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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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mem_interior_iff_mem_nhds {s : set α} {a : α} :
a ∈ interior s ↔ s ∈ 𝓝 a | by rw [interior_eq_nhds', mem_set_of_eq] | lemma | mem_interior_iff_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_eq_nhds'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_mem_nhds {s : set α} {a : α} :
interior s ∈ 𝓝 a ↔ s ∈ 𝓝 a | ⟨λ h, mem_of_superset h interior_subset,
λ h, is_open.mem_nhds is_open_interior (mem_interior_iff_mem_nhds.2 h)⟩ | lemma | interior_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_subset",
"is_open.mem_nhds",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_set_of_eq {p : α → Prop} :
interior {x | p x} = {x | ∀ᶠ y in 𝓝 x, p y} | interior_eq_nhds' | lemma | interior_set_of_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_eq_nhds'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_set_of_eventually_nhds {p : α → Prop} :
is_open {x | ∀ᶠ y in 𝓝 x, p y} | by simp only [← interior_set_of_eq, is_open_interior] | lemma | is_open_set_of_eventually_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior_set_of_eq",
"is_open",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_iff_nhds {s V : set α} : s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x | show (∀ x, x ∈ s → x ∈ _) ↔ _, by simp_rw mem_interior_iff_mem_nhds | lemma | subset_interior_iff_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_nhds {s : set α} : is_open s ↔ ∀a∈s, 𝓝 a ≤ 𝓟 s | calc is_open s ↔ s ⊆ interior s : subset_interior_iff_is_open.symm
... ↔ (∀a∈s, 𝓝 a ≤ 𝓟 s) : by rw [interior_eq_nhds]; refl | lemma | is_open_iff_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_eq_nhds",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_mem_nhds {s : set α} : is_open s ↔ ∀a∈s, s ∈ 𝓝 a | is_open_iff_nhds.trans $ forall_congr $ λ _, imp_congr_right $ λ _, le_principal_iff | lemma | is_open_iff_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_iff_eventually {s : set α} : is_open s ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, y ∈ s | is_open_iff_mem_nhds | lemma | is_open_iff_eventually | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_iff_mem_nhds"
] | A set `s` is open iff for every point `x` in `s` and every `y` close to `x`, `y` is in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_iff_ultrafilter {s : set α} :
is_open s ↔ (∀ (x ∈ s) (l : ultrafilter α), ↑l ≤ 𝓝 x → s ∈ l) | by simp_rw [is_open_iff_mem_nhds, ← mem_iff_ultrafilter] | theorem | is_open_iff_ultrafilter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_iff_mem_nhds",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_singleton_iff_nhds_eq_pure (a : α) :
is_open ({a} : set α) ↔ 𝓝 a = pure a | begin
split,
{ intros h,
apply le_antisymm _ (pure_le_nhds a),
rw le_pure_iff,
exact h.mem_nhds (mem_singleton a) },
{ intros h,
simp [is_open_iff_nhds, h] }
end | lemma | is_open_singleton_iff_nhds_eq_pure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_iff_nhds",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_singleton_iff_punctured_nhds {α : Type*} [topological_space α] (a : α) :
is_open ({a} : set α) ↔ (𝓝[≠] a) = ⊥ | by rw [is_open_singleton_iff_nhds_eq_pure, nhds_within, ← mem_iff_inf_principal_compl,
← le_pure_iff, nhds_ne_bot.le_pure_iff] | lemma | is_open_singleton_iff_punctured_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_singleton_iff_nhds_eq_pure",
"nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_frequently {s : set α} {a : α} : a ∈ closure s ↔ ∃ᶠ x in 𝓝 a, x ∈ s | by rw [filter.frequently, filter.eventually, ← mem_interior_iff_mem_nhds,
closure_eq_compl_interior_compl]; refl | lemma | mem_closure_iff_frequently | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_eq_compl_interior_compl",
"filter.eventually",
"filter.frequently",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_frequently {s : set α} : is_closed s ↔ ∀ x, (∃ᶠ y in 𝓝 x, y ∈ s) → x ∈ s | begin
rw ← closure_subset_iff_is_closed,
apply forall_congr (λ x, _),
rw mem_closure_iff_frequently
