statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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continuous_on_extend_from [regular_space Y] {f : X → Y} {A B : set X} (hB : B ⊆ closure A)
(hf : ∀ x ∈ B, ∃ y, tendsto f (𝓝[A] x) (𝓝 y)) : continuous_on (extend_from A f) B | begin
set φ := extend_from A f,
intros x x_in,
suffices : ∀ V' ∈ 𝓝 (φ x), is_closed V' → φ ⁻¹' V' ∈ 𝓝[B] x,
by simpa [continuous_within_at, (closed_nhds_basis _).tendsto_right_iff],
intros V' V'_in V'_closed,
obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, is_open V ∧ V ∩ A ⊆ f ⁻¹' V',
{ have := tendsto_exte... | lemma | continuous_on_extend_from | topology | src/topology/extend_from.lean | [
"topology.separation"
] | [
"closed_nhds_basis",
"closure",
"continuous_on",
"continuous_within_at",
"extend_from",
"inter_mem_nhds_within",
"is_closed",
"is_open",
"is_open.mem_nhds",
"nhds_within_basis_open",
"regular_space",
"tendsto_extend_from"
] | If `f` is a function to a T₃ space `Y` which has a limit within `A` at any
point of a set `B ⊆ closure A`, then `extend_from A f` is continuous on `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_extend_from [regular_space Y] {f : X → Y} {A : set X} (hA : dense A)
(hf : ∀ x, ∃ y, tendsto f (𝓝[A] x) (𝓝 y)) : continuous (extend_from A f) | begin
rw continuous_iff_continuous_on_univ,
exact continuous_on_extend_from (λ x _, hA x) (by simpa using hf)
end | lemma | continuous_extend_from | topology | src/topology/extend_from.lean | [
"topology.separation"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_extend_from",
"dense",
"extend_from",
"regular_space"
] | If a function `f` to a T₃ space `Y` has a limit within a
dense set `A` for any `x`, then `extend_from A f` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extremally_disconnected : Prop | (open_closure : ∀ U : set X, is_open U → is_open (closure U)) | class | extremally_disconnected | topology | src/topology/extremally_disconnected.lean | [
"topology.stone_cech"
] | [
"closure",
"is_open"
] | An extremally disconnected topological space is a space
in which the closure of every open set is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_t2.projective : Prop | Π {Y Z : Type u} [topological_space Y] [topological_space Z],
by exactI Π [compact_space Y] [t2_space Y] [compact_space Z] [t2_space Z],
Π {f : X → Z} {g : Y → Z} (hf : continuous f) (hg : continuous g) (g_sur : surjective g),
∃ h : X → Y, continuous h ∧ g ∘ h = f | def | compact_t2.projective | topology | src/topology/extremally_disconnected.lean | [
"topology.stone_cech"
] | [
"compact_space",
"continuous",
"t2_space",
"topological_space"
] | The assertion `compact_t2.projective` states that given continuous maps
`f : X → Z` and `g : Y → Z` with `g` surjective between `t_2`, compact topological spaces,
there exists a continuous lift `h : X → Y`, such that `f = g ∘ h`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stone_cech.projective [discrete_topology X] : compact_t2.projective (stone_cech X) | begin
introsI Y Z _tsY _tsZ _csY _t2Y _csZ _csZ f g hf hg g_sur,
let s : Z → Y := λ z, classical.some $ g_sur z,
have hs : g ∘ s = id := funext (λ z, classical.some_spec (g_sur z)),
let t := s ∘ f ∘ stone_cech_unit,
have ht : continuous t := continuous_of_discrete_topology,
let h : stone_cech X → Y := stone... | lemma | stone_cech.projective | topology | src/topology/extremally_disconnected.lean | [
"topology.stone_cech"
] | [
"compact_t2.projective",
"continuous",
"continuous_of_discrete_topology",
"continuous_stone_cech_extend",
"discrete_topology",
"stone_cech",
"stone_cech_extend",
"stone_cech_extend_extends",
"stone_cech_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_t2.projective.extremally_disconnected [compact_space X] [t2_space X]
(h : compact_t2.projective X) :
extremally_disconnected X | begin
refine { open_closure := λ U hU, _ },
let Z₁ : set (X × bool) := Uᶜ ×ˢ {tt},
let Z₂ : set (X × bool) := closure U ×ˢ {ff},
let Z : set (X × bool) := Z₁ ∪ Z₂,
have hZ₁₂ : disjoint Z₁ Z₂ := disjoint_left.2 (λ x hx₁ hx₂, by cases hx₁.2.symm.trans hx₂.2),
have hZ₁ : is_closed Z₁ := hU.is_closed_compl.prod... | lemma | compact_t2.projective.extremally_disconnected | topology | src/topology/extremally_disconnected.lean | [
"topology.stone_cech"
] | [
"closure",
"closure_minimal",
"compact_space",
"compact_t2.projective",
"continuous",
"continuous_id",
"continuous_subtype_val",
"disjoint",
"extremally_disconnected",
"is_closed",
"set.mem_singleton",
"set.preimage_comp",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Iic_principal {s : set α} : is_open (Iic (𝓟 s)) | generate_open.basic _ (mem_range_self _) | lemma | filter.is_open_Iic_principal | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_set_of_mem {s : set α} : is_open {l : filter α | s ∈ l} | by simpa only [Iic_principal] using is_open_Iic_principal | lemma | filter.is_open_set_of_mem | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_Iic_principal :
