻¹' (U i)), by { ext, simp },
refine ⟨(F : set ι).pi U, ⟨U, F, h1, rfl⟩, _⟩,
rw [this, h2, set.pre... | lemma | is_topological_basis_infi | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"finset",
"inducing",
"inducing_infi_to_pi",
"is_topological_basis_pi",
"set.preimage_Inter",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_singletons (α : Type*) [topological_space α] [discrete_topology α] :
is_topological_basis {s | ∃ (x : α), (s : set α) = {x}} | is_topological_basis_of_open_of_nhds (λ u hu, is_open_discrete _) $
λ x u hx u_open, ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩ | lemma | is_topological_basis_singletons | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"discrete_topology",
"is_open_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.separable_space {α β : Type*} [topological_space α] [separable_space α]
[topological_space β] {f : α → β} (h : dense_range f) (h' : continuous f) :
separable_space β | let ⟨s, s_cnt, s_dense⟩ := exists_countable_dense α in
⟨⟨f '' s, countable.image s_cnt f, h.dense_image h' s_dense⟩⟩ | lemma | dense_range.separable_space | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"continuous",
"dense_range",
"topological_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.exists_countable_dense_subset {α : Type*} [topological_space α]
{s : set α} [separable_space s] (hs : dense s) :
∃ t ⊆ s, t.countable ∧ dense t | let ⟨t, htc, htd⟩ := exists_countable_dense s
in ⟨coe '' t, image_subset_iff.2 $ λ x _, mem_preimage.2 $ subtype.coe_prop _, htc.image coe,
hs.dense_range_coe.dense_image continuous_subtype_val htd⟩ | lemma | dense.exists_countable_dense_subset | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"continuous_subtype_val",
"dense",
"subtype.coe_prop",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_countable_dense_subset_bot_top {α : Type*} [topological_space α]
[partial_order α] {s : set α} [separable_space s] (hs : dense s) :
∃ t ⊆ s, t.countable ∧ dense t ∧ (∀ x, is_bot x → x ∈ s → x ∈ t) ∧
(∀ x, is_top x → x ∈ s → x ∈ t) | begin
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩,
refine ⟨(t ∪ ({x | is_bot x} ∪ {x | is_top x})) ∩ s, _, _, _, _, _⟩,
exacts [inter_subset_right _ _,
(htc.union ((countable_is_bot α).union (countable_is_top α))).mono (inter_subset_left _ _),
htd.mono (subset_inter (subset_union_left ... | lemma | dense.exists_countable_dense_subset_bot_top | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense",
"is_bot",
"is_top",
"topological_space"
] | Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
to `s`. For a dense subset containing... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separable_space_univ {α : Type*} [topological_space α] [separable_space α] :
separable_space (univ : set α) | (equiv.set.univ α).symm.surjective.dense_range.separable_space (continuous_id.subtype_mk _) | instance | separable_space_univ | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"equiv.set.univ",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_countable_dense_bot_top (α : Type*) [topological_space α] [separable_space α]
[partial_order α] :
∃ s : set α, s.countable ∧ dense s ∧ (∀ x, is_bot x → x ∈ s) ∧ (∀ x, is_top x → x ∈ s) | by simpa using dense_univ.exists_countable_dense_subset_bot_top | lemma | exists_countable_dense_bot_top | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"dense",
"is_bot",
"is_top",
"topological_space"
] | If `α` is a separable topological space with a partial order, then there exists a countable
dense set `s : set α` that contains those of both bottom and top elements of `α` that actually
exist. For a dense set containing neither bot nor top elements, see
`exists_countable_dense_no_bot_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
first_countable_topology : Prop | (nhds_generated_countable : ∀a:α, (𝓝 a).is_countably_generated) | class | topological_space.first_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | A first-countable space is one in which every point has a
countable neighborhood basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_subseq [first_countable_topology α] {u : ℕ → α} {x : α}
(hx : map_cluster_pt x at_top u) :
∃ (ψ : ℕ → ℕ), (strict_mono ψ) ∧ (tendsto (u ∘ ψ) at_top (𝓝 x)) | subseq_tendsto_of_ne_bot hx | lemma | topological_space.first_countable_topology.tendsto_subseq | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"map_cluster_pt",