end | lemma | is_closed_iff_frequently | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure_subset_iff_is_closed",
"is_closed",
"mem_closure_iff_frequently"
] | A set `s` is closed iff for every point `x`, if there is a point `y` close to `x` that belongs
to `s` then `x` is in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_set_of_cluster_pt {f : filter α} : is_closed {x | cluster_pt x f} | begin
simp only [cluster_pt, inf_ne_bot_iff_frequently_left, set_of_forall, imp_iff_not_or],
refine is_closed_Inter (λ p, is_closed.union _ _); apply is_closed_compl_iff.2,
exacts [is_open_set_of_eventually_nhds, is_open_const]
end | lemma | is_closed_set_of_cluster_pt | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"filter",
"imp_iff_not_or",
"is_closed",
"is_closed.union",
"is_closed_Inter",
"is_open_const",
"is_open_set_of_eventually_nhds"
] | The set of cluster points of a filter is closed. In particular, the set of limit points
of a sequence is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff_cluster_pt {s : set α} {a : α} : a ∈ closure s ↔ cluster_pt a (𝓟 s) | mem_closure_iff_frequently.trans cluster_pt_principal_iff_frequently.symm | theorem | mem_closure_iff_cluster_pt | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"cluster_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds_ne_bot {s : set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ | mem_closure_iff_cluster_pt.trans ne_bot_iff | lemma | mem_closure_iff_nhds_ne_bot | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds_within_ne_bot {s : set α} {x : α} :
x ∈ closure s ↔ ne_bot (𝓝[s] x) | mem_closure_iff_cluster_pt | lemma | mem_closure_iff_nhds_within_ne_bot | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"mem_closure_iff_cluster_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_compl_singleton (x : α) [ne_bot (𝓝[≠] x)] : dense ({x}ᶜ : set α) | begin
intro y,
unfreezingI { rcases eq_or_ne y x with rfl|hne },
{ rwa mem_closure_iff_nhds_within_ne_bot },
{ exact subset_closure hne }
end | lemma | dense_compl_singleton | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"eq_or_ne",
"mem_closure_iff_nhds_within_ne_bot",
"subset_closure"
] | If `x` is not an isolated point of a topological space, then `{x}ᶜ` is dense in the whole
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_compl_singleton (x : α) [ne_bot (𝓝[≠] x)] :
closure {x}ᶜ = (univ : set α) | (dense_compl_singleton x).closure_eq | lemma | closure_compl_singleton | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense_compl_singleton"
] | If `x` is not an isolated point of a topological space, then the closure of `{x}ᶜ` is the whole
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_singleton (x : α) [ne_bot (𝓝[≠] x)] :
interior {x} = (∅ : set α) | interior_eq_empty_iff_dense_compl.2 (dense_compl_singleton x) | lemma | interior_singleton | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_compl_singleton",
"interior"
] | If `x` is not an isolated point of a topological space, then the interior of `{x}` is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_open_singleton (x : α) [ne_bot (𝓝[≠] x)] : ¬ is_open ({x} : set α) | dense_compl_singleton_iff_not_open.1 (dense_compl_singleton x) | lemma | not_is_open_singleton | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_compl_singleton",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq_cluster_pts {s : set α} : closure s = {a | cluster_pt a (𝓟 s)} | set.ext $ λ x, mem_closure_iff_cluster_pt | lemma | closure_eq_cluster_pts | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"cluster_pt",
"mem_closure_iff_cluster_pt",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds {s : set α} {a : α} :
a ∈ closure s ↔ ∀ t ∈ 𝓝 a, (t ∩ s).nonempty | mem_closure_iff_cluster_pt.trans cluster_pt_principal_iff | theorem | mem_closure_iff_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"cluster_pt_principal_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds' {s : set α} {a : α} :
a ∈ closure s ↔ ∀ t ∈ 𝓝 a, ∃ y : s, ↑y ∈ t | by simp only [mem_closure_iff_nhds, set.inter_nonempty_iff_exists_right,
