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
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|---|---|---|---|---|---|---|---|---|---|---|
continuous_iff_coinduced_le {t₁ : tspace α} {t₂ : tspace β} :
cont t₁ t₂ f ↔ coinduced f t₁ ≤ t₂ | continuous_def.trans iff.rfl | lemma | continuous_iff_coinduced_le | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_le_induced {t₁ : tspace α} {t₂ : tspace β} :
cont t₁ t₂ f ↔ t₁ ≤ induced f t₂ | iff.trans continuous_iff_coinduced_le (gc_coinduced_induced f _ _) | lemma | continuous_iff_le_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_coinduced_le",
"gc_coinduced_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_generated_from {t : tspace α} {b : set (set β)}
(h : ∀s∈b, is_open (f ⁻¹' s)) : cont t (generate_from b) f | continuous_iff_coinduced_le.2 $ le_generate_from h | theorem | continuous_generated_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"is_open",
"le_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_induced_dom {t : tspace β} : cont (induced f t) t f | by { rw continuous_def, assume s h, exact ⟨_, h, rfl⟩ } | lemma | continuous_induced_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_induced_rng {g : γ → α} {t₂ : tspace β} {t₁ : tspace γ} :
cont t₁ (induced f t₂) g ↔ cont t₁ t₂ (f ∘ g) | by simp only [continuous_iff_le_induced, induced_compose] | lemma | continuous_induced_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_le_induced",
"induced_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coinduced_rng {t : tspace α} : cont t (coinduced f t) f | by { rw continuous_def, assume s h, exact h } | lemma | continuous_coinduced_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coinduced_dom {g : β → γ} {t₁ : tspace α} {t₂ : tspace γ} :
cont (coinduced f t₁) t₂ g ↔ cont t₁ t₂ (g ∘ f) | by simp only [continuous_iff_coinduced_le, coinduced_compose] | lemma | continuous_coinduced_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"coinduced_compose",
"cont",
"continuous_iff_coinduced_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_le_dom {t₁ t₂ : tspace α} {t₃ : tspace β}
(h₁ : t₂ ≤ t₁) (h₂ : cont t₁ t₃ f) : cont t₂ t₃ f | begin
rw continuous_def at h₂ ⊢,
assume s h,
exact h₁ _ (h₂ s h)
end | lemma | continuous_le_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_le_rng {t₁ : tspace α} {t₂ t₃ : tspace β}
(h₁ : t₂ ≤ t₃) (h₂ : cont t₁ t₂ f) : cont t₁ t₃ f | begin
rw continuous_def at h₂ ⊢,
assume s h,
exact h₂ s (h₁ s h)
end | lemma | continuous_le_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sup_dom {t₁ t₂ : tspace α} {t₃ : tspace β} :
cont (t₁ ⊔ t₂) t₃ f ↔ cont t₁ t₃ f ∧ cont t₂ t₃ f | by simp only [continuous_iff_le_induced, sup_le_iff] | lemma | continuous_sup_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_le_induced",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sup_rng_left {t₁ : tspace α} {t₃ t₂ : tspace β} :
cont t₁ t₂ f → cont t₁ (t₂ ⊔ t₃) f | continuous_le_rng le_sup_left | lemma | continuous_sup_rng_left | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_le_rng",
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sup_rng_right {t₁ : tspace α} {t₃ t₂ : tspace β} :
cont t₁ t₃ f → cont t₁ (t₂ ⊔ t₃) f | continuous_le_rng le_sup_right | lemma | continuous_sup_rng_right | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_le_rng",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_Sup_dom {T : set (tspace α)} {t₂ : tspace β} :
cont (Sup T) t₂ f ↔ ∀ t ∈ T, cont t t₂ f | by simp only [continuous_iff_le_induced, Sup_le_iff] | lemma | continuous_Sup_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"Sup_le_iff",
"cont",
"continuous_iff_le_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_Sup_rng {t₁ : tspace α} {t₂ : set (tspace β)} {t : tspace β}
(h₁ : t ∈ t₂) (hf : cont t₁ t f) : cont t₁ (Sup t₂) f | continuous_iff_coinduced_le.2 $ le_Sup_of_le h₁ $ continuous_iff_coinduced_le.1 hf | lemma | continuous_Sup_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"le_Sup_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_supr_dom {t₁ : ι → tspace α} {t₂ : tspace β} :
cont (supr t₁) t₂ f ↔ ∀ i, cont (t₁ i) t₂ f | by simp only [continuous_iff_le_induced, supr_le_iff] | lemma | continuous_supr_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_le_induced",
"supr",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_supr_rng {t₁ : tspace α} {t₂ : ι → tspace β} {i : ι}
(h : cont t₁ (t₂ i) f) : cont t₁ (supr t₂) f | continuous_Sup_rng ⟨i, rfl⟩ h | lemma | continuous_supr_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_Sup_rng",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inf_rng {t₁ : tspace α} {t₂ t₃ : tspace β} :
cont t₁ (t₂ ⊓ t₃) f ↔ cont t₁ t₂ f ∧ cont t₁ t₃ f | by simp only [continuous_iff_coinduced_le, le_inf_iff] | lemma | continuous_inf_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_coinduced_le",
"le_inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inf_dom_left {t₁ t₂ : tspace α} {t₃ : tspace β} :