is_topological_basis (range (Iic ∘ 𝓟 : set α → set (filter α))) | { exists_subset_inter :=
begin
rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl,
exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, subset.rfl⟩
end,
sUnion_eq := sUnion_eq_univ_iff.2 $ λ l, ⟨Iic ⊤, ⟨univ, congr_arg Iic principal_univ⟩, le_top⟩,
eq_generate_from := rfl } | lemma | filter.is_topological_basis_Iic_principal | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff {s : set (filter α)} :
is_open s ↔ ∃ T : set (set α), s = ⋃ t ∈ T, Iic (𝓟 t) | is_topological_basis_Iic_principal.open_iff_eq_sUnion.trans $
by simp only [exists_subset_range_iff, sUnion_image] | lemma | filter.is_open_iff | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq (l : filter α) : 𝓝 l = l.lift' (Iic ∘ 𝓟) | nhds_generate_from.trans $ by simp only [mem_set_of_eq, and_comm (l ∈ _), infi_and, infi_range,
filter.lift', filter.lift, (∘), mem_Iic, le_principal_iff] | lemma | filter.nhds_eq | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"filter.lift",
"filter.lift'",
"infi_and",
"infi_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq' (l : filter α) : 𝓝 l = l.lift' (λ s, {l' | s ∈ l'}) | by simpa only [(∘), Iic_principal] using nhds_eq l | lemma | filter.nhds_eq' | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds {la : filter α} {lb : filter β} {f : α → filter β} :
tendsto f la (𝓝 lb) ↔ ∀ s ∈ lb, ∀ᶠ a in la, s ∈ f a | by simp only [nhds_eq', tendsto_lift', mem_set_of_eq] | lemma | filter.tendsto_nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis.nhds {l : filter α} {p : ι → Prop} {s : ι → set α} (h : has_basis l p s) :
has_basis (𝓝 l) p (λ i, Iic (𝓟 (s i))) | by { rw nhds_eq, exact h.lift' monotone_principal.Iic } | lemma | filter.has_basis.nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis.nhds' {l : filter α} {p : ι → Prop} {s : ι → set α} (h : has_basis l p s) :
has_basis (𝓝 l) p (λ i, {l' | s i ∈ l'}) | by simpa only [Iic_principal] using h.nhds | lemma | filter.has_basis.nhds' | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_iff {l : filter α} {S : set (filter α)} :
S ∈ 𝓝 l ↔ ∃ t ∈ l, Iic (𝓟 t) ⊆ S | l.basis_sets.nhds.mem_iff | lemma | filter.mem_nhds_iff | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_iff' {l : filter α} {S : set (filter α)} :
S ∈ 𝓝 l ↔ ∃ t ∈ l, ∀ ⦃l' : filter α⦄, t ∈ l' → l' ∈ S | l.basis_sets.nhds'.mem_iff | lemma | filter.mem_nhds_iff' | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bot : 𝓝 (⊥ : filter α) = pure ⊥ | by simp [nhds_eq, lift'_bot monotone_principal.Iic] | lemma | filter.nhds_bot | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top : 𝓝 (⊤ : filter α) = ⊤ | by simp [nhds_eq] | lemma | filter.nhds_top | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"nhds_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_principal (s : set α) : 𝓝 (𝓟 s) = 𝓟 (Iic (𝓟 s)) | (has_basis_principal s).nhds.eq_of_same_basis (has_basis_principal _) | lemma | filter.nhds_principal | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_pure (x : α) : 𝓝 (pure x : filter α) = 𝓟 {⊥, pure x} | by rw [← principal_singleton, nhds_principal, principal_singleton, Iic_pure] | lemma | filter.nhds_pure | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_infi (f : ι → filter α) : 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i) | by { simp only [nhds_eq], apply lift'_infi_of_map_univ; simp } | lemma | filter.nhds_infi | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"nhds_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inf (l₁ l₂ : filter α) : 𝓝 (l₁ ⊓ l₂) = 𝓝 l₁ ⊓ 𝓝 l₂ | by simpa only [infi_bool_eq] using nhds_infi (λ b, cond b l₁ l₂) | lemma | filter.nhds_inf | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"infi_bool_eq",
"nhds_inf",
"nhds_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_nhds : monotone (𝓝 : filter α → filter (filter α)) | monotone.of_map_inf nhds_inf | lemma | filter.monotone_nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"monotone",
"monotone.of_map_inf",
"nhds_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inter_nhds (l : filter α) : ⋂₀ {s | s ∈ 𝓝 l} = Iic l | by simp only [nhds_eq, sInter_lift'_sets monotone_principal.Iic, Iic, le_principal_iff,
← set_of_forall, ← filter.le_def] | lemma | filter.Inter_nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"filter.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_mono {l₁ l₂ : filter α} : 𝓝 l₁ ≤ 𝓝 l₂ ↔ l₁ ≤ l₂ | begin
refine ⟨λ h, _, λ h, monotone_nhds h⟩,
rw [← Iic_subset_Iic, ← Inter_nhds, ← Inter_nhds],
exact sInter_subset_sInter h
end | lemma | filter.nhds_mono | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"nhds_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_interior {s : set (filter α)} {l : filter α} :
l ∈ interior s ↔ ∃ t ∈ l, Iic (𝓟 t) ⊆ s | by rw [mem_interior_iff_mem_nhds, mem_nhds_iff] | lemma | filter.mem_interior | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"interior",