set_coe.exists, subtype.coe_mk] | theorem | mem_closure_iff_nhds' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"mem_closure_iff_nhds",
"set.inter_nonempty_iff_exists_right",
"set_coe.exists",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_comap_ne_bot {A : set α} {x : α} :
x ∈ closure A ↔ ne_bot (comap (coe : A → α) (𝓝 x)) | by simp_rw [mem_closure_iff_nhds, comap_ne_bot_iff, set.inter_nonempty_iff_exists_right,
set_coe.exists, subtype.coe_mk] | theorem | mem_closure_iff_comap_ne_bot | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"mem_closure_iff_nhds",
"set.inter_nonempty_iff_exists_right",
"set_coe.exists",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds_basis' {a : α} {p : ι → Prop} {s : ι → set α} (h : (𝓝 a).has_basis p s)
{t : set α} :
a ∈ closure t ↔ ∀ i, p i → (s i ∩ t).nonempty | mem_closure_iff_cluster_pt.trans $ (h.cluster_pt_iff (has_basis_principal _)).trans $
by simp only [exists_prop, forall_const] | theorem | mem_closure_iff_nhds_basis' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"exists_prop",
"forall_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → set α} (h : (𝓝 a).has_basis p s)
{t : set α} :
a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i | (mem_closure_iff_nhds_basis' h).trans $
by simp only [set.nonempty, mem_inter_iff, exists_prop, and_comm] | theorem | mem_closure_iff_nhds_basis | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"exists_prop",
"mem_closure_iff_nhds_basis'",
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_ultrafilter {s : set α} {x : α} :
x ∈ closure s ↔ ∃ (u : ultrafilter α), s ∈ u ∧ ↑u ≤ 𝓝 x | by simp [closure_eq_cluster_pts, cluster_pt, ← exists_ultrafilter_iff, and.comm] | lemma | mem_closure_iff_ultrafilter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_eq_cluster_pts",
"cluster_pt",
"ultrafilter"
] | `x` belongs to the closure of `s` if and only if some ultrafilter
supported on `s` converges to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_iff_cluster_pt {s : set α} : is_closed s ↔ ∀a, cluster_pt a (𝓟 s) → a ∈ s | calc is_closed s ↔ closure s ⊆ s : closure_subset_iff_is_closed.symm
... ↔ (∀a, cluster_pt a (𝓟 s) → a ∈ s) : by simp only [subset_def, mem_closure_iff_cluster_pt] | lemma | is_closed_iff_cluster_pt | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"cluster_pt",
"is_closed",
"mem_closure_iff_cluster_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_nhds {s : set α} : is_closed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).nonempty) → x ∈ s | by simp_rw [is_closed_iff_cluster_pt, cluster_pt, inf_principal_ne_bot_iff] | lemma | is_closed_iff_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"is_closed",
"is_closed_iff_cluster_pt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.interior_union_left {s t : set α} (h : is_closed s) :
interior (s ∪ t) ⊆ s ∪ interior t | λ a ⟨u, ⟨⟨hu₁, hu₂⟩, ha⟩⟩, (classical.em (a ∈ s)).imp_right $ λ h, mem_interior.mpr
⟨u ∩ sᶜ, λ x hx, (hu₂ hx.1).resolve_left hx.2, is_open.inter hu₁ is_closed.is_open_compl, ⟨ha, h⟩⟩ | lemma | is_closed.interior_union_left | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"is_closed",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.interior_union_right {s t : set α} (h : is_closed t) :
interior (s ∪ t) ⊆ interior s ∪ t | by simpa only [union_comm] using h.interior_union_left | lemma | is_closed.interior_union_right | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.inter_closure {s t : set α} (h : is_open s) : s ∩ closure t ⊆ closure (s ∩ t) | compl_subset_compl.mp $ by simpa only [← interior_compl, compl_inter]
using is_closed.interior_union_left h.is_closed_compl | lemma | is_open.inter_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"interior_compl",
"is_closed.interior_union_left",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.closure_inter {s t : set α} (h : is_open t) : closure s ∩ t ⊆ closure (s ∩ t) | by simpa only [inter_comm] using h.inter_closure | lemma | is_open.closure_inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.open_subset_closure_inter {s t : set α} (hs : dense s) (ht : is_open t) :