cont t₁ t₃ f → cont (t₁ ⊓ t₂) t₃ f | continuous_le_dom inf_le_left | lemma | continuous_inf_dom_left | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_le_dom",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inf_dom_right {t₁ t₂ : tspace α} {t₃ : tspace β} :
cont t₂ t₃ f → cont (t₁ ⊓ t₂) t₃ f | continuous_le_dom inf_le_right | lemma | continuous_inf_dom_right | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_le_dom",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_Inf_dom {t₁ : set (tspace α)} {t₂ : tspace β} {t : tspace α} (h₁ : t ∈ t₁) :
cont t t₂ f → cont (Inf t₁) t₂ f | continuous_le_dom $ Inf_le h₁ | lemma | continuous_Inf_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"Inf_le",
"cont",
"continuous_le_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_Inf_rng {t₁ : tspace α} {T : set (tspace β)} :
cont t₁ (Inf T) f ↔ ∀ t ∈ T, cont t₁ t f | by simp only [continuous_iff_coinduced_le, le_Inf_iff] | lemma | continuous_Inf_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_coinduced_le",
"le_Inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_infi_dom {t₁ : ι → tspace α} {t₂ : tspace β} {i : ι} :
cont (t₁ i) t₂ f → cont (infi t₁) t₂ f | continuous_le_dom $ infi_le _ _ | lemma | continuous_infi_dom | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_le_dom",
"infi",
"infi_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_infi_rng {t₁ : tspace α} {t₂ : ι → tspace β} :
cont t₁ (infi t₂) f ↔ ∀ i, cont t₁ (t₂ i) f | by simp only [continuous_iff_coinduced_le, le_infi_iff] | lemma | continuous_infi_rng | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_iff_coinduced_le",
"infi",
"le_infi_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_bot {t : tspace β} : cont ⊥ t f | continuous_iff_le_induced.2 $ bot_le | lemma | continuous_bot | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"bot_le",
"cont"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_top {t : tspace α} : cont t ⊤ f | continuous_iff_coinduced_le.2 $ le_top | lemma | continuous_top | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_id_iff_le {t t' : tspace α} : cont t t' id ↔ t ≤ t' | @continuous_def _ _ t t' id | lemma | continuous_id_iff_le | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont",
"continuous_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_id_of_le {t t' : tspace α} (h : t ≤ t') : cont t t' id | continuous_id_iff_le.2 h | lemma | continuous_id_of_le | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"cont"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_induced [T : topological_space α] (f : β → α) (a : β) (s : set β) :
s ∈ @nhds β (topological_space.induced f T) a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s | begin
simp only [mem_nhds_iff, is_open_induced_iff, exists_prop, set.mem_set_of_eq],
split,
{ rintros ⟨u, usub, ⟨v, openv, ueq⟩, au⟩,
exact ⟨v, ⟨v, set.subset.refl v, openv, by rwa ←ueq at au⟩, by rw ueq; exact usub⟩ },
rintros ⟨u, ⟨v, vsubu, openv, amem⟩, finvsub⟩,
exact ⟨f ⁻¹' v, set.subset.trans (set.p... | theorem | mem_nhds_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"exists_prop",
"is_open_induced_iff",
"mem_nhds_iff",
"nhds",
"set.preimage_mono",
"set.subset.refl",
"set.subset.trans",
"topological_space",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_induced [T : topological_space α] (f : β → α) (a : β) :
@nhds β (topological_space.induced f T) a = comap f (𝓝 (f a)) | by { ext s, rw [mem_nhds_induced, mem_comap] } | theorem | nhds_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"mem_nhds_induced",
"nhds",
"topological_space",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_iff_nhds_eq [tα : topological_space α] [tβ : topological_space β] (f : β → α) :
tβ = tα.induced f ↔ ∀ b, 𝓝 b = comap f (𝓝 $ f b) | ⟨λ h a, h.symm ▸ nhds_induced f a, λ h, eq_of_nhds_eq_nhds $ λ x, by rw [h, nhds_induced]⟩ | lemma | induced_iff_nhds_eq | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"eq_of_nhds_eq_nhds",
"nhds_induced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_induced_of_surjective [T : topological_space α]
{f : β → α} (hf : surjective f) (a : β) :
map f (@nhds β (topological_space.induced f T) a) = 𝓝 (f a) | by rw [nhds_induced, map_comap_of_surjective hf] | theorem | map_nhds_induced_of_surjective | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"nhds",
"nhds_induced",
"topological_space",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_induced_eq {s : set α} :
is_open[induced f t] s ↔ s ∈ preimage f '' {s | is_open s} | iff.rfl | theorem | is_open_induced_eq | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_induced {s : set β} (h : is_open s) : is_open[induced f t] (f ⁻¹' s) | ⟨s, h, rfl⟩ | theorem | is_open_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_induced_eq (a : α) : map f (@nhds α (induced f t) a) = 𝓝[range f] (f a) | by rw [nhds_induced, filter.map_comap, nhds_within] | lemma | map_nhds_induced_eq | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter.map_comap",