"mem_interior",
"mem_interior_iff_mem_nhds",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure {s : set (filter α)} {l : filter α} :
l ∈ closure s ↔ ∀ t ∈ l, ∃ l' ∈ s, t ∈ l' | by simp only [closure_eq_compl_interior_compl, filter.mem_interior, mem_compl_iff, not_exists,
not_forall, not_not, exists_prop, not_and, and_comm, subset_def, mem_Iic, le_principal_iff] | lemma | filter.mem_closure | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"closure",
"closure_eq_compl_interior_compl",
"exists_prop",
"filter",
"filter.mem_interior",
"not_and",
"not_exists",
"not_forall",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_singleton (l : filter α) : closure {l} = Ici l | by { ext l', simp [filter.mem_closure, filter.le_def] } | lemma | filter.closure_singleton | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"closure",
"closure_singleton",
"filter",
"filter.le_def",
"filter.mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_le {l₁ l₂ : filter α} : l₁ ⤳ l₂ ↔ l₁ ≤ l₂ | by simp only [specializes_iff_closure_subset, filter.closure_singleton, Ici_subset_Ici] | lemma | filter.specializes_iff_le | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"filter.closure_singleton",
"specializes_iff_closure_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_at_top [preorder α] : 𝓝 at_top = ⨅ x : α, 𝓟 (Iic (𝓟 (Ici x))) | by simp only [at_top, nhds_infi, nhds_principal] | lemma | filter.nhds_at_top | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"nhds_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_at_top_iff [preorder β] {l : filter α} {f : α → filter β} :
tendsto f l (𝓝 at_top) ↔ ∀ y, ∀ᶠ a in l, Ici y ∈ f a | by simp only [nhds_at_top, tendsto_infi, tendsto_principal, mem_Iic, le_principal_iff] | lemma | filter.tendsto_nhds_at_top_iff | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_at_bot [preorder α] : 𝓝 at_bot = ⨅ x : α, 𝓟 (Iic (𝓟 (Iic x))) | @nhds_at_top αᵒᵈ _ | lemma | filter.nhds_at_bot | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_at_bot_iff [preorder β] {l : filter α} {f : α → filter β} :
tendsto f l (𝓝 at_bot) ↔ ∀ y, ∀ᶠ a in l, Iic y ∈ f a | @filter.tendsto_nhds_at_top_iff α βᵒᵈ _ _ _ | lemma | filter.tendsto_nhds_at_bot_iff | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"filter.tendsto_nhds_at_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_nhds (x : X) :
𝓝 (𝓝 x) = ⨅ (s : set X) (hs : is_open s) (hx : x ∈ s), 𝓟 (Iic (𝓟 s)) | by simp only [(nhds_basis_opens x).nhds.eq_binfi, infi_and, @infi_comm _ (_ ∈ _)] | lemma | filter.nhds_nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"infi_and",
"infi_comm",
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_nhds : inducing (𝓝 : X → filter X) | inducing_iff_nhds.2 $ λ x, (nhds_def' _).trans $
by simp only [nhds_nhds, comap_infi, comap_principal, Iic_principal, preimage_set_of_eq,
← mem_interior_iff_mem_nhds, set_of_mem_eq, is_open.interior_eq] { contextual := tt } | lemma | filter.inducing_nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter",
"inducing",
"is_open.interior_eq",
"mem_interior_iff_mem_nhds",
"nhds_def'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_nhds : continuous (𝓝 : X → filter X) | inducing_nhds.continuous | lemma | filter.continuous_nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"continuous",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.nhds {f : α → X} {l : filter α} {x : X} (h : tendsto f l (𝓝 x)) :
tendsto (𝓝 ∘ f) l (𝓝 (𝓝 x)) | (continuous_nhds.tendsto _).comp h | lemma | filter.tendsto.nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.nhds (h : continuous_within_at f s x) :
continuous_within_at (𝓝 ∘ f) s x | h.nhds | lemma | continuous_within_at.nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.nhds (h : continuous_at f x) : continuous_at (𝓝 ∘ f) x | h.nhds | lemma | continuous_at.nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.nhds (h : continuous_on f s) : continuous_on (𝓝 ∘ f) s | λ x hx, (h x hx).nhds | lemma | continuous_on.nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"continuous_on",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.nhds (h : continuous f) : continuous (𝓝 ∘ f) | filter.continuous_nhds.comp h | lemma | continuous.nhds | topology | src/topology/filter.lean | [
"order.filter.lift",
"topology.separation",
"data.set.intervals.monotone"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_data extends glue_data Top | (f_open : ∀ i j, open_embedding (f i j))
(f_mono := λ i j, (Top.mono_iff_injective _).mpr (f_open i j).to_embedding.inj) | structure | Top.glue_data | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top",
"Top.mono_iff_injective",
"open_embedding"
] | A family of gluing data consists of
1. An index type `J`
2. An object `U i` for each `i : J`.
3. An object `V i j` for each `i j : J`.
(Note that this is `J × J → Top` rather than `J → J → Top` to connect to the
limits library easier.)
