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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interior_eq_empty_iff_dense_compl {s : set α} : interior s = ∅ ↔ dense sᶜ | by rw [dense_iff_closure_eq, closure_compl, compl_univ_iff] | lemma | interior_eq_empty_iff_dense_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure_compl",
"dense",
"dense_iff_closure_eq",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.interior_compl {s : set α} (h : dense s) : interior sᶜ = ∅ | interior_eq_empty_iff_dense_compl.2 $ by rwa compl_compl | lemma | dense.interior_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"compl_compl",
"dense",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_closure {s : set α} : dense (closure s) ↔ dense s | by rw [dense, dense, closure_closure] | lemma | dense_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_closure",
"dense"
] | The closure of a set `s` is dense if and only if `s` is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_univ : dense (univ : set α) | λ x, subset_closure trivial | lemma | dense_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_iff_inter_open {s : set α} :
dense s ↔ ∀ U, is_open U → U.nonempty → (U ∩ s).nonempty | begin
split ; intro h,
{ rintros U U_op ⟨x, x_in⟩,
exact mem_closure_iff.1 (by simp only [h.closure_eq]) U U_op x_in },
{ intro x,
rw mem_closure_iff,
intros U U_op x_in,
exact h U U_op ⟨_, x_in⟩ },
end | lemma | dense_iff_inter_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"is_open",
"mem_closure_iff"
] | A set is dense if and only if it has a nonempty intersection with each nonempty open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.exists_mem_open {s : set α} (hs : dense s) {U : set α} (ho : is_open U)
(hne : U.nonempty) :
∃ x ∈ s, x ∈ U | let ⟨x, hx⟩ := hs.inter_open_nonempty U ho hne in ⟨x, hx.2, hx.1⟩ | lemma | dense.exists_mem_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.nonempty_iff {s : set α} (hs : dense s) :
s.nonempty ↔ nonempty α | ⟨λ ⟨x, hx⟩, ⟨x⟩, λ ⟨x⟩,
let ⟨y, hy⟩ := hs.inter_open_nonempty _ is_open_univ ⟨x, trivial⟩ in ⟨y, hy.2⟩⟩ | lemma | dense.nonempty_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.nonempty [h : nonempty α] {s : set α} (hs : dense s) : s.nonempty | hs.nonempty_iff.2 h | lemma | dense.nonempty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.mono {s₁ s₂ : set α} (h : s₁ ⊆ s₂) (hd : dense s₁) : dense s₂ | λ x, closure_mono h (hd x) | lemma | dense.mono | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure_mono",
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_compl_singleton_iff_not_open {x : α} : dense ({x}ᶜ : set α) ↔ ¬is_open ({x} : set α) | begin
fsplit,
{ intros hd ho,
exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _) },
{ refine λ ho, dense_iff_inter_open.2 (λ U hU hne, inter_compl_nonempty_iff.2 $ λ hUx, _),
obtain rfl : U = {x}, from eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩,
exact ho hU... | lemma | dense_compl_singleton_iff_not_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"is_open"
] | Complement to a singleton is dense if and only if the singleton is not an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frontier (s : set α) : set α | closure s \ interior s | def | frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"interior"
] | The frontier of a set is the set of points between the closure and interior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_diff_interior (s : set α) : closure s \ interior s = frontier s | rfl | lemma | closure_diff_interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_diff_frontier (s : set α) : closure s \ frontier s = interior s | by rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure] | lemma | closure_diff_frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"interior",
"interior_subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_diff_frontier (s : set α) : s \ frontier s = interior s | by rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,