"nhds",
"nhds_induced",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_induced_of_mem {a : α} (h : range f ∈ 𝓝 (f a)) :
map f (@nhds α (induced f t) a) = 𝓝 (f a) | by rw [nhds_induced, filter.map_comap_of_mem h] | lemma | map_nhds_induced_of_mem | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter.map_comap_of_mem",
"nhds",
"nhds_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_induced [t : topological_space β] {f : α → β} {a : α} {s : set α} :
a ∈ @closure α (topological_space.induced f t) s ↔ f a ∈ closure (f '' s) | by simp only [mem_closure_iff_frequently, nhds_induced, frequently_comap, mem_image, and_comm] | lemma | closure_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"closure",
"mem_closure_iff_frequently",
"nhds_induced",
"topological_space",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_induced_iff' [t : topological_space β] {f : α → β} {s : set α} :
is_closed[t.induced f] s ↔ ∀ a, f a ∈ closure (f '' s) → a ∈ s | by simp only [← closure_subset_iff_is_closed, subset_def, closure_induced] | lemma | is_closed_induced_iff' | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"closure",
"closure_induced",
"closure_subset_iff_is_closed",
"is_closed",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_singleton_true : is_open ({true} : set Prop) | topological_space.generate_open.basic _ (mem_singleton _) | lemma | is_open_singleton_true | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_true : 𝓝 true = pure true | le_antisymm (le_pure_iff.2 $ is_open_singleton_true.mem_nhds $ mem_singleton _) (pure_le_nhds _) | lemma | nhds_true | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_false : 𝓝 false = ⊤ | topological_space.nhds_generate_from.trans $ by simp [@and.comm (_ ∈ _)] | lemma | nhds_false | topology | src/topology/order.lean | [
"topology.tactic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_Prop {p : α → Prop} : continuous p ↔ is_open {x | p x} | ⟨assume h : continuous p,
have is_open (p ⁻¹' {true}),
from is_open_singleton_true.preimage h,
by simpa [preimage, eq_true_iff] using this,
assume h : is_open {x | p x},
continuous_generated_from $ assume s (hs : s = {true}),
by simp [hs, preimage, eq_true_iff, h]⟩ | lemma | continuous_Prop | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous",
"continuous_generated_from",
"eq_true_iff",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_continuous_mem {s : set α} : is_open s ↔ continuous (λ x, x ∈ s) | continuous_Prop.symm | lemma | is_open_iff_continuous_mem | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_union (a₁ a₂ : set (set α)) :
topological_space.generate_from (a₁ ∪ a₂) =
topological_space.generate_from a₁ ⊓ topological_space.generate_from a₂ | (topological_space.gc_generate_from α).u_inf | lemma | generate_from_union | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space.gc_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_is_open_sup (t₁ t₂ : topological_space α) :
{s | is_open[t₁ ⊔ t₂] s} = {s | is_open[t₁] s} ∩ {s | is_open[t₂] s} | rfl | lemma | set_of_is_open_sup | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_Union {f : ι → set (set α)} :
topological_space.generate_from (⋃ i, f i) = (⨅ i, topological_space.generate_from (f i)) | (topological_space.gc_generate_from α).u_infi | lemma | generate_from_Union | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space.gc_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_is_open_supr {t : ι → topological_space α} :
{s | is_open[⨆ i, t i] s} = ⋂ i, {s | is_open[t i] s} | (topological_space.gc_generate_from α).l_supr | lemma | set_of_is_open_supr | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.gc_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_sUnion {S : set (set (set α))} :
topological_space.generate_from (⋃₀ S) = (⨅ s ∈ S, topological_space.generate_from s) | (topological_space.gc_generate_from α).u_Inf | lemma | generate_from_sUnion | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space.gc_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_is_open_Sup {T : set (topological_space α)} :
{s | is_open[Sup T] s} = ⋂ t ∈ T, {s | is_open[t] s} | (topological_space.gc_generate_from α).l_Sup | lemma | set_of_is_open_Sup | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.gc_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_union_is_open (a b : topological_space α) :
topological_space.generate_from ({s | is_open[a] s} ∪ {s | is_open[b] s}) = a ⊓ b | (topological_space.gci_generate_from α).u_inf_l a b | lemma | generate_from_union_is_open | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.gci_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_Union_is_open (f : ι → topological_space α) :
topological_space.generate_from (⋃ i, {s | is_open[f i] s}) = ⨅ i, (f i) | (topological_space.gci_generate_from α).u_infi_l f | lemma | generate_from_Union_is_open | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.gci_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_inter (a b : topological_space α) :