4. An open embedding `f i j : V i j ⟶ U i` for each `i j : ι`.
5. A transition m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_surjective : function.surjective 𝖣 .π | (Top.epi_iff_surjective 𝖣 .π).mp infer_instance | lemma | Top.glue_data.π_surjective | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.epi_iff_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff (U : set 𝖣 .glued) : is_open U ↔ ∀ i, is_open (𝖣 .ι i ⁻¹' U) | begin
delta category_theory.glue_data.ι,
simp_rw ← multicoequalizer.ι_sigma_π 𝖣 .diagram,
rw ← (homeo_of_iso (multicoequalizer.iso_coequalizer 𝖣 .diagram).symm).is_open_preimage,
rw [coequalizer_is_open_iff, colimit_is_open_iff.{u}],
split,
{ intros h j, exact h ⟨j⟩, },
{ intros h j, cases j, exact h j,... | lemma | Top.glue_data.is_open_iff | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"category_theory.glue_data.ι",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_jointly_surjective (x : 𝖣 .glued) : ∃ i (y : D.U i), 𝖣 .ι i y = x | 𝖣 .ι_jointly_surjective (forget Top) x | lemma | Top.glue_data.ι_jointly_surjective | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel (a b : Σ i, ((D.U i : Top) : Type*)) : Prop | a = b ∨ ∃ (x : D.V (a.1, b.1)) , D.f _ _ x = a.2 ∧ D.f _ _ (D.t _ _ x) = b.2 | def | Top.glue_data.rel | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top",
"rel"
] | An equivalence relation on `Σ i, D.U i` that holds iff `𝖣 .ι i x = 𝖣 .ι j y`.
See `Top.glue_data.ι_eq_iff_rel`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_equiv : equivalence D.rel | ⟨ λ x, or.inl (refl x),
begin
rintros a b (⟨⟨⟩⟩|⟨x,e₁,e₂⟩),
exacts [or.inl rfl, or.inr ⟨D.t _ _ x, by simp [e₁, e₂]⟩]
end,
begin
rintros ⟨i,a⟩ ⟨j,b⟩ ⟨k,c⟩ (⟨⟨⟩⟩|⟨x,e₁,e₂⟩), exact id,
rintro (⟨⟨⟩⟩|⟨y,e₃,e₄⟩), exact or.inr ⟨x,e₁,e₂⟩,
let z := (pullback_iso_prod_subtype (D.f j i) (D.f j k)).inv ⟨... | lemma | Top.glue_data.rel_equiv | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"continuous_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eqv_gen_of_π_eq {x y : ∐ D.U} (h : 𝖣 .π x = 𝖣 .π y) :
eqv_gen (types.coequalizer_rel 𝖣 .diagram.fst_sigma_map 𝖣 .diagram.snd_sigma_map) x y | begin
delta glue_data.π multicoequalizer.sigma_π at h,
simp_rw comp_app at h,
replace h := (Top.mono_iff_injective (multicoequalizer.iso_coequalizer 𝖣 .diagram).inv).mp _ h,
let diagram := parallel_pair 𝖣 .diagram.fst_sigma_map 𝖣 .diagram.snd_sigma_map ⋙ forget _,
have : colimit.ι diagram one x = colimit.ι... | lemma | Top.glue_data.eqv_gen_of_π_eq | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_eq_iff_rel (i j : D.J) (x : D.U i) (y : D.U j) :
𝖣 .ι i x = 𝖣 .ι j y ↔ D.rel ⟨i, x⟩ ⟨j, y⟩ | begin
split,
{ delta glue_data.ι,
simp_rw ← multicoequalizer.ι_sigma_π,
intro h,
rw ← (show _ = sigma.mk i x,
from concrete_category.congr_hom (sigma_iso_sigma.{u} D.U).inv_hom_id _),
rw ← (show _ = sigma.mk j y,
from concrete_category.congr_hom (sigma_iso_sigma.{u} D.U).inv_hom_id _),
... | lemma | Top.glue_data.ι_eq_iff_rel | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.comp_app",
"Top.sigma_iso_sigma_inv_apply",
"continuous_map.to_fun_eq_coe",
"eqv_gen.mono",
"inv_image.equivalence"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_injective (i : D.J) : function.injective (𝖣 .ι i) | begin
intros x y h,
rcases (D.ι_eq_iff_rel _ _ _ _).mp h with (⟨⟨⟩⟩|⟨_,e₁,e₂⟩),
{ refl },
{ dsimp only at *, cases e₁, cases e₂, simp }
end | lemma | Top.glue_data.ι_injective | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_mono (i : D.J) : mono (𝖣 .ι i) | (Top.mono_iff_injective _).mpr (D.ι_injective _) | instance | Top.glue_data.ι_mono | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_inter (i j : D.J) :