inter_eq_self_of_subset_right interior_subset, empty_union] | lemma | self_diff_frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"interior",
"interior_subset",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_eq_closure_inter_closure {s : set α} :
frontier s = closure s ∩ closure sᶜ | by rw [closure_compl, frontier, diff_eq] | lemma | frontier_eq_closure_inter_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_compl",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_subset_closure {s : set α} : frontier s ⊆ closure s | diff_subset _ _ | lemma | frontier_subset_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.frontier_subset (hs : is_closed s) : frontier s ⊆ s | frontier_subset_closure.trans hs.closure_eq.subset | lemma | is_closed.frontier_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_closure_subset {s : set α} : frontier (closure s) ⊆ frontier s | diff_subset_diff closure_closure.subset $ interior_mono subset_closure | lemma | frontier_closure_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"interior_mono",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_interior_subset {s : set α} : frontier (interior s) ⊆ frontier s | diff_subset_diff (closure_mono interior_subset) interior_interior.symm.subset | lemma | frontier_interior_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure_mono",
"frontier",
"interior",
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_compl (s : set α) : frontier sᶜ = frontier s | by simp only [frontier_eq_closure_inter_closure, compl_compl, inter_comm] | lemma | frontier_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"compl_compl",
"frontier",
"frontier_eq_closure_inter_closure"
] | The complement of a set has the same frontier as the original set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frontier_univ : frontier (univ : set α) = ∅ | by simp [frontier] | lemma | frontier_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_empty : frontier (∅ : set α) = ∅ | by simp [frontier] | lemma | frontier_empty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_inter_subset (s t : set α) :
frontier (s ∩ t) ⊆ (frontier s ∩ closure t) ∪ (closure s ∩ frontier t) | begin
simp only [frontier_eq_closure_inter_closure, compl_inter, closure_union],
convert inter_subset_inter_left _ (closure_inter_subset_inter_closure s t),
simp only [inter_distrib_left, inter_distrib_right, inter_assoc],
congr' 2,
apply inter_comm
end | lemma | frontier_inter_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_inter_subset_inter_closure",
"closure_union",
"frontier",
"frontier_eq_closure_inter_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_union_subset (s t : set α) :
frontier (s ∪ t) ⊆ (frontier s ∩ closure tᶜ) ∪ (closure sᶜ ∩ frontier t) | by simpa only [frontier_compl, ← compl_union]
using frontier_inter_subset sᶜ tᶜ | lemma | frontier_union_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"frontier_compl",
"frontier_inter_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.frontier_eq {s : set α} (hs : is_closed s) : frontier s = s \ interior s | by rw [frontier, hs.closure_eq] | lemma | is_closed.frontier_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"interior",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.frontier_eq {s : set α} (hs : is_open s) : frontier s = closure s \ s | by rw [frontier, hs.interior_eq] | lemma | is_open.frontier_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.inter_frontier_eq {s : set α} (hs : is_open s) : s ∩ frontier s = ∅ | by rw [hs.frontier_eq, inter_diff_self] | lemma | is_open.inter_frontier_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_frontier {s : set α} : is_closed (frontier s) | by rw frontier_eq_closure_inter_closure; exact is_closed.inter is_closed_closure is_closed_closure | lemma | is_closed_frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"frontier_eq_closure_inter_closure",
"is_closed",