topological_space.generate_from ({s | is_open[a] s} ∩ {s | is_open[b] s}) = a ⊔ b | (topological_space.gci_generate_from α).u_sup_l a b | lemma | generate_from_inter | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.gci_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_Inter (f : ι → topological_space α) :
topological_space.generate_from (⋂ i, {s | is_open[f i] s}) = ⨆ i, (f i) | (topological_space.gci_generate_from α).u_supr_l f | lemma | generate_from_Inter | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.gci_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_Inter_of_generate_from_eq_self (f : ι → set (set α))
(hf : ∀ i, {s | is_open[topological_space.generate_from (f i)] s} = f i) :
topological_space.generate_from (⋂ i, (f i)) = ⨆ i, topological_space.generate_from (f i) | (topological_space.gci_generate_from α).u_supr_of_lu_eq_self f hf | lemma | generate_from_Inter_of_generate_from_eq_self | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space.gci_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_supr_iff {s : set α} : is_open[⨆ i, t i] s ↔ ∀ i, is_open[t i] s | show s ∈ set_of (is_open[supr t]) ↔ s ∈ {x : set α | ∀ (i : ι), is_open[t i] x},
by simp [set_of_is_open_supr] | lemma | is_open_supr_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"set_of_is_open_supr",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_supr_iff {s : set α} : is_closed[⨆ i, t i] s ↔ ∀ i, is_closed[t i] s | by simp [← is_open_compl_iff, is_open_supr_iff] | lemma | is_closed_supr_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_supr_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
paracompact_space (X : Type v) [topological_space X] : Prop | (locally_finite_refinement :
∀ (α : Type v) (s : α → set X) (ho : ∀ a, is_open (s a)) (hc : (⋃ a, s a) = univ),
∃ (β : Type v) (t : β → set X) (ho : ∀ b, is_open (t b)) (hc : (⋃ b, t b) = univ),
locally_finite t ∧ ∀ b, ∃ a, t b ⊆ s a) | class | paracompact_space | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"is_open",
"locally_finite",
"topological_space"
] | A topological space is called paracompact, if every open covering of this space admits a locally
finite refinement. We use the same universe for all types in the definition to avoid creating a
class like `paracompact_space.{u v}`. Due to lemma `precise_refinement` below, every open covering
`s : α → set X` indexed on `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precise_refinement [paracompact_space X] (u : ι → set X) (uo : ∀ a, is_open (u a))
(uc : (⋃ i, u i) = univ) :
∃ v : ι → set X, (∀ a, is_open (v a)) ∧ (⋃ i, v i) = univ ∧ locally_finite v ∧ (∀ a, v a ⊆ u a) | begin
-- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
have := paracompact_space.locally_finite_refinement (range u) coe
(set_coe.forall.2 $ forall_range_iff.2 uo) (by rwa [← sUnion_range, subtype.range_coe]),
simp only [set_coe.exists, subtype.coe_mk, exists_ran... | lemma | precise_refinement | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"exists_prop",
"is_open",
"is_open_Union",
"locally_finite",
"paracompact_space",
"set.nonempty",
"set_coe.exists",
"subtype.coe_mk",
"subtype.range_coe"
] | Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precise_refinement_set [paracompact_space X] {s : set X} (hs : is_closed s)
(u : ι → set X) (uo : ∀ i, is_open (u i)) (us : s ⊆ ⋃ i, u i) :
∃ v : ι → set X, (∀ i, is_open (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ locally_finite v ∧ (∀ i, v i ⊆ u i) | begin
rcases precise_refinement (option.elim sᶜ u)
(option.forall.2 ⟨is_open_compl_iff.2 hs, uo⟩) _ with ⟨v, vo, vc, vf, vu⟩,
refine ⟨v ∘ some, λ i, vo _, _, vf.comp_injective (option.some_injective _), λ i, vu _⟩,
{ simp only [Union_option, ← compl_subset_iff_union] at vc,
exact subset.trans (subset_comp... | lemma | precise_refinement_set | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"compl_compl",
"is_closed",
"is_open",
"locally_finite",
"option.elim",
"option.some_injective",
"paracompact_space",
"precise_refinement"
] | In a paracompact space, every open covering of a closed set admits a locally finite refinement
indexed by the same type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
paracompact_of_compact [compact_space X] : paracompact_space X | begin
-- the proof is trivial: we choose a finite subcover using compactness, and use it
refine ⟨λ ι s ho hu, _⟩,
rcases is_compact_univ.elim_finite_subcover _ ho hu.ge with ⟨T, hT⟩,
have := hT, simp only [subset_def, mem_Union] at this,
choose i hiT hi using λ x, this x (mem_univ x),
refine ⟨(T : set ι), λ... | instance | paracompact_of_compact | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"compact_space",
"locally_finite_of_finite",
"paracompact_space"
] | A compact space is paracompact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refinement_of_locally_compact_sigma_compact_of_nhds_basis_set
[locally_compact_space X] [sigma_compact_space X] [t2_space X]