"strict_mono"
] | In a first-countable space, a cluster point `x` of a sequence
is the limit of some subsequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_countably_generated_nhds_within (x : α) [is_countably_generated (𝓝 x)] (s : set α) :
is_countably_generated (𝓝[s] x) | inf.is_countably_generated _ _ | instance | topological_space.is_countably_generated_nhds_within | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_countable_topology : Prop | (is_open_generated_countable [] :
∃ b : set (set α), b.countable ∧ t = topological_space.generate_from b) | class | topological_space.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"topological_space.generate_from"
] | A second-countable space is one with a countable basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.second_countable_topology
{b : set (set α)} (hb : is_topological_basis b) (hc : b.countable) :
second_countable_topology α | ⟨⟨b, hc, hb.eq_generate_from⟩⟩ | lemma | topological_space.is_topological_basis.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_countable_basis [second_countable_topology α] :
∃ b : set (set α), b.countable ∧ ∅ ∉ b ∧ is_topological_basis b | begin
obtain ⟨b, hb₁, hb₂⟩ := second_countable_topology.is_open_generated_countable α,
refine ⟨_, _, not_mem_diff_of_mem _, (is_topological_basis_of_subbasis hb₂).diff_empty⟩,
exacts [((countable_set_of_finite_subset hb₁).image _).mono (diff_subset _ _), rfl],
end | lemma | topological_space.exists_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_basis [second_countable_topology α] : set (set α) | (exists_countable_basis α).some | def | topological_space.countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | A countable topological basis of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_countable_basis [second_countable_topology α] : (countable_basis α).countable | (exists_countable_basis α).some_spec.1 | lemma | topological_space.countable_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
encodable_countable_basis [second_countable_topology α] :
encodable (countable_basis α) | (countable_countable_basis α).to_encodable | instance | topological_space.encodable_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"encodable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_nmem_countable_basis [second_countable_topology α] : ∅ ∉ countable_basis α | (exists_countable_basis α).some_spec.2.1 | lemma | topological_space.empty_nmem_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_basis_countable_basis [second_countable_topology α] :
is_topological_basis (countable_basis α) | (exists_countable_basis α).some_spec.2.2 | lemma | topological_space.is_basis_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_generate_from_countable_basis [second_countable_topology α] :
‹topological_space α› = generate_from (countable_basis α) | (is_basis_countable_basis α).eq_generate_from | lemma | topological_space.eq_generate_from_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_of_mem_countable_basis [second_countable_topology α] {s : set α}
(hs : s ∈ countable_basis α) : is_open s | (is_basis_countable_basis α).is_open hs | lemma | topological_space.is_open_of_mem_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_of_mem_countable_basis [second_countable_topology α] {s : set α}
(hs : s ∈ countable_basis α) : s.nonempty | nonempty_iff_ne_empty.2 $ ne_of_mem_of_not_mem hs $ empty_nmem_countable_basis α | lemma | topological_space.nonempty_of_mem_countable_basis | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_countable_topology.to_first_countable_topology
[second_countable_topology α] : first_countable_topology α | ⟨λ x, has_countable_basis.is_countably_generated $
⟨(is_basis_countable_basis α).nhds_has_basis, (countable_countable_basis α).mono $
inter_subset_left _ _⟩⟩ | instance | topological_space.second_countable_topology.to_first_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_countable_topology_induced (β)
[t : topological_space β] [second_countable_topology β] (f : α → β) :
@second_countable_topology α (t.induced f) | begin
rcases second_countable_topology.is_open_generated_countable β with ⟨b, hb, eq⟩,