set.range (𝖣 .ι i) ∩ set.range (𝖣 .ι j) = set.range (D.f i j ≫ 𝖣 .ι _) | begin
ext x,
split,
{ rintro ⟨⟨x₁, eq₁⟩, ⟨x₂, eq₂⟩⟩,
obtain (⟨⟨⟩⟩|⟨y,e₁,e₂⟩) := (D.ι_eq_iff_rel _ _ _ _).mp (eq₁.trans eq₂.symm),
{ exact ⟨inv (D.f i i) x₁, by simp [eq₁]⟩ },
{ dsimp only at *, substs e₁ eq₁, exact ⟨y, by simp⟩ } },
{ rintro ⟨x, hx⟩,
exact ⟨⟨D.f i j x, hx⟩, ⟨D.f j i (D.t _ _ x),... | lemma | Top.glue_data.image_inter | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_range (i j : D.J) :
𝖣 .ι j ⁻¹' (set.range (𝖣 .ι i)) = set.range (D.f j i) | by rw [← set.preimage_image_eq (set.range (D.f j i)) (D.ι_injective j), ← set.image_univ,
← set.image_univ, ←set.image_comp, ←coe_comp, set.image_univ,set.image_univ,
← image_inter, set.preimage_range_inter] | lemma | Top.glue_data.preimage_range | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"set.image_univ",
"set.preimage_image_eq",
"set.preimage_range_inter",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_image_eq_image (i j : D.J) (U : set (𝖣 .U i)) :
𝖣 .ι j ⁻¹' (𝖣 .ι i '' U) = D.f _ _ '' ((D.t j i ≫ D.f _ _) ⁻¹' U) | begin
have : D.f _ _ ⁻¹' (𝖣 .ι j ⁻¹' (𝖣 .ι i '' U)) = (D.t j i ≫ D.f _ _) ⁻¹' U,
{ ext x,
conv_rhs { rw ← set.preimage_image_eq U (D.ι_injective _) },
generalize : 𝖣 .ι i '' U = U',
simp },
rw [← this, set.image_preimage_eq_inter_range],
symmetry,
apply set.inter_eq_self_of_subset_left,
rw ← ... | lemma | Top.glue_data.preimage_image_eq_image | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"set.image_preimage_eq_inter_range",
"set.image_subset_range",
"set.inter_eq_self_of_subset_left",
"set.preimage_image_eq",
"set.preimage_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_image_eq_image' (i j : D.J) (U : set (𝖣 .U i)) :
𝖣 .ι j ⁻¹' (𝖣 .ι i '' U) = (D.t i j ≫ D.f _ _) '' ((D.f _ _) ⁻¹' U) | begin
convert D.preimage_image_eq_image i j U using 1,
rw [coe_comp, coe_comp, ← set.image_image],
congr' 1,
rw [← set.eq_preimage_iff_image_eq, set.preimage_preimage],
change _ = (D.t i j ≫ D.t j i ≫ _) ⁻¹' _,
rw 𝖣 .t_inv_assoc,
rw ← is_iso_iff_bijective,
apply (forget Top).map_is_iso
end | lemma | Top.glue_data.preimage_image_eq_image' | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top",
"set.eq_preimage_iff_image_eq",
"set.image_image",
"set.preimage_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_image_open (i : D.J) (U : opens (𝖣 .U i)) : is_open (𝖣 .ι i '' U) | begin
rw is_open_iff,
intro j,
rw preimage_image_eq_image,
apply (D.f_open _ _).is_open_map,
apply (D.t j i ≫ D.f i j).continuous_to_fun.is_open_preimage,
exact U.is_open
end | lemma | Top.glue_data.open_image_open | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"is_open",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_open_embedding (i : D.J) : open_embedding (𝖣 .ι i) | open_embedding_of_continuous_injective_open
(𝖣 .ι i).continuous_to_fun (D.ι_injective i) (λ U h, D.open_image_open i ⟨U, h⟩) | lemma | Top.glue_data.ι_open_embedding | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_core | {J : Type u}
(U : J → Top.{u})
(V : Π i, J → opens (U i))
(t : Π i j, (opens.to_Top _).obj (V i j) ⟶ (opens.to_Top _).obj (V j i))
(V_id : ∀ i, V i i = ⊤)
(t_id : ∀ i, ⇑(t i i) = id)
(t_inter : ∀ ⦃i j⦄ k (x : V i j), ↑x ∈ V i k → @coe (V j i) (U j) _ (t i j x) ∈ V j k)
(cocycle : ∀ i j k (x : V i j) (h : ↑x ∈ V i k),
... | structure | Top.glue_data.mk_core | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [] | A family of gluing data consists of
1. An index type `J`
2. A bundled topological space `U i` for each `i : J`.
3. An open set `V i j ⊆ U i` for each `i j : J`.
4. A transition map `t i j : V i j ⟶ V j i` for each `i j : ι`.
such that
6. `V i i = U i`.
7. `t i i` is the identity.