{ι : X → Type u} {p : Π x, ι x → Prop} {B : Π x, ι x → set X} {s : set X}
(hs : is_closed s) (hB : ∀ x ∈ s, (𝓝 x).has_basis (p x) (B x)) :
∃ (α : Type v) (c : α → X) (r : Π a, ι (c a)), ... | begin
classical,
-- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
set K' : compact_exhaustion X := compact_exhaustion.choice X,
set K : compact_exhaustion X := K'.shiftr.shiftr,
set Kdiff := λ n, K (n + 1) \ interior (K n),
-- Now we restate some properties of... | theorem | refinement_of_locally_compact_sigma_compact_of_nhds_basis_set | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"compact_exhaustion",
"compact_exhaustion.choice",
"finite",
"interior",
"interior_subset",
"is_closed",
"is_compact",
"is_open.mem_nhds",
"is_open_interior",
"locally_compact_space",
"locally_finite",
"sigma_compact_space",
"subtype.coe_mk",
"t2_space"
] | Let `X` be a locally compact sigma compact Hausdorff topological space, let `s` be a closed set
in `X`. Suppose that for each `x ∈ s` the sets `B x : ι x → set X` with the predicate
`p x : ι x → Prop` form a basis of the filter `𝓝 x`. Then there exists a locally finite covering
`λ i, B (c i) (r i)` of `s` such that al... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refinement_of_locally_compact_sigma_compact_of_nhds_basis
[locally_compact_space X] [sigma_compact_space X] [t2_space X]
{ι : X → Type u} {p : Π x, ι x → Prop} {B : Π x, ι x → set X}
(hB : ∀ x, (𝓝 x).has_basis (p x) (B x)) :
∃ (α : Type v) (c : α → X) (r : Π a, ι (c a)), (∀ a, p (c a) (r a)) ∧
(⋃ a, B (c a... | let ⟨α, c, r, hp, hU, hfin⟩ := refinement_of_locally_compact_sigma_compact_of_nhds_basis_set
is_closed_univ (λ x _, hB x)
in ⟨α, c, r, λ a, (hp a).2, univ_subset_iff.1 hU, hfin⟩ | theorem | refinement_of_locally_compact_sigma_compact_of_nhds_basis | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"is_closed_univ",
"locally_compact_space",
"locally_finite",
"refinement_of_locally_compact_sigma_compact_of_nhds_basis_set",
"sigma_compact_space",
"t2_space"
] | Let `X` be a locally compact sigma compact Hausdorff topological space. Suppose that for each
`x` the sets `B x : ι x → set X` with the predicate `p x : ι x → Prop` form a basis of the filter
`𝓝 x`. Then there exists a locally finite covering `λ i, B (c i) (r i)` of `X` such that each `r i`
satisfies `p (c i)`
The no... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
paracompact_of_locally_compact_sigma_compact [locally_compact_space X]
[sigma_compact_space X] [t2_space X] : paracompact_space X | begin
refine ⟨λ α s ho hc, _⟩,
choose i hi using Union_eq_univ_iff.1 hc,
have : ∀ x : X, (𝓝 x).has_basis (λ t : set X, (x ∈ t ∧ is_open t) ∧ t ⊆ s (i x)) id,
from λ x : X, (nhds_basis_opens x).restrict_subset (is_open.mem_nhds (ho (i x)) (hi x)),
rcases refinement_of_locally_compact_sigma_compact_of_nhds_b... | instance | paracompact_of_locally_compact_sigma_compact | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"is_open",
"is_open.mem_nhds",
"locally_compact_space",
"nhds_basis_opens",
"paracompact_space",
"refinement_of_locally_compact_sigma_compact_of_nhds_basis",
"sigma_compact_space",
"t2_space"
] | A locally compact sigma compact Hausdorff space is paracompact. See also
`refinement_of_locally_compact_sigma_compact_of_nhds_basis` for a more precise statement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_of_paracompact_t2 [t2_space X] [paracompact_space X] : normal_space X | begin
/- First we show how to go from points to a set on one side. -/
have : ∀ (s t : set X), is_closed s → is_closed t →
(∀ x ∈ s, ∃ u v, is_open u ∧ is_open v ∧ x ∈ u ∧ t ⊆ v ∧ disjoint u v) →
∃ u v, is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ disjoint u v,
{ /- For each `x ∈ s` we choose open disjoint `u x... | lemma | normal_of_paracompact_t2 | topology | src/topology/paracompact.lean | [
"topology.subset_properties",
"topology.separation",
"data.option.basic"
] | [
"closure",
"closure_minimal",
"compl_le_compl",
"disjoint",
"is_closed",
"is_closed_singleton",
"is_open",
"is_open_Union",
"le_rfl",
"normal_space",
"paracompact_space",
"precise_refinement_set",
"subset_closure",
"t2_separation",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rtendsto_nhds {r : rel β α} {l : filter β} {a : α} :
rtendsto r l (𝓝 a) ↔ (∀ s, is_open s → a ∈ s → r.core s ∈ l) | all_mem_nhds_filter _ _ (λ s t, id) _ | theorem | rtendsto_nhds | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"all_mem_nhds_filter",
"filter",
"is_open",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rtendsto'_nhds {r : rel β α} {l : filter β} {a : α} :
rtendsto' r l (𝓝 a) ↔ (∀ s, is_open s → a ∈ s → r.preimage s ∈ l) | by { rw [rtendsto'_def], apply all_mem_nhds_filter, apply rel.preimage_mono } | theorem | rtendsto'_nhds | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"all_mem_nhds_filter",
"filter",
"is_open",
"rel",
"rel.preimage_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ptendsto_nhds {f : β →. α} {l : filter β} {a : α} :