t ⊆ closure (t ∩ s) | calc t = t ∩ closure s : by rw [hs.closure_eq, inter_univ]
... ⊆ closure (t ∩ s) : ht.inter_closure | lemma | dense.open_subset_closure_inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_of_mem_closure_union {s₁ s₂ : set α} {x : α} (h : x ∈ closure (s₁ ∪ s₂))
(h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ | begin
rw mem_closure_iff_nhds_ne_bot at *,
rwa ← calc
𝓝 x ⊓ principal (s₁ ∪ s₂) = 𝓝 x ⊓ (principal s₁ ⊔ principal s₂) : by rw sup_principal
... = (𝓝 x ⊓ principal s₁) ⊔ (𝓝 x ⊓ principal s₂) : inf_sup_left
... = ⊥ ⊔ 𝓝 x ⊓ principal s₂ : by rw inf_principal_eq_bot.mpr h₁
... = 𝓝 x ⊓ principal s₂... | lemma | mem_closure_of_mem_closure_union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"bot_sup_eq",
"closure",
"inf_sup_left",
"mem_closure_iff_nhds_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.inter_of_open_left {s t : set α} (hs : dense s) (ht : dense t) (hso : is_open s) :
dense (s ∩ t) | λ x, (closure_minimal hso.inter_closure is_closed_closure) $
by simp [hs.closure_eq, ht.closure_eq] | lemma | dense.inter_of_open_left | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure_minimal",
"dense",
"is_closed_closure",
"is_open"
] | The intersection of an open dense set with a dense set is a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.inter_of_open_right {s t : set α} (hs : dense s) (ht : dense t) (hto : is_open t) :
dense (s ∩ t) | inter_comm t s ▸ ht.inter_of_open_left hs hto | lemma | dense.inter_of_open_right | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"is_open"
] | The intersection of a dense set with an open dense set is a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.inter_nhds_nonempty {s t : set α} (hs : dense s) {x : α} (ht : t ∈ 𝓝 x) :
(s ∩ t).nonempty | let ⟨U, hsub, ho, hx⟩ := mem_nhds_iff.1 ht in
(hs.inter_open_nonempty U ho ⟨x, hx⟩).mono $ λ y hy, ⟨hy.2, hsub hy.1⟩ | lemma | dense.inter_nhds_nonempty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_diff {s t : set α} : closure s \ closure t ⊆ closure (s \ t) | calc closure s \ closure t = (closure t)ᶜ ∩ closure s : by simp only [diff_eq, inter_comm]
... ⊆ closure ((closure t)ᶜ ∩ s) : (is_open_compl_iff.mpr $ is_closed_closure).inter_closure
... = closure (s \ closure t) : by simp only [diff_eq, inter_comm]
... ⊆ closure (s \ t) : closure_mono $ diff_subset_diff (subset... | lemma | closure_diff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_mono",
"is_closed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.frequently.mem_of_closed {a : α} {s : set α} (h : ∃ᶠ x in 𝓝 a, x ∈ s)
(hs : is_closed s) : a ∈ s | hs.closure_subset h.mem_closure | lemma | filter.frequently.mem_of_closed | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.mem_of_frequently_of_tendsto {f : β → α} {b : filter β} {a : α} {s : set α}
(hs : is_closed s) (h : ∃ᶠ x in b, f x ∈ s) (hf : tendsto f b (𝓝 a)) : a ∈ s | (hf.frequently $ show ∃ᶠ x in b, (λ y, y ∈ s) (f x), from h).mem_of_closed hs | lemma | is_closed.mem_of_frequently_of_tendsto | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.mem_of_tendsto {f : β → α} {b : filter β} {a : α} {s : set α}
[ne_bot b] (hs : is_closed s) (hf : tendsto f b (𝓝 a)) (h : ∀ᶠ x in b, f x ∈ s) : a ∈ s | hs.mem_of_frequently_of_tendsto h.frequently hf | lemma | is_closed.mem_of_tendsto | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_of_frequently_of_tendsto {f : β → α} {b : filter β} {a : α} {s : set α}
(h : ∃ᶠ x in b, f x ∈ s) (hf : tendsto f b (𝓝 a)) : a ∈ closure s | filter.frequently.mem_closure $ hf.frequently h | lemma | mem_closure_of_frequently_of_tendsto | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_of_tendsto {f : β → α} {b : filter β} {a : α} {s : set α}
[ne_bot b] (hf : tendsto f b (𝓝 a)) (h : ∀ᶠ x in b, f x ∈ s) : a ∈ closure s | mem_closure_of_frequently_of_tendsto h.frequently hf | lemma | mem_closure_of_tendsto | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"filter",
"mem_closure_of_frequently_of_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : filter β} {s : set β}
{a : α} (h : ∀ x ∉ s, f x = a) :