"is_closed.inter",
"is_closed_closure"
] | The frontier of a set is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_frontier {s : set α} (h : is_closed s) : interior (frontier s) = ∅ | begin
have A : frontier s = s \ interior s, from h.frontier_eq,
have B : interior (frontier s) ⊆ interior s, by rw A; exact interior_mono (diff_subset _ _),
have C : interior (frontier s) ⊆ frontier s := interior_subset,
have : interior (frontier s) ⊆ (interior s) ∩ (s \ interior s) :=
subset_inter B (by si... | lemma | interior_frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"interior",
"interior_mono",
"interior_subset",
"is_closed"
] | The frontier of a closed set has no interior point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_interior_union_frontier (s : set α) : closure s = interior s ∪ frontier s | (union_diff_cancel interior_subset_closure).symm | lemma | closure_eq_interior_union_frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"interior",
"interior_subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq_self_union_frontier (s : set α) : closure s = s ∪ frontier s | (union_diff_cancel' interior_subset subset_closure).symm | lemma | closure_eq_self_union_frontier | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"frontier",
"interior_subset",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.frontier_left (ht : is_open t) (hd : disjoint s t) : disjoint (frontier s) t | subset_compl_iff_disjoint_right.1 $ frontier_subset_closure.trans $ closure_minimal
(disjoint_left.1 hd) $ is_closed_compl_iff.2 ht | lemma | disjoint.frontier_left | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure_minimal",
"disjoint",
"frontier",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.frontier_right (hs : is_open s) (hd : disjoint s t) : disjoint s (frontier t) | (hd.symm.frontier_left hs).symm | lemma | disjoint.frontier_right | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"disjoint",
"frontier",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_eq_inter_compl_interior {s : set α} :
frontier s = (interior s)ᶜ ∩ (interior (sᶜ))ᶜ | by { rw [←frontier_compl, ←closure_compl], refl } | lemma | frontier_eq_inter_compl_interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"frontier",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_frontier_eq_union_interior {s : set α} :
(frontier s)ᶜ = interior s ∪ interior sᶜ | begin
rw frontier_eq_inter_compl_interior,
simp only [compl_inter, compl_compl],
end | lemma | compl_frontier_eq_union_interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"compl_compl",
"frontier",
"frontier_eq_inter_compl_interior",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds (a : α) : filter α | (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, 𝓟 s) | def | nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"is_open"
] | A set is called a neighborhood of `a` if it contains an open set around `a`. The set of all
neighborhoods of `a` forms a filter, the neighborhood filter at `a`, is here defined as the
infimum over the principal filters of all open sets containing `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_within (a : α) (s : set α) : filter α | 𝓝 a ⊓ 𝓟 s | def | nhds_within | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter"
] | The "neighborhood within" filter. Elements of `𝓝[s] a` are sets containing the
intersection of `s` and a neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_def (a : α) : 𝓝 a = (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, 𝓟 s) | by rw nhds | lemma | nhds_def | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_def' (a : α) : 𝓝 a = ⨅ (s : set α) (hs : is_open s) (ha : a ∈ s), 𝓟 s | by simp only [nhds_def, mem_set_of_eq, and_comm (a ∈ _), infi_and] | lemma | nhds_def' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"infi_and",
"is_open",
"nhds_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_opens (a : α) : (𝓝 a).has_basis (λ s : set α, a ∈ s ∧ is_open s) (λ s, s) | begin