ptendsto f l (𝓝 a) ↔ (∀ s, is_open s → a ∈ s → f.core s ∈ l) | rtendsto_nhds | theorem | ptendsto_nhds | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"filter",
"is_open",
"rtendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ptendsto'_nhds {f : β →. α} {l : filter β} {a : α} :
ptendsto' f l (𝓝 a) ↔ (∀ s, is_open s → a ∈ s → f.preimage s ∈ l) | rtendsto'_nhds | theorem | ptendsto'_nhds | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"filter",
"is_open",
"rtendsto'_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pcontinuous (f : α →. β) | ∀ s, is_open s → is_open (f.preimage s) | def | pcontinuous | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"is_open"
] | Continuity of a partial function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_dom_of_pcontinuous {f : α →. β} (h : pcontinuous f) : is_open f.dom | by rw [←pfun.preimage_univ]; exact h _ is_open_univ | lemma | open_dom_of_pcontinuous | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"is_open",
"is_open_univ",
"pcontinuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pcontinuous_iff' {f : α →. β} :
pcontinuous f ↔ ∀ {x y} (h : y ∈ f x), ptendsto' f (𝓝 x) (𝓝 y) | begin
split,
{ intros h x y h',
simp only [ptendsto'_def, mem_nhds_iff],
rintros s ⟨t, tsubs, opent, yt⟩,
exact ⟨f.preimage t, pfun.preimage_mono _ tsubs, h _ opent, ⟨y, yt, h'⟩⟩ },
intros hf s os,
rw is_open_iff_nhds,
rintros x ⟨y, ys, fxy⟩ t,
rw [mem_principal],
assume h : f.preimage s ⊆ t,
... | lemma | pcontinuous_iff' | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"is_open_iff_nhds",
"mem_nhds_iff",
"pcontinuous",
"pfun.preimage_mono",
"set.subset.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff_ptendsto_res (f : α → β) {x : α} {s : set α} :
continuous_within_at f s x ↔ ptendsto (pfun.res f s) (𝓝 x) (𝓝 (f x)) | tendsto_iff_ptendsto _ _ _ _ | theorem | continuous_within_at_iff_ptendsto_res | topology | src/topology/partial.lean | [
"topology.continuous_on",
"order.filter.partial"
] | [
"continuous_within_at",
"pfun.res"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partition_of_unity (ι X : Type*) [topological_space X] (s : set X := univ) | (to_fun : ι → C(X, ℝ))
(locally_finite' : locally_finite (λ i, support (to_fun i)))
(nonneg' : 0 ≤ to_fun)
(sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, to_fun i x = 1)
(sum_le_one' : ∀ x, ∑ᶠ i, to_fun i x ≤ 1) | structure | partition_of_unity | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"locally_finite",
"topological_space"
] | A continuous partition of unity on a set `s : set X` is a collection of continuous functions
`f i` such that
* the supports of `f i` form a locally finite family of sets, i.e., for every point `x : X` there
exists a neighborhood `U ∋ x` such that all but finitely many functions `f i` are zero on `U`;
* the functions... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bump_covering (ι X : Type*) [topological_space X] (s : set X := univ) | (to_fun : ι → C(X, ℝ))
(locally_finite' : locally_finite (λ i, support (to_fun i)))
(nonneg' : 0 ≤ to_fun)
(le_one' : to_fun ≤ 1)
(eventually_eq_one' : ∀ x ∈ s, ∃ i, to_fun i =ᶠ[𝓝 x] 1) | structure | bump_covering | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"locally_finite",
"topological_space"
] | A `bump_covering ι X s` is an indexed family of functions `f i`, `i : ι`, such that
* the supports of `f i` form a locally finite family of sets, i.e., for every point `x : X` there
exists a neighborhood `U ∋ x` such that all but finitely many functions `f i` are zero on `U`;
* for all `i`, `x` we have `0 ≤ f i x ≤ ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_finite : locally_finite (λ i, support (f i)) | f.locally_finite' | lemma | partition_of_unity.locally_finite | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite_tsupport : locally_finite (λ i, tsupport (f i)) | f.locally_finite.closure | lemma | partition_of_unity.locally_finite_tsupport | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"locally_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg (i : ι) (x : X) : 0 ≤ f i x | f.nonneg' i x | lemma | partition_of_unity.nonneg | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_eq_one {x : X} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 | f.sum_eq_one' x hx | lemma | partition_of_unity.sum_eq_one | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x | begin
have H := f.sum_eq_one hx,
contrapose! H,
simpa only [λ i, (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
end | lemma | partition_of_unity.exists_pos | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"zero_ne_one"
] | If `f` is a partition of unity on `s`, then for every `x ∈ s` there exists an index `i` such