tendsto f (l ⊓ 𝓟 s) (𝓝 a) ↔ tendsto f l (𝓝 a) | begin
rw [tendsto_iff_comap, tendsto_iff_comap],
replace h : 𝓟 sᶜ ≤ comap f (𝓝 a),
{ rintros U ⟨t, ht, htU⟩ x hx,
have : f x ∈ t, from (h x hx).symm ▸ mem_of_mem_nhds ht,
exact htU this },
refine ⟨λ h', _, le_trans inf_le_left⟩,
have := sup_le h' h,
rw [sup_inf_right, sup_principal, union_compl_se... | lemma | tendsto_inf_principal_nhds_iff_of_forall_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"inf_top_eq",
"mem_of_mem_nhds",
"sup_inf_right",
"sup_le",
"sup_le_iff"
] | Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lim [nonempty α] (f : filter α) : α | epsilon $ λa, f ≤ 𝓝 a | def | Lim | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter"
] | If `f` is a filter, then `Lim f` is a limit of the filter, if it exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lim' (f : filter α) [ne_bot f] : α | @Lim _ _ (nonempty_of_ne_bot f) f | def | Lim' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"Lim",
"filter"
] | If `f` is a filter satisfying `ne_bot f`, then `Lim' f` is a limit of the filter, if it exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter.Lim : ultrafilter α → α | λ F, Lim' F | def | ultrafilter.Lim | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"Lim'",
"ultrafilter"
] | If `F` is an ultrafilter, then `filter.ultrafilter.Lim F` is a limit of the filter, if it exists.
Note that dot notation `F.Lim` can be used for `F : ultrafilter α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim [nonempty α] (f : filter β) (g : β → α) : α | Lim (f.map g) | def | lim | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"Lim",
"filter"
] | If `f` is a filter in `β` and `g : β → α` is a function, then `lim f` is a limit of `g` at `f`,
if it exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_nhds_Lim {f : filter α} (h : ∃a, f ≤ 𝓝 a) : f ≤ 𝓝 (@Lim _ _ (nonempty_of_exists h) f) | epsilon_spec h | lemma | le_nhds_Lim | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"Lim",
"filter"
] | If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Lim f)`. We formulate
this lemma with a `[nonempty α]` argument of `Lim` derived from `h` to make it useful for types
without a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify
this instance with any other insta... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds_lim {f : filter β} {g : β → α} (h : ∃ a, tendsto g f (𝓝 a)) :
tendsto g f (𝓝 $ @lim _ _ _ (nonempty_of_exists h) f g) | le_nhds_Lim h | lemma | tendsto_nhds_lim | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"filter",
"le_nhds_Lim",
"lim"
] | If `g` tends to some `𝓝 a` along `f`, then it tends to `𝓝 (lim f g)`. We formulate
this lemma with a `[nonempty α]` argument of `lim` derived from `h` to make it useful for types
without a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify
this instance with any other instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (f : α → β) : Prop | (is_open_preimage : ∀s, is_open s → is_open (f ⁻¹' s)) | structure | continuous | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | A function between topological spaces is continuous if the preimage
of every open set is open. Registered as a structure to make sure it is not unfolded by Lean. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_def {f : α → β} : continuous f ↔ (∀s, is_open s → is_open (f ⁻¹' s)) | ⟨λ hf s hs, hf.is_open_preimage s hs, λ h, ⟨h⟩⟩ | lemma | continuous_def | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.preimage {f : α → β} (hf : continuous f) {s : set β} (h : is_open s) :
is_open (f ⁻¹' s) | hf.is_open_preimage s h | lemma | is_open.preimage | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.congr {f g : α → β} (h : continuous f) (h' : ∀ x, f x = g x) : continuous g | by { convert h, ext, rw h' } | lemma | continuous.congr | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at (f : α → β) (x : α) | tendsto f (𝓝 x) (𝓝 (f x)) | def | continuous_at | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [] | A function between topological spaces is continuous at a point `x₀`