8. For each `x ∈ V i j ∩ V i k`, `t i j... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_core.t_inv (h : mk_core) (i j : h.J) (x : h.V j i) : h.t i j ((h.t j i) x) = x | begin
have := h.cocycle j i j x _,
rw h.t_id at this,
convert subtype.eq this,
{ ext, refl },
all_goals { rw h.V_id, trivial }
end | lemma | Top.glue_data.mk_core.t_inv | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_core.t' (h : mk_core.{u}) (i j k : h.J) : pullback (h.V i j).inclusion (h.V i k).inclusion ⟶
pullback (h.V j k).inclusion (h.V j i).inclusion | begin
refine (pullback_iso_prod_subtype _ _).hom ≫ ⟨_, _⟩ ≫ (pullback_iso_prod_subtype _ _).inv,
{ intro x,
refine ⟨⟨⟨(h.t i j x.1.1).1, _⟩, h.t i j x.1.1⟩, rfl⟩,
rcases x with ⟨⟨⟨x, hx⟩, ⟨x', hx'⟩⟩, (rfl : x = x')⟩,
exact h.t_inter _ ⟨x, hx⟩ hx' },
continuity,
end | def | Top.glue_data.mk_core.t' | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"continuity"
] | (Implementation) the restricted transition map to be fed into `glue_data`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' (h : mk_core.{u}) : Top.glue_data | { J := h.J,
U := h.U,
V := λ i, (opens.to_Top _).obj (h.V i.1 i.2),
f := λ i j, (h.V i j).inclusion ,
f_id := λ i, (h.V_id i).symm ▸ is_iso.of_iso (opens.inclusion_top_iso (h.U i)),
f_open := λ (i j : h.J), (h.V i j).open_embedding,
t := h.t,
t_id := λ i, by { ext, rw h.t_id, refl },
t' := h.t',
t_fac... | def | Top.glue_data.mk' | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.comp_app",
"Top.glue_data",
"Top.id_app",
"continuous_map.coe_mk",
"mk'",
"open_embedding",
"prod.mk.inj_iff",
"subtype.coe_mk",
"subtype.mk_eq_mk",
"subtype.val_eq_coe"
] | This is a constructor of `Top.glue_data` whose arguments are in terms of elements and
intersections rather than subobjects and pullbacks. Please refer to `Top.glue_data.mk_core` for
details. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_open_subsets : Top.glue_data.{u} | mk'.{u}
{ J := J,
U := λ i, (opens.to_Top $ Top.of α).obj (U i),
V := λ i j, (opens.map $ opens.inclusion _).obj (U j),
t := λ i j, ⟨λ x, ⟨⟨x.1.1, x.2⟩, x.1.2⟩, by continuity⟩,
V_id := λ i, by { ext, cases U i, simp },
t_id := λ i, by { ext, refl },
t_inter := λ i j k x hx, hx,
cocycle := λ i j k x h, rfl... | def | Top.glue_data.of_open_subsets | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.of"
] | We may construct a glue data from a family of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_open_subsets_glue : (of_open_subsets U).to_glue_data.glued ⟶ Top.of α | multicoequalizer.desc _ _ (λ x, opens.inclusion _) (by { rintro ⟨i, j⟩, ext x, refl }) | def | Top.glue_data.from_open_subsets_glue | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.of"
] | The canonical map from the glue of a family of open subsets `α` into `α`.
This map is an open embedding (`from_open_subsets_glue_open_embedding`),
and its range is `⋃ i, (U i : set α)` (`range_from_open_subsets_glue`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_from_open_subsets_glue (i : J) :
(of_open_subsets U).to_glue_data.ι i ≫ from_open_subsets_glue U = opens.inclusion _ | multicoequalizer.π_desc _ _ _ _ _ | lemma | Top.glue_data.ι_from_open_subsets_glue | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_open_subsets_glue_injective : function.injective (from_open_subsets_glue U) | begin
intros x y e,
obtain ⟨i, ⟨x, hx⟩, rfl⟩ := (of_open_subsets U).ι_jointly_surjective x,
obtain ⟨j, ⟨y, hy⟩, rfl⟩ := (of_open_subsets U).ι_jointly_surjective y,
rw [ι_from_open_subsets_glue_apply, ι_from_open_subsets_glue_apply] at e,
change x = y at e,
subst e,
rw (of_open_subsets U).ι_eq_iff_rel,
r... | lemma | Top.glue_data.from_open_subsets_glue_injective | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_open_subsets_glue_is_open_map : is_open_map (from_open_subsets_glue U) | begin
intros s hs,
rw (of_open_subsets U).is_open_iff at hs,
rw is_open_iff_forall_mem_open,
rintros _ ⟨x, hx, rfl⟩,
obtain ⟨i, ⟨x, hx'⟩, rfl⟩ := (of_open_subsets U).ι_jointly_surjective x,
use from_open_subsets_glue U '' s ∩ set.range (@opens.inclusion (Top.of α) (U i)),
use set.inter_subset_left _ _,
... | lemma | Top.glue_data.from_open_subsets_glue_is_open_map | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"Top.of",
"is_open_iff_forall_mem_open",
"is_open_map",
"set.image_preimage_eq_inter_range",
"set.inter_subset_left",
"set.mem_range_self",
"set.preimage_comp",
"set.preimage_image_eq",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_open_subsets_glue_open_embedding : open_embedding (from_open_subsets_glue U) | open_embedding_of_continuous_injective_open (continuous_map.continuous_to_fun _)
(from_open_subsets_glue_injective U) (from_open_subsets_glue_is_open_map U) | lemma | Top.glue_data.from_open_subsets_glue_open_embedding | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_from_open_subsets_glue : set.range (from_open_subsets_glue U) = ⋃ i, (U i : set α) | begin
ext,
split,
{ rintro ⟨x, rfl⟩,
obtain ⟨i, ⟨x, hx'⟩, rfl⟩ := (of_open_subsets U).ι_jointly_surjective x,
rw ι_from_open_subsets_glue_apply,
exact set.subset_Union _ i hx' },
{ rintro ⟨_, ⟨i, rfl⟩, hx⟩,
refine ⟨(of_open_subsets U).to_glue_data.ι i ⟨x, hx⟩, ι_from_open_subsets_glue_apply _ _ ... | lemma | Top.glue_data.range_from_open_subsets_glue | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"set.range",
"set.subset_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_cover_glue_homeo (h : (⋃ i, (U i : set α)) = set.univ) :