rw nhds_def,
exact has_basis_binfi_principal
(λ s ⟨has, hs⟩ t ⟨hat, ht⟩, ⟨s ∩ t, ⟨⟨has, hat⟩, is_open.inter hs ht⟩,
⟨inter_subset_left _ _, inter_subset_right _ _⟩⟩)
⟨univ, ⟨mem_univ a, is_open_univ⟩⟩
end | lemma | nhds_basis_opens | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open.inter",
"nhds_def"
] | The open sets containing `a` are a basis for the neighborhood filter. See `nhds_basis_opens'`
for a variant using open neighborhoods instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_closeds (a : α) : (𝓝 a).has_basis (λ s : set α, a ∉ s ∧ is_closed s) compl | ⟨λ t, (nhds_basis_opens a).mem_iff.trans $ compl_surjective.exists.trans $
by simp only [is_open_compl_iff, mem_compl_iff]⟩ | lemma | nhds_basis_closeds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open_compl_iff",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : set α, a ∈ s → is_open s → s ∈ f | by simp [nhds_def] | lemma | le_nhds_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"nhds_def"
] | A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_le_of_le {f a} {s : set α} (h : a ∈ s) (o : is_open s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f | by rw nhds_def; exact infi_le_of_le s (infi_le_of_le ⟨h, o⟩ sf) | lemma | nhds_le_of_le | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"infi_le_of_le",
"is_open",
"nhds_def"
] | To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above
the principal filter of some open set `s` containing `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_iff {a : α} {s : set α} :
s ∈ 𝓝 a ↔ ∃ t ⊆ s, is_open t ∧ a ∈ t | (nhds_basis_opens a).mem_iff.trans
⟨λ ⟨t, ⟨hat, ht⟩, hts⟩, ⟨t, hts, ht, hat⟩, λ ⟨t, hts, ht, hat⟩, ⟨t, ⟨hat, ht⟩, hts⟩⟩ | lemma | mem_nhds_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_iff {a : α} {p : α → Prop} :
(∀ᶠ x in 𝓝 a, p x) ↔ ∃ (t : set α), (∀ x ∈ t, p x) ∧ is_open t ∧ a ∈ t | mem_nhds_iff.trans $ by simp only [subset_def, exists_prop, mem_set_of_eq] | lemma | eventually_nhds_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"exists_prop",
"is_open"
] | A predicate is true in a neighborhood of `a` iff it is true for all the points in an open set
containing `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_nhds {a : α} {f : α → β} :
map f (𝓝 a) = (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, 𝓟 (image f s)) | ((nhds_basis_opens a).map f).eq_binfi | lemma | map_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_mem_nhds {a : α} {s : set α} : s ∈ 𝓝 a → a ∈ s | λ H, let ⟨t, ht, _, hs⟩ := mem_nhds_iff.1 H in ht hs | lemma | mem_of_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.self_of_nhds {p : α → Prop} {a : α}
(h : ∀ᶠ y in 𝓝 a, p y) : p a | mem_of_mem_nhds h | lemma | filter.eventually.self_of_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"mem_of_mem_nhds"
] | If a predicate is true in a neighborhood of `a`, then it is true for `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open.mem_nhds {a : α} {s : set α} (hs : is_open s) (ha : a ∈ s) :
s ∈ 𝓝 a | mem_nhds_iff.2 ⟨s, subset.refl _, hs, ha⟩ | lemma | is_open.mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.mem_nhds_iff {a : α} {s : set α} (hs : is_open s) : s ∈ 𝓝 a ↔ a ∈ s | ⟨mem_of_mem_nhds, λ ha, mem_nhds_iff.2 ⟨s, subset.refl _, hs, ha⟩⟩ | lemma | is_open.mem_nhds_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.compl_mem_nhds {a : α} {s : set α} (hs : is_closed s) (ha : a ∉ s) : sᶜ ∈ 𝓝 a | hs.is_open_compl.mem_nhds (mem_compl ha) | lemma | is_closed.compl_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.eventually_mem {a : α} {s : set α} (hs : is_open s) (ha : a ∈ s) :
∀ᶠ x in 𝓝 a, x ∈ s | is_open.mem_nhds hs ha | lemma | is_open.eventually_mem | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_opens' (a : α) : (𝓝 a).has_basis (λ s : set α, s ∈ 𝓝 a ∧ is_open s) (λ x, x) | begin
convert nhds_basis_opens a,
ext s,
exact and.congr_left_iff.2 is_open.mem_nhds_iff