if `f x` tends to `f x₀` when `x` tends to `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at.tendsto {f : α → β} {x : α} (h : continuous_at f x) :
tendsto f (𝓝 x) (𝓝 (f x)) | h | lemma | continuous_at.tendsto | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_def {f : α → β} {x : α} : continuous_at f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x | iff.rfl | lemma | continuous_at_def | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_congr {f g : α → β} {x : α} (h : f =ᶠ[𝓝 x] g) :
continuous_at f x ↔ continuous_at g x | by simp only [continuous_at, tendsto_congr' h, h.eq_of_nhds] | lemma | continuous_at_congr | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.congr {f g : α → β} {x : α} (hf : continuous_at f x) (h : f =ᶠ[𝓝 x] g) :
continuous_at g x | (continuous_at_congr h).1 hf | lemma | continuous_at.congr | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at",
"continuous_at_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.preimage_mem_nhds {f : α → β} {x : α} {t : set β} (h : continuous_at f x)
(ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝 x | h ht | lemma | continuous_at.preimage_mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_zero_nhds {M₀} [has_zero M₀] {a : α} {f : α → M₀} :
f =ᶠ[𝓝 a] 0 ↔ a ∉ closure (function.support f) | by rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, function.compl_support]; refl | lemma | eventually_eq_zero_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"function.support",
"interior_compl",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_pt.map {x : α} {la : filter α} {lb : filter β} (H : cluster_pt x la)
{f : α → β} (hfc : continuous_at f x) (hf : tendsto f la lb) :
cluster_pt (f x) lb | ⟨ne_bot_of_le_ne_bot ((map_ne_bot_iff f).2 H).ne $ hfc.tendsto.inf hf⟩ | lemma | cluster_pt.map | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"cluster_pt",
"continuous_at",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_interior_subset_interior_preimage {f : α → β} {s : set β}
(hf : continuous f) : f⁻¹' (interior s) ⊆ interior (f⁻¹' s) | interior_maximal (preimage_mono interior_subset) (is_open_interior.preimage hf) | lemma | preimage_interior_subset_interior_preimage | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"interior",
"interior_maximal",
"interior_subset"
] | See also `interior_preimage_subset_preimage_interior`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_id : continuous (id : α → α) | continuous_def.2 $ assume s h, h | lemma | continuous_id | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.comp {g : β → γ} {f : α → β} (hg : continuous g) (hf : continuous f) :
continuous (g ∘ f) | continuous_def.2 $ assume s h, (h.preimage hg).preimage hf | lemma | continuous.comp | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.iterate {f : α → α} (h : continuous f) (n : ℕ) : continuous (f^[n]) | nat.rec_on n continuous_id (λ n ihn, ihn.comp h) | lemma | continuous.iterate | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp {g : β → γ} {f : α → β} {x : α}
(hg : continuous_at g (f x)) (hf : continuous_at f x) :
continuous_at (g ∘ f) x | hg.comp hf | lemma | continuous_at.comp | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.comp_of_eq {g : β → γ} {f : α → β} {x : α} {y : β}
(hg : continuous_at g y) (hf : continuous_at f x) (hy : f x = y) :
continuous_at (g ∘ f) x | by { subst hy, exact hg.comp hf } | lemma | continuous_at.comp_of_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | See note [comp_of_eq lemmas] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.tendsto {f : α → β} (hf : continuous f) (x) :
tendsto f (𝓝 x) (𝓝 (f x)) | ((nhds_basis_opens x).tendsto_iff $ nhds_basis_opens $ f x).2 $
λ t ⟨hxt, ht⟩, ⟨f ⁻¹' t, ⟨hxt, ht.preimage hf⟩, subset.refl _⟩ | lemma | continuous.tendsto | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.tendsto' {f : α → β} (hf : continuous f) (x : α) (y : β) (h : f x = y) :
tendsto f (𝓝 x) (𝓝 y) | h ▸ hf.tendsto x | lemma | continuous.tendsto' | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous"
] | A version of `continuous.tendsto` that allows one to specify a simpler form of the limit.