refine { is_open_generated_countable := ⟨preimage f '' b, hb.image _, _⟩ },
rw [eq, induced_generate_from_eq]
end | lemma | topological_space.second_countable_topology_induced | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"induced_generate_from_eq",
"topological_space"
] | If `β` is a second-countable space, then its induced topology
via `f` on `α` is also second-countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.second_countable_topology (s : set α) [second_countable_topology α] :
second_countable_topology s | second_countable_topology_induced s α coe | instance | topological_space.subtype.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_countable_topology.to_separable_space
[second_countable_topology α] : separable_space α | begin
choose p hp using λ s : countable_basis α, nonempty_of_mem_countable_basis s.2,
exact ⟨⟨range p, countable_range _,
(is_basis_countable_basis α).dense_iff.2 $ λ o ho _, ⟨p ⟨o, ho⟩, hp _, mem_range_self _⟩⟩⟩
end | instance | topological_space.second_countable_topology.to_separable_space | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_countable_topology_of_countable_cover {ι} [encodable ι] {U : ι → set α}
[∀ i, second_countable_topology (U i)] (Uo : ∀ i, is_open (U i)) (hc : (⋃ i, U i) = univ) :
second_countable_topology α | begin
have : is_topological_basis (⋃ i, image (coe : U i → α) '' (countable_basis (U i))),
from is_topological_basis_of_cover Uo hc (λ i, is_basis_countable_basis (U i)),
exact this.second_countable_topology
(countable_Union $ λ i, (countable_countable_basis _).image _)
end | lemma | topological_space.second_countable_topology_of_countable_cover | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"encodable",
"is_open"
] | A countable open cover induces a second-countable topology if all open covers
are themselves second countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_Union_countable [second_countable_topology α]
{ι} (s : ι → set α) (H : ∀ i, is_open (s i)) :
∃ T : set ι, T.countable ∧ (⋃ i ∈ T, s i) = ⋃ i, s i | begin
let B := {b ∈ countable_basis α | ∃ i, b ⊆ s i},
choose f hf using λ b : B, b.2.2,
haveI : encodable B := ((countable_countable_basis α).mono (sep_subset _ _)).to_encodable,
refine ⟨_, countable_range f, (Union₂_subset_Union _ _).antisymm (sUnion_subset _)⟩,
rintro _ ⟨i, rfl⟩ x xs,
rcases (is_basis_co... | lemma | topological_space.is_open_Union_countable | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"encodable",
"is_open"
] | In a second-countable space, an open set, given as a union of open sets,
is equal to the union of countably many of those sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_sUnion_countable [second_countable_topology α]
(S : set (set α)) (H : ∀ s ∈ S, is_open s) :
∃ T : set (set α), T.countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S | let ⟨T, cT, hT⟩ := is_open_Union_countable (λ s:S, s.1) (λ s, H s.1 s.2) in
⟨subtype.val '' T, cT.image _,
image_subset_iff.2 $ λ ⟨x, xs⟩ xt, xs,
by rwa [sUnion_image, sUnion_eq_Union]⟩ | lemma | topological_space.is_open_sUnion_countable | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_cover_nhds [second_countable_topology α] {f : α → set α}
(hf : ∀ x, f x ∈ 𝓝 x) : ∃ s : set α, s.countable ∧ (⋃ x ∈ s, f x) = univ | begin
rcases is_open_Union_countable (λ x, interior (f x)) (λ x, is_open_interior) with ⟨s, hsc, hsU⟩,
suffices : (⋃ x ∈ s, interior (f x)) = univ,
from ⟨s, hsc, flip eq_univ_of_subset this $ Union₂_mono $ λ _ _, interior_subset⟩,
simp only [hsU, eq_univ_iff_forall, mem_Union],
exact λ x, ⟨x, mem_interior_i... | lemma | topological_space.countable_cover_nhds | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"interior",
"is_open_interior"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_cover_nhds_within [second_countable_topology α] {f : α → set α} {s : set α}
(hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t ⊆ s, t.countable ∧ s ⊆ (⋃ x ∈ t, f x) | begin
have : ∀ x : s, coe ⁻¹' (f x) ∈ 𝓝 x, from λ x, preimage_coe_mem_nhds_subtype.2 (hf x x.2),
rcases countable_cover_nhds this with ⟨t, htc, htU⟩,
refine ⟨coe '' t, subtype.coe_image_subset _ _, htc.image _, λ x hx, _⟩,
simp only [bUnion_image, eq_univ_iff_forall, ← preimage_Union, mem_preimage] at htU ⊢,