end | lemma | nhds_basis_opens' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open.mem_nhds_iff",
"nhds_basis_opens"
] | The open neighborhoods of `a` are a basis for the neighborhood filter. See `nhds_basis_opens`
for a variant using open sets around `a` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_open_set_nhds {s U : set α} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
∃ V : set α, s ⊆ V ∧ is_open V ∧ V ⊆ U | begin
have := λ x hx, (nhds_basis_opens x).mem_iff.1 (h x hx),
choose! Z hZ hZU using this, choose hZmem hZo using hZ,
exact ⟨⋃ x ∈ s, Z x, λ x hx, mem_bUnion hx (hZmem x hx), is_open_bUnion hZo, Union₂_subset hZU⟩
end | lemma | exists_open_set_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_bUnion",
"nhds_basis_opens"
] | If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of `s`:
it contains an open set containing `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_open_set_nhds' {s U : set α} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
∃ V : set α, s ⊆ V ∧ is_open V ∧ V ⊆ U | exists_open_set_nhds (by simpa using h) | lemma | exists_open_set_nhds' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"exists_open_set_nhds",
"is_open"
] | If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:
it contains an open set containing `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.eventually.eventually_nhds {p : α → Prop} {a : α} (h : ∀ᶠ y in 𝓝 a, p y) :
∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝 y, p x | let ⟨t, htp, hto, ha⟩ := eventually_nhds_iff.1 h in
eventually_nhds_iff.2 ⟨t, λ x hx, eventually_nhds_iff.2 ⟨t, htp, hto, hx⟩, hto, ha⟩ | lemma | filter.eventually.eventually_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | If a predicate is true in a neighbourhood of `a`, then for `y` sufficiently close
to `a` this predicate is true in a neighbourhood of `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_eventually_nhds {p : α → Prop} {a : α} :
(∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝 a, p x | ⟨λ h, h.self_of_nhds, λ h, h.eventually_nhds⟩ | lemma | eventually_eventually_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_frequently_nhds {p : α → Prop} {a : α} :
(∃ᶠ y in 𝓝 a, ∃ᶠ x in 𝓝 y, p x) ↔ (∃ᶠ x in 𝓝 a, p x) | begin
rw ← not_iff_not,
simp_rw not_frequently,
exact eventually_eventually_nhds,
end | lemma | frequently_frequently_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"eventually_eventually_nhds",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_mem_nhds {s : set α} {a : α} :
(∀ᶠ x in 𝓝 a, s ∈ 𝓝 x) ↔ s ∈ 𝓝 a | eventually_eventually_nhds | lemma | eventually_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"eventually_eventually_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bind_nhds : (𝓝 a).bind 𝓝 = 𝓝 a | filter.ext $ λ s, eventually_eventually_nhds | lemma | nhds_bind_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"eventually_eventually_nhds",
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eventually_eq_nhds {f g : α → β} {a : α} :
(∀ᶠ y in 𝓝 a, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 a] g | eventually_eventually_nhds | lemma | eventually_eventually_eq_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"eventually_eventually_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.eq_of_nhds {f g : α → β} {a : α} (h : f =ᶠ[𝓝 a] g) : f a = g a | h.self_of_nhds | lemma | filter.eventually_eq.eq_of_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eventually_le_nhds [has_le β] {f g : α → β} {a : α} :
(∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 a] g | eventually_eventually_nhds | lemma | eventually_eventually_le_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"eventually_eventually_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.eventually_eq_nhds {f g : α → β} {a : α} (h : f =ᶠ[𝓝 a] g) :
∀ᶠ y in 𝓝 a, f =ᶠ[𝓝 y] g | h.eventually_nhds | lemma | filter.eventually_eq.eventually_eq_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | If two functions are equal in a neighbourhood of `a`, then for `y` sufficiently close