E.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.continuous_at {f : α → β} {x : α} (h : continuous f) :
continuous_at f x | h.tendsto x | lemma | continuous.continuous_at | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_continuous_at {f : α → β} : continuous f ↔ ∀ x, continuous_at f x | ⟨continuous.tendsto,
assume hf : ∀x, tendsto f (𝓝 x) (𝓝 (f x)),
continuous_def.2 $
assume s, assume hs : is_open s,
have ∀a, f a ∈ s → s ∈ 𝓝 (f a),
from λ a ha, is_open.mem_nhds hs ha,
show is_open (f ⁻¹' s),
from is_open_iff_nhds.2 $ λ a ha, le_principal_iff.2 $ hf _ (this a ha)⟩ | lemma | continuous_iff_continuous_at | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"continuous_at",
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const {x : α} {b : β} : continuous_at (λ a:α, b) x | tendsto_const_nhds | lemma | continuous_at_const | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const {b : β} : continuous (λa:α, b) | continuous_iff_continuous_at.mpr $ assume a, continuous_at_const | lemma | continuous_const | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"continuous_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.continuous_at {x : α} {f : α → β} {y : β} (h : f =ᶠ[𝓝 x] (λ _, y)) :
continuous_at f x | (continuous_at_congr h).2 tendsto_const_nhds | lemma | filter.eventually_eq.continuous_at | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at",
"continuous_at_congr",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_const {f : α → β} (h : ∀ x y, f x = f y) : continuous f | continuous_iff_continuous_at.mpr $ λ x, filter.eventually_eq.continuous_at $
eventually_of_forall (λ y, h y x) | lemma | continuous_of_const | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"filter.eventually_eq.continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_id {x : α} : continuous_at id x | continuous_id.continuous_at | lemma | continuous_at_id | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.iterate {f : α → α} {x : α} (hf : continuous_at f x) (hx : f x = x) (n : ℕ) :
continuous_at (f^[n]) x | nat.rec_on n continuous_at_id $ λ n ihn,
show continuous_at (f^[n] ∘ f) x,
from continuous_at.comp (hx.symm ▸ ihn) hf | lemma | continuous_at.iterate | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_is_closed {f : α → β} :
continuous f ↔ (∀s, is_closed s → is_closed (f ⁻¹' s)) | ⟨assume hf s hs, by simpa using (continuous_def.1 hf sᶜ hs.is_open_compl).is_closed_compl,
assume hf, continuous_def.2 $ assume s,
by rw [←is_closed_compl_iff, ←is_closed_compl_iff]; exact hf _⟩ | lemma | continuous_iff_is_closed | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.preimage {f : α → β} (hf : continuous f) {s : set β} (h : is_closed s) :
is_closed (f ⁻¹' s) | continuous_iff_is_closed.mp hf s h | lemma | is_closed.preimage | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_image {f : α → β} {x : α} {s : set α} (hf : continuous_at f x)
(hx : x ∈ closure s) : f x ∈ closure (f '' s) | mem_closure_of_frequently_of_tendsto
((mem_closure_iff_frequently.1 hx).mono (λ x, mem_image_of_mem _)) hf | lemma | mem_closure_image | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous_at",
"mem_closure_of_frequently_of_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff_ultrafilter {f : α → β} {x} : continuous_at f x ↔
∀ g : ultrafilter α, ↑g ≤ 𝓝 x → tendsto f g (𝓝 (f x)) | tendsto_iff_ultrafilter f (𝓝 x) (𝓝 (f x)) | lemma | continuous_at_iff_ultrafilter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous_at",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_ultrafilter {f : α → β} :
continuous f ↔ ∀ x (g : ultrafilter α), ↑g ≤ 𝓝 x → tendsto f g (𝓝 (f x)) | by simp only [continuous_iff_continuous_at, continuous_at_iff_ultrafilter] | lemma | continuous_iff_ultrafilter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"continuous_at_iff_ultrafilter",
"continuous_iff_continuous_at",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.closure_preimage_subset {f : α → β}
(hf : continuous f) (t : set β) :
closure (f ⁻¹' t) ⊆ f ⁻¹' (closure t) | begin
rw ← (is_closed_closure.preimage hf).closure_eq,
exact closure_mono (preimage_mono subset_closure),
end | lemma | continuous.closure_preimage_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_mono",