that `0 < f i x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_le_one (x : X) : ∑ᶠ i, f i x ≤ 1 | f.sum_le_one' x | lemma | partition_of_unity.sum_le_one | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_nonneg (x : X) : 0 ≤ ∑ᶠ i, f i x | finsum_nonneg $ λ i, f.nonneg i x | lemma | partition_of_unity.sum_nonneg | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one (i : ι) (x : X) : f i x ≤ 1 | (single_le_finsum i (f.locally_finite.point_finite x) (λ j, f.nonneg j x)).trans (f.sum_le_one x) | lemma | partition_of_unity.le_one | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), continuous_at g x) :
continuous (λ x, f i x • g x) | continuous_of_tsupport $ λ x hx, ((f i).continuous_at x).smul $
hg x $ tsupport_smul_subset_left _ _ hx | lemma | partition_of_unity.continuous_smul | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"continuous",
"continuous_at",
"tsupport_smul_subset_left"
] | If `f` is a partition of unity on `s : set X` and `g : X → E` is continuous at every point of
the topological support of some `f i`, then `λ x, f i x • g x` is continuous on the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_finsum_smul [has_continuous_add E] {g : ι → X → E}
(hg : ∀ i (x ∈ tsupport (f i)), continuous_at (g i) x) :
continuous (λ x, ∑ᶠ i, f i x • g i x) | continuous_finsum (λ i, f.continuous_smul (hg i)) $
f.locally_finite.subset $ λ i, support_smul_subset_left _ _ | lemma | partition_of_unity.continuous_finsum_smul | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"continuous",
"continuous_at",
"has_continuous_add"
] | If `f` is a partition of unity on a set `s : set X` and `g : ι → X → E` is a family of functions
such that each `g i` is continuous at every point of the topological support of `f i`, then the sum
`λ x, ∑ᶠ i, f i x • g i x` is continuous on the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subordinate (U : ι → set X) : Prop | ∀ i, tsupport (f i) ⊆ U i | def | partition_of_unity.is_subordinate | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | A partition of unity `f i` is subordinate to a family of sets `U i` indexed by the same type if
for each `i` the closure of the support of `f i` is a subset of `U i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_finset_nhd_support_subset {U : ι → set X}
(hso : f.is_subordinate U) (ho : ∀ i, is_open (U i)) (x : X) :
∃ (is : finset ι) {n : set X} (hn₁ : n ∈ 𝓝 x) (hn₂ : n ⊆ ⋂ i ∈ is, U i), ∀ (z ∈ n),
support (λ i, f i z) ⊆ is | f.locally_finite.exists_finset_nhd_support_subset hso ho x | lemma | partition_of_unity.exists_finset_nhd_support_subset | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"finset",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.continuous_finsum_smul [has_continuous_add E] {U : ι → set X}
(ho : ∀ i, is_open (U i)) (hf : f.is_subordinate U) {g : ι → X → E}
(hg : ∀ i, continuous_on (g i) (U i)) :
continuous (λ x, ∑ᶠ i, f i x • g i x) | f.continuous_finsum_smul $ λ i x hx, (hg i).continuous_at $ (ho i).mem_nhds $ hf i hx | lemma | partition_of_unity.is_subordinate.continuous_finsum_smul | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"continuous",
"continuous_at",
"continuous_on",
"has_continuous_add",
"is_open"
] | If `f` is a partition of unity that is subordinate to a family of open sets `U i` and
`g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum
`λ x, ∑ᶠ i, f i x • g i x` is a continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_finite (x : X) : {i | f i x ≠ 0}.finite | f.locally_finite.point_finite x | lemma | bump_covering.point_finite | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one (i : ι) (x : X) : f i x ≤ 1 | f.le_one' i x | lemma | bump_covering.le_one | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single (i : ι) (s : set X) : bump_covering ι X s | { to_fun := pi.single i 1,
locally_finite' := λ x,
begin
refine ⟨univ, univ_mem, (finite_singleton i).subset _⟩,
rintro j ⟨x, hx, -⟩,
contrapose! hx,
rw [mem_singleton_iff] at hx,
simp [hx]
end,
nonneg' := le_update_iff.2 ⟨λ x, zero_le_one, λ _ _, le_rfl⟩,
le_one' := update_l... | def | bump_covering.single | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering",
"zero_le_one"
] | A `bump_covering` that consists of a single function, uniformly equal to one, defined as an
example for `inhabited` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_single (i : ι) (s : set X) : ⇑(bump_covering.single i s) = pi.single i 1 | rfl | lemma | bump_covering.coe_single | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate (f : bump_covering ι X s) (U : ι → set X) : Prop | ∀ i, tsupport (f i) ⊆ U i | def | bump_covering.is_subordinate | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering"
] | A collection of bump functions `f i` is subordinate to a family of sets `U i` indexed by the
same type if for each `i` the closure of the support of `f i` is a subset of `U i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subordinate.mono {f : bump_covering ι X s} {U V : ι → set X} (hU : f.is_subordinate U)
(hV : ∀ i, U i ⊆ V i) :
f.is_subordinate V | λ i, subset.trans (hU i) (hV i) | lemma | bump_covering.is_subordinate.mono | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_is_subordinate_of_locally_finite_of_prop [normal_space X] (p : (X → ℝ) → Prop)