to `a` these functions are equal in a neighbourhood of `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.eventually_le.eventually_le_nhds [has_le β] {f g : α → β} {a : α} (h : f ≤ᶠ[𝓝 a] g) :
∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g | h.eventually_nhds | lemma | filter.eventually_le.eventually_le_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | If `f x ≤ g x` in a neighbourhood of `a`, then for `y` sufficiently close to `a` we have
`f x ≤ g x` in a neighbourhood of `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
all_mem_nhds (x : α) (P : set α → Prop) (hP : ∀ s t, s ⊆ t → P s → P t) :
(∀ s ∈ 𝓝 x, P s) ↔ (∀ s, is_open s → x ∈ s → P s) | ((nhds_basis_opens x).forall_iff hP).trans $ by simp only [and_comm (x ∈ _), and_imp] | theorem | all_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"and_imp",
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
all_mem_nhds_filter (x : α) (f : set α → set β) (hf : ∀ s t, s ⊆ t → f s ⊆ f t)
(l : filter β) :
(∀ s ∈ 𝓝 x, f s ∈ l) ↔ (∀ s, is_open s → x ∈ s → f s ∈ l) | all_mem_nhds _ _ (λ s t ssubt h, mem_of_superset h (hf s t ssubt)) | theorem | all_mem_nhds_filter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"all_mem_nhds",
"filter",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds {f : β → α} {l : filter β} {a : α} :
tendsto f l (𝓝 a) ↔ (∀ s, is_open s → a ∈ s → f ⁻¹' s ∈ l) | all_mem_nhds_filter _ _ (λ s t h, preimage_mono h) _ | theorem | tendsto_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"all_mem_nhds_filter",
"filter",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_nhds [nonempty β] [semilattice_sup β] {f : β → α} {a : α} :
(tendsto f at_top (𝓝 a)) ↔ ∀ U : set α, a ∈ U → is_open U → ∃ N, ∀ n, N ≤ n → f n ∈ U | (at_top_basis.tendsto_iff (nhds_basis_opens a)).trans $
by simp only [and_imp, exists_prop, true_and, mem_Ici, ge_iff_le] | lemma | tendsto_at_top_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"and_imp",
"exists_prop",
"ge_iff_le",
"is_open",
"nhds_basis_opens",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_nhds {a : α} {f : filter β} : tendsto (λb:β, a) f (𝓝 a) | tendsto_nhds.mpr $ assume s hs ha, univ_mem' $ assume _, ha | lemma | tendsto_const_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_of_eventually_const {ι : Type*} [semilattice_sup ι] [nonempty ι]
{x : α} {u : ι → α} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : tendsto u at_top (𝓝 x) | tendsto.congr' (eventually_eq.symm (eventually_at_top.mpr ⟨i₀, h⟩)) tendsto_const_nhds | lemma | tendsto_at_top_of_eventually_const | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"semilattice_sup",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_of_eventually_const {ι : Type*} [semilattice_inf ι] [nonempty ι]
{x : α} {u : ι → α} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : tendsto u at_bot (𝓝 x) | tendsto.congr' (eventually_eq.symm (eventually_at_bot.mpr ⟨i₀, h⟩)) tendsto_const_nhds | lemma | tendsto_at_bot_of_eventually_const | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"semilattice_inf",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_le_nhds : pure ≤ (𝓝 : α → filter α) | assume a s hs, mem_pure.2 $ mem_of_mem_nhds hs | lemma | pure_le_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pure_nhds {α : Type*} [topological_space β] (f : α → β) (a : α) :
tendsto f (pure a) (𝓝 (f a)) | (tendsto_pure_pure f a).mono_right (pure_le_nhds _) | lemma | tendsto_pure_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"pure_le_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_top.tendsto_at_top_nhds {α : Type*} [partial_order α] [order_top α]
[topological_space β] (f : α → β) : tendsto f at_top (𝓝 $ f ⊤) | (tendsto_at_top_pure f).mono_right (pure_le_nhds _) | lemma | order_top.tendsto_at_top_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"order_top",