(of_open_subsets U).to_glue_data.glued ≃ₜ α | homeomorph.homeomorph_of_continuous_open
(equiv.of_bijective (from_open_subsets_glue U)
⟨from_open_subsets_glue_injective U,
set.range_iff_surjective.mp ((range_from_open_subsets_glue U).symm ▸ h)⟩)
(from_open_subsets_glue U).2 (from_open_subsets_glue_is_open_map U) | def | Top.glue_data.open_cover_glue_homeo | topology | src/topology/gluing.lean | [
"category_theory.glue_data",
"category_theory.concrete_category.elementwise",
"topology.category.Top.limits.pullbacks",
"topology.category.Top.opens"
] | [
"equiv.of_bijective",
"homeomorph.homeomorph_of_continuous_open"
] | The gluing of an open cover is homeomomorphic to the original space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ (s : set α) : Prop | ∃T : set (set α), (∀t ∈ T, is_open t) ∧ T.countable ∧ s = (⋂₀ T) | def | is_Gδ | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_open"
] | A Gδ set is a countable intersection of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open.is_Gδ {s : set α} (h : is_open s) : is_Gδ s | ⟨{s}, by simp [h], countable_singleton _, (set.sInter_singleton _).symm⟩ | lemma | is_open.is_Gδ | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_open",
"set.sInter_singleton"
] | An open set is a Gδ set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ_empty : is_Gδ (∅ : set α) | is_open_empty.is_Gδ | lemma | is_Gδ_empty | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_univ : is_Gδ (univ : set α) | is_open_univ.is_Gδ | lemma | is_Gδ_univ | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_bInter_of_open {I : set ι} (hI : I.countable) {f : ι → set α}
(hf : ∀i ∈ I, is_open (f i)) : is_Gδ (⋂i∈I, f i) | ⟨f '' I, by rwa ball_image_iff, hI.image _, by rw sInter_image⟩ | lemma | is_Gδ_bInter_of_open | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_Inter_of_open [encodable ι] {f : ι → set α}
(hf : ∀i, is_open (f i)) : is_Gδ (⋂i, f i) | ⟨range f, by rwa forall_range_iff, countable_range _, by rw sInter_range⟩ | lemma | is_Gδ_Inter_of_open | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"encodable",
"is_Gδ",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_Inter [encodable ι] {s : ι → set α} (hs : ∀ i, is_Gδ (s i)) : is_Gδ (⋂ i, s i) | begin
choose T hTo hTc hTs using hs,
obtain rfl : s = λ i, ⋂₀ T i := funext hTs,
refine ⟨⋃ i, T i, _, countable_Union hTc, (sInter_Union _).symm⟩,
simpa [@forall_swap ι] using hTo
end | lemma | is_Gδ_Inter | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"encodable",
"forall_swap",
"is_Gδ"
] | The intersection of an encodable family of Gδ sets is a Gδ set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ_bInter {s : set ι} (hs : s.countable) {t : Π i ∈ s, set α}
(ht : ∀ i ∈ s, is_Gδ (t i ‹_›)) : is_Gδ (⋂ i ∈ s, t i ‹_›) | begin
rw [bInter_eq_Inter],
haveI := hs.to_encodable,
exact is_Gδ_Inter (λ x, ht x x.2)
end | lemma | is_Gδ_bInter | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_sInter {S : set (set α)} (h : ∀s∈S, is_Gδ s) (hS : S.countable) : is_Gδ (⋂₀ S) | by simpa only [sInter_eq_bInter] using is_Gδ_bInter hS h | lemma | is_Gδ_sInter | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_bInter"
] | A countable intersection of Gδ sets is a Gδ set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ.inter {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) : is_Gδ (s ∩ t) | by { rw inter_eq_Inter, exact is_Gδ_Inter (bool.forall_bool.2 ⟨ht, hs⟩) } | lemma | is_Gδ.inter | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ.union {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) : is_Gδ (s ∪ t) | begin
rcases hs with ⟨S, Sopen, Scount, rfl⟩,
rcases ht with ⟨T, Topen, Tcount, rfl⟩,
rw [sInter_union_sInter],
apply is_Gδ_bInter_of_open (Scount.prod Tcount),
rintros ⟨a, b⟩ ⟨ha, hb⟩,
exact (Sopen a ha).union (Topen b hb)
end | lemma | is_Gδ.union | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_bInter_of_open"
] | The union of two Gδ sets is a Gδ set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Gδ_bUnion {s : set ι} (hs : s.finite) {f : ι → set α} (h : ∀ i ∈ s, is_Gδ (f i)) :
is_Gδ (⋃ i ∈ s, f i) | begin
refine finite.induction_on hs (by simp) _ h,
simp only [ball_insert_iff, bUnion_insert],
exact λ a s _ _ ihs H, H.1.union (ihs H.2)
end | lemma | is_Gδ_bUnion | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ"
] | The union of finitely many Gδ sets is a Gδ set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.is_Gδ {α} [uniform_space α] [is_countably_generated (𝓤 α)]
{s : set α} (hs : is_closed s) : is_Gδ s | begin
rcases (@uniformity_has_basis_open α _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩,
rw [← hs.closure_eq, ← hU.bInter_bUnion_ball],
refine is_Gδ_bInter (to_countable _) (λ n hn, is_open.is_Gδ _),
exact is_open_bUnion (λ x hx, uniform_space.is_open_ball _ (hUo _).2)
end | lemma | is_closed.is_Gδ | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_bInter",
"is_closed",
"is_open.is_Gδ",
"is_open_bUnion",
"uniform_space",
"uniform_space.is_open_ball",
"uniformity_has_basis_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_compl_singleton (a : α) : is_Gδ ({a}ᶜ : set α) | is_open_compl_singleton.is_Gδ | lemma | is_Gδ_compl_singleton | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.countable.is_Gδ_compl {s : set α} (hs : s.countable) : is_Gδ sᶜ | begin
rw [← bUnion_of_singleton s, compl_Union₂],
exact is_Gδ_bInter hs (λ x _, is_Gδ_compl_singleton x)
end | lemma | set.countable.is_Gδ_compl | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_bInter",