(h01 : ∀ s t, is_closed s → is_closed t → disjoint s t →
∃ f : C(X, ℝ), p f ∧ eq_on f 0 s ∧ eq_on f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
(hs : is_closed s) (U : ι → set X) (ho : ∀ i, is_open (U i)) (hf : locally_finite U)
(hU : s... | begin
rcases exists_subset_Union_closure_subset hs ho (λ x _, hf.point_finite x) hU
with ⟨V, hsV, hVo, hVU⟩,
have hVU' : ∀ i, V i ⊆ U i, from λ i, subset.trans subset_closure (hVU i),
rcases exists_subset_Union_closure_subset hs hVo
(λ x _, (hf.subset hVU').point_finite x) hsV with ⟨W, hsW, hWo, hWV⟩,
c... | lemma | bump_covering.exists_is_subordinate_of_locally_finite_of_prop | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering",
"closure_mono",
"disjoint",
"exists_subset_Union_closure_subset",
"is_closed",
"is_closed_closure",
"is_open",
"locally_finite",
"normal_space",
"subset_closure"
] | If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
`bump_covering.exists_is_subordinate`. This... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_subordinate_of_locally_finite [normal_space X] (hs : is_closed s)
(U : ι → set X) (ho : ∀ i, is_open (U i)) (hf : locally_finite U)
(hU : s ⊆ ⋃ i, U i) :
∃ f : bump_covering ι X s, f.is_subordinate U | let ⟨f, _, hfU⟩ :=
exists_is_subordinate_of_locally_finite_of_prop (λ _, true)
(λ s t hs ht hd, (exists_continuous_zero_one_of_closed hs ht hd).imp $ λ f hf, ⟨trivial, hf⟩)
hs U ho hf hU
in ⟨f, hfU⟩ | lemma | bump_covering.exists_is_subordinate_of_locally_finite | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering",
"exists_continuous_zero_one_of_closed",
"is_closed",
"is_open",
"locally_finite",
"normal_space"
] | If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a
closed set `s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. If `X` is a
paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
`bump_covering.exists_is_subordinate`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_subordinate_of_prop [normal_space X] [paracompact_space X] (p : (X → ℝ) → Prop)
(h01 : ∀ s t, is_closed s → is_closed t → disjoint s t →
∃ f : C(X, ℝ), p f ∧ eq_on f 0 s ∧ eq_on f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
(hs : is_closed s) (U : ι → set X) (ho : ∀ i, is_open (U i)) (hU : s ⊆ ⋃ i, U i) :
∃ f :... | begin
rcases precise_refinement_set hs _ ho hU with ⟨V, hVo, hsV, hVf, hVU⟩,
rcases exists_is_subordinate_of_locally_finite_of_prop p h01 hs V hVo hVf hsV with ⟨f, hfp, hf⟩,
exact ⟨f, hfp, hf.mono hVU⟩
end | lemma | bump_covering.exists_is_subordinate_of_prop | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering",
"disjoint",
"is_closed",
"is_open",
"normal_space",
"paracompact_space",
"precise_refinement_set"
] | If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
`s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. This version assumes that
`p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a
`bump_covering` such that each function of ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_subordinate [normal_space X] [paracompact_space X]
(hs : is_closed s) (U : ι → set X) (ho : ∀ i, is_open (U i)) (hU : s ⊆ ⋃ i, U i) :
∃ f : bump_covering ι X s, f.is_subordinate U | begin
rcases precise_refinement_set hs _ ho hU with ⟨V, hVo, hsV, hVf, hVU⟩,
rcases exists_is_subordinate_of_locally_finite hs V hVo hVf hsV with ⟨f, hf⟩,
exact ⟨f, hf.mono hVU⟩
end | lemma | bump_covering.exists_is_subordinate | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering",
"is_closed",
"is_open",
"normal_space",
"paracompact_space",
"precise_refinement_set"
] | If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
`s`, then there exists a `bump_covering ι X s` that is subordinate to `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ind (x : X) (hx : x ∈ s) : ι | (f.eventually_eq_one' x hx).some | def | bump_covering.ind | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | Index of a bump function such that `fs i =ᶠ[𝓝 x] 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_eq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 | (f.eventually_eq_one' x hx).some_spec | lemma | bump_covering.eventually_eq_one | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 | (f.eventually_eq_one x hx).eq_of_nhds | lemma | bump_covering.ind_apply | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pou_fun (i : ι) (x : X) : ℝ | f i x * ∏ᶠ j (hj : well_ordering_rel j i), (1 - f j x) | def | bump_covering.to_pou_fun | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"well_ordering_rel"
] | Partition of unity defined by a `bump_covering`. We use this auxiliary definition to prove some
properties of the new family of functions before bundling it into a `partition_of_unity`. Do not use
this definition, use `bump_function.to_partition_of_unity` instead.
The partition of unity is given by the formula `g i x ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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