"pure_le_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_ne_bot {a : α} : ne_bot (𝓝 a) | ne_bot_of_le (pure_le_nhds a) | instance | nhds_ne_bot | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt (x : α) (F : filter α) : Prop | ne_bot (𝓝 x ⊓ F) | def | cluster_pt | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter"
] | A point `x` is a cluster point of a filter `F` if `𝓝 x ⊓ F ≠ ⊥`. Also known as
an accumulation point or a limit point, but beware that terminology varies. This
is *not* the same as asking `𝓝[≠] x ⊓ F ≠ ⊥`. See `mem_closure_iff_cluster_pt` in particular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cluster_pt.ne_bot {x : α} {F : filter α} (h : cluster_pt x F) : ne_bot (𝓝 x ⊓ F) | h | lemma | cluster_pt.ne_bot | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.cluster_pt_iff {ιa ιF} {pa : ιa → Prop} {sa : ιa → set α}
{pF : ιF → Prop} {sF : ιF → set α} {F : filter α}
(ha : (𝓝 a).has_basis pa sa) (hF : F.has_basis pF sF) :
cluster_pt a F ↔ ∀ ⦃i⦄ (hi : pa i) ⦃j⦄ (hj : pF j), (sa i ∩ sF j).nonempty | ha.inf_basis_ne_bot_iff hF | lemma | filter.has_basis.cluster_pt_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt_iff {x : α} {F : filter α} :
cluster_pt x F ↔ ∀ ⦃U : set α⦄ (hU : U ∈ 𝓝 x) ⦃V⦄ (hV : V ∈ F), (U ∩ V).nonempty | inf_ne_bot_iff | lemma | cluster_pt_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt_principal_iff {x : α} {s : set α} :
cluster_pt x (𝓟 s) ↔ ∀ U ∈ 𝓝 x, (U ∩ s).nonempty | inf_principal_ne_bot_iff | lemma | cluster_pt_principal_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt"
] | `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty
set. See also `mem_closure_iff_cluster_pt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cluster_pt_principal_iff_frequently {x : α} {s : set α} :
cluster_pt x (𝓟 s) ↔ ∃ᶠ y in 𝓝 x, y ∈ s | by simp only [cluster_pt_principal_iff, frequently_iff, set.nonempty, exists_prop, mem_inter_iff] | lemma | cluster_pt_principal_iff_frequently | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"cluster_pt_principal_iff",
"exists_prop",
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.of_le_nhds {x : α} {f : filter α} (H : f ≤ 𝓝 x) [ne_bot f] : cluster_pt x f | by rwa [cluster_pt, inf_eq_right.mpr H] | lemma | cluster_pt.of_le_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.of_le_nhds' {x : α} {f : filter α} (H : f ≤ 𝓝 x) (hf : ne_bot f) :
cluster_pt x f | cluster_pt.of_le_nhds H | lemma | cluster_pt.of_le_nhds' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"cluster_pt.of_le_nhds",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.of_nhds_le {x : α} {f : filter α} (H : 𝓝 x ≤ f) : cluster_pt x f | by simp only [cluster_pt, inf_eq_left.mpr H, nhds_ne_bot] | lemma | cluster_pt.of_nhds_le | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter",
"nhds_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.mono {x : α} {f g : filter α} (H : cluster_pt x f) (h : f ≤ g) :
cluster_pt x g | ⟨ne_bot_of_le_ne_bot H.ne $ inf_le_inf_left _ h⟩ | lemma | cluster_pt.mono | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter",
"inf_le_inf_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.of_inf_left {x : α} {f g : filter α} (H : cluster_pt x $ f ⊓ g) :
cluster_pt x f | H.mono inf_le_left | lemma | cluster_pt.of_inf_left | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.of_inf_right {x : α} {f g : filter α} (H : cluster_pt x $ f ⊓ g) :
cluster_pt x g | H.mono inf_le_right | lemma | cluster_pt.of_inf_right | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter.cluster_pt_iff {x : α} {f : ultrafilter α} : cluster_pt x f ↔ ↑f ≤ 𝓝 x | ⟨f.le_of_inf_ne_bot', λ h, cluster_pt.of_le_nhds h⟩ | lemma | ultrafilter.cluster_pt_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"cluster_pt.of_le_nhds",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cluster_pt {ι :Type*} (x : α) (F : filter ι) (u : ι → α) : Prop | cluster_pt x (map u F) | def | map_cluster_pt | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter"