"is_Gδ_compl_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.is_Gδ_compl {s : set α} (hs : s.finite) : is_Gδ sᶜ | hs.countable.is_Gδ_compl | lemma | set.finite.is_Gδ_compl | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.is_Gδ_compl {s : set α} (hs : s.subsingleton) : is_Gδ sᶜ | hs.finite.is_Gδ_compl | lemma | set.subsingleton.is_Gδ_compl | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.is_Gδ_compl (s : finset α) : is_Gδ (sᶜ : set α) | s.finite_to_set.is_Gδ_compl | lemma | finset.is_Gδ_compl | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"finset",
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_singleton (a : α) : is_Gδ ({a} : set α) | begin
rcases (nhds_basis_opens a).exists_antitone_subbasis with ⟨U, hU, h_basis⟩,
rw [← bInter_basis_nhds h_basis.to_has_basis],
exact is_Gδ_bInter (to_countable _) (λ n hn, (hU n).2.is_Gδ),
end | lemma | is_Gδ_singleton | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"bInter_basis_nhds",
"is_Gδ",
"is_Gδ_bInter",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.is_Gδ {s : set α} (hs : s.finite) : is_Gδ s | finite.induction_on hs is_Gδ_empty $ λ a s _ _ hs, (is_Gδ_singleton a).union hs | lemma | set.finite.is_Gδ | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"is_Gδ",
"is_Gδ_empty",
"is_Gδ_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_set_of_continuous_at [uniform_space β] [is_countably_generated (𝓤 β)] (f : α → β) :
is_Gδ {x | continuous_at f x} | begin
obtain ⟨U, hUo, hU⟩ := (@uniformity_has_basis_open_symmetric β _).exists_antitone_subbasis,
simp only [uniform.continuous_at_iff_prod, nhds_prod_eq],
simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.to_has_basis, forall_prop_of_true,
set_of_forall, id],
refine is_Gδ_Inter (λ k, is_open.is_Gδ $... | lemma | is_Gδ_set_of_continuous_at | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"continuous_at",
"forall_prop_of_true",
"is_Gδ",
"is_Gδ_Inter",
"is_open.is_Gδ",
"is_open.mem_nhds",
"nhds_basis_opens",
"nhds_prod_eq",
"uniform.continuous_at_iff_prod",
"uniform_space",
"uniformity_has_basis_open_symmetric"
] | The set of points where a function is continuous is a Gδ set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
residual (α : Type*) [topological_space α] : filter α | filter.countable_generate {t | is_open t ∧ dense t} | def | residual | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"dense",
"filter",
"filter.countable_generate",
"is_open",
"topological_space"
] | A set `s` is called *residual* if it includes a countable intersection of dense open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_Inter_filter_residual : countable_Inter_filter (residual α) | by rw [residual]; apply_instance | instance | countable_Inter_filter_residual | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"countable_Inter_filter",
"residual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
residual_of_dense_open {s : set α} (ho : is_open s) (hd : dense s) : s ∈ residual α | countable_generate_sets.basic ⟨ho, hd⟩ | lemma | residual_of_dense_open | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"dense",
"is_open",
"residual"
] | Dense open sets are residual. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
residual_of_dense_Gδ {s : set α} (ho : is_Gδ s) (hd : dense s) : s ∈ residual α | begin
rcases ho with ⟨T, To, Tct, rfl⟩,
exact (countable_sInter_mem Tct).mpr (λ t tT, residual_of_dense_open (To t tT)
(hd.mono (sInter_subset_of_mem tT))),
end | lemma | residual_of_dense_Gδ | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"countable_sInter_mem",
"dense",
"is_Gδ",
"residual",
"residual_of_dense_open"
] | Dense Gδ sets are residual. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_residual_iff {s : set α} : s ∈ residual α ↔
∃ (S : set (set α)), (∀ t ∈ S, is_open t) ∧ (∀ t ∈ S, dense t) ∧ S.countable ∧ ⋂₀ S ⊆ s | mem_countable_generate_iff.trans $ by simp_rw
[subset_def, mem_set_of, forall_and_distrib, and_assoc] | lemma | mem_residual_iff | topology | src/topology/G_delta.lean | [
"topology.uniform_space.basic",
"topology.separation",
"order.filter.countable_Inter"
] | [
"dense",
"forall_and_distrib",
"is_open",
"residual"
] | A set is residual iff it includes a countable intersection of dense open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph (α : Type*) (β : Type*) [topological_space α] [topological_space β]
extends α ≃ β | (continuous_to_fun : continuous to_fun . tactic.interactive.continuity')
(continuous_inv_fun : continuous inv_fun . tactic.interactive.continuity') | structure | homeomorph | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous",
"inv_fun",
"tactic.interactive.continuity'",
"topological_space"
] | Homeomorphism between `α` and `β`, also called topological isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph_mk_coe (a : equiv α β) (b c) :
((homeomorph.mk a b c) : α → β) = a | rfl | lemma | homeomorph.homeomorph_mk_coe | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : α ≃ₜ β) : β ≃ₜ α | { continuous_to_fun := h.continuous_inv_fun,
continuous_inv_fun := h.continuous_to_fun,
to_equiv := h.to_equiv.symm } | def | homeomorph.symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | Inverse of a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : α ≃ₜ β) : α → β | h | def | homeomorph.simps.apply | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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