... | lemma | topological_space.countable_cover_nhds_within | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"subtype.coe_image_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis.sigma
{s : Π (i : ι), set (set (E i))} (hs : ∀ i, is_topological_basis (s i)) :
is_topological_basis (⋃ (i : ι), (λ u, ((sigma.mk i) '' u : set (Σ i, E i))) '' (s i)) | begin
apply is_topological_basis_of_open_of_nhds,
{ assume u hu,
obtain ⟨i, t, ts, rfl⟩ : ∃ (i : ι) (t : set (E i)), t ∈ s i ∧ sigma.mk i '' t = u,
by simpa only [mem_Union, mem_image] using hu,
exact is_open_map_sigma_mk _ ((hs i).is_open ts) },
{ rintros ⟨i, x⟩ u hxu u_open,
have hx : x ∈ sigm... | lemma | topological_space.is_topological_basis.sigma | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"is_open_map_sigma_mk"
] | In a disjoint union space `Σ i, E i`, one can form a topological basis by taking the union of
topological bases on each of the parts of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.sum
{s : set (set α)} (hs : is_topological_basis s) {t : set (set β)} (ht : is_topological_basis t) :
is_topological_basis (((λ u, sum.inl '' u) '' s) ∪ ((λ u, sum.inr '' u) '' t)) | begin
apply is_topological_basis_of_open_of_nhds,
{ assume u hu,
cases hu,
{ rcases hu with ⟨w, hw, rfl⟩,
exact open_embedding_inl.is_open_map w (hs.is_open hw) },
{ rcases hu with ⟨w, hw, rfl⟩,
exact open_embedding_inr.is_open_map w (ht.is_open hw) } },
{ rintros x u hxu u_open,
cases... | lemma | topological_space.is_topological_basis.sum | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [] | In a sum space `α ⊕ β`, one can form a topological basis by taking the union of
topological bases on each of the two components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.quotient_map {V : set (set X)} (hV : is_topological_basis V)
(h' : quotient_map π) (h : is_open_map π) :
is_topological_basis (set.image π '' V) | begin
apply is_topological_basis_of_open_of_nhds,
{ rintros - ⟨U, U_in_V, rfl⟩,
apply h U (hV.is_open U_in_V), },
{ intros y U y_in_U U_open,
obtain ⟨x, rfl⟩ := h'.surjective y,
let W := π ⁻¹' U,
have x_in_W : x ∈ W := y_in_U,
have W_open : is_open W := U_open.preimage h'.continuous,
obtai... | lemma | topological_space.is_topological_basis.quotient_map | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open",
"is_open_map",
"quotient_map",
"set.image",
"set.image_subset"
] | The image of a topological basis under an open quotient map is a topological basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map.second_countable_topology [second_countable_topology X] (h' : quotient_map π)
(h : is_open_map π) :
second_countable_topology Y | { is_open_generated_countable :=
begin
obtain ⟨V, V_countable, V_no_empty, V_generates⟩ := exists_countable_basis X,
exact ⟨set.image π '' V, V_countable.image (set.image π),
(V_generates.quotient_map h' h).eq_generate_from⟩,
end } | lemma | topological_space.quotient_map.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open_map",
"quotient_map",
"set.image"
] | A second countable space is mapped by an open quotient map to a second countable space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis.quotient {V : set (set X)}
(hV : is_topological_basis V) (h : is_open_map (quotient.mk : X → quotient S)) :
is_topological_basis (set.image (quotient.mk : X → quotient S) '' V) | hV.quotient_map quotient_map_quotient_mk h | lemma | topological_space.is_topological_basis.quotient | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open_map",
"quotient_map_quotient_mk",
"set.image"
] | The image of a topological basis "downstairs" in an open quotient is a topological basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.second_countable_topology [second_countable_topology X]
(h : is_open_map (quotient.mk : X → quotient S)) :
second_countable_topology (quotient S) | quotient_map_quotient_mk.second_countable_topology h | lemma | topological_space.quotient.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"is_open_map"
] | An open quotient of a second countable space is second countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inducing.second_countable_topology [second_countable_topology β]
(hf : inducing f) : second_countable_topology α | by { rw hf.1, exact second_countable_topology_induced α β f } | lemma | inducing.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.