] | A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point
of `map u F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cluster_pt_iff {ι :Type*} (x : α) (F : filter ι) (u : ι → α) :
map_cluster_pt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s | by { simp_rw [map_cluster_pt, cluster_pt, inf_ne_bot_iff_frequently_left, frequently_map], refl } | lemma | map_cluster_pt_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter",
"map_cluster_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cluster_pt_of_comp {ι δ :Type*} {F : filter ι} {φ : δ → ι} {p : filter δ}
{x : α} {u : ι → α} [ne_bot p] (h : tendsto φ p F) (H : tendsto (u ∘ φ) p (𝓝 x)) :
map_cluster_pt x F u | begin
have := calc
map (u ∘ φ) p = map u (map φ p) : map_map
... ≤ map u F : map_mono h,
have : map (u ∘ φ) p ≤ 𝓝 x ⊓ map u F,
from le_inf H this,
exact ne_bot_of_le this
end | lemma | map_cluster_pt_of_comp | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"le_inf",
"map_cluster_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc_pt (x : α) (F : filter α) : Prop | ne_bot (𝓝[≠] x ⊓ F) | def | acc_pt | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter"
] | A point `x` is an accumulation point of a filter `F` if `𝓝[≠] x ⊓ F ≠ ⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
acc_iff_cluster (x : α) (F : filter α) : acc_pt x F ↔ cluster_pt x (𝓟 {x}ᶜ ⊓ F) | by rw [acc_pt, nhds_within, cluster_pt, inf_assoc] | lemma | acc_iff_cluster | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"acc_pt",
"cluster_pt",
"filter",
"inf_assoc",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
acc_principal_iff_cluster (x : α) (C : set α) :
acc_pt x (𝓟 C) ↔ cluster_pt x (𝓟(C \ {x})) | by rw [acc_iff_cluster, inf_principal, inter_comm]; refl | lemma | acc_principal_iff_cluster | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"acc_iff_cluster",
"acc_pt",
"cluster_pt"
] | `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
acc_pt_iff_nhds (x : α) (C : set α) : acc_pt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x | by simp [acc_principal_iff_cluster, cluster_pt_principal_iff, set.nonempty, exists_prop,
and_assoc, and_comm (¬ _ = x)] | lemma | acc_pt_iff_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"acc_principal_iff_cluster",
"acc_pt",
"cluster_pt_principal_iff",
"exists_prop",
"set.nonempty"
] | `x` is an accumulation point of a set `C` iff every neighborhood
of `x` contains a point of `C` other than `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
acc_pt_iff_frequently (x : α) (C : set α) : acc_pt x (𝓟 C) ↔ ∃ᶠ y in 𝓝 x, y ≠ x ∧ y ∈ C | by simp [acc_principal_iff_cluster, cluster_pt_principal_iff_frequently, and_comm] | lemma | acc_pt_iff_frequently | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"acc_principal_iff_cluster",
"acc_pt",
"cluster_pt_principal_iff_frequently"
] | `x` is an accumulation point of a set `C` iff
there are points near `x` in `C` and different from `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
acc_pt.mono {x : α} {F G : filter α} (h : acc_pt x F) (hFG : F ≤ G) : acc_pt x G | ⟨ne_bot_of_le_ne_bot h.ne (inf_le_inf_left _ hFG)⟩ | lemma | acc_pt.mono | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"acc_pt",
"filter",
"inf_le_inf_left"
] | If `x` is an accumulation point of `F` and `F ≤ G`, then
`x` is an accumulation point of `D. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_eq_nhds' {s : set α} : interior s = {a | s ∈ 𝓝 a} | set.ext $ λ x, by simp only [mem_interior, mem_nhds_iff, mem_set_of_eq] | lemma | interior_eq_nhds' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"mem_interior",
"mem_nhds_iff",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_eq_nhds {s : set α} : interior s = {a | 𝓝 a ≤ 𝓟 s} | interior_eq_nhds'.trans $ by simp only [le_principal_iff] | lemma | interior_eq_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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