"continuous",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.frontier_preimage_subset
{f : α → β} (hf : continuous f) (t : set β) :
frontier (f ⁻¹' t) ⊆ f ⁻¹' (frontier t) | diff_subset_diff (hf.closure_preimage_subset t) (preimage_interior_subset_interior_preimage hf) | lemma | continuous.frontier_preimage_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"frontier",
"preimage_interior_subset_interior_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.maps_to.closure {s : set α} {t : set β} {f : α → β} (h : maps_to f s t)
(hc : continuous f) : maps_to f (closure s) (closure t) | begin
simp only [maps_to, mem_closure_iff_cluster_pt],
exact λ x hx, hx.map hc.continuous_at (tendsto_principal_principal.2 h)
end | lemma | set.maps_to.closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous",
"mem_closure_iff_cluster_pt"
] | If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_closure_subset_closure_image {f : α → β} {s : set α} (h : continuous f) :
f '' closure s ⊆ closure (f '' s) | ((maps_to_image f s).closure h).image_subset | lemma | image_closure_subset_closure_image | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_subset_preimage_closure_image {f : α → β} {s : set α} (h : continuous f) :
closure s ⊆ f ⁻¹' (closure (f '' s)) | by { rw ← set.image_subset_iff, exact image_closure_subset_closure_image h } | lemma | closure_subset_preimage_closure_image | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous",
"image_closure_subset_closure_image",
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mem_closure {s : set α} {t : set β} {f : α → β} {a : α}
(hf : continuous f) (ha : a ∈ closure s) (ht : maps_to f s t) : f a ∈ closure t | ht.closure hf ha | lemma | map_mem_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.maps_to.closure_left {s : set α} {t : set β} {f : α → β} (h : maps_to f s t)
(hc : continuous f) (ht : is_closed t) : maps_to f (closure s) t | ht.closure_eq ▸ h.closure hc | lemma | set.maps_to.closure_left | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous",
"is_closed"
] | If a continuous map `f` maps `s` to a closed set `t`, then it maps `closure s` to `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range | dense (range f) | def | dense_range | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense"
] | `f : ι → β` has dense range if its range (image) is a dense subset of β. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.dense_range (hf : function.surjective f) : dense_range f | λ x, by simp [hf.range_eq] | lemma | function.surjective.dense_range | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range"
] | A surjective map has dense range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range_id : dense_range (id : α → α) | function.surjective.dense_range function.surjective_id | lemma | dense_range_id | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range",
"function.surjective.dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_iff_closure_range : dense_range f ↔ closure (range f) = univ | dense_iff_closure_eq | lemma | dense_range_iff_closure_range | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense_iff_closure_eq",
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.closure_range (h : dense_range f) : closure (range f) = univ | h.closure_eq | lemma | dense_range.closure_range | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.dense_range_coe {s : set α} (h : dense s) : dense_range (coe : s → α) | by simpa only [dense_range, subtype.range_coe_subtype] | lemma | dense.dense_range_coe | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense",
"dense_range",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.range_subset_closure_image_dense {f : α → β} (hf : continuous f)
{s : set α} (hs : dense s) :
range f ⊆ closure (f '' s) | by { rw [← image_univ, ← hs.closure_eq], exact image_closure_subset_closure_image hf } | lemma | continuous.range_subset_closure_image_dense | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"continuous",
"dense",
"image_closure_subset_closure_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.dense_image {f : α → β} (hf' : dense_range f) (hf : continuous f)
{s : set α} (hs : dense s) :
dense (f '' s) | (hf'.mono $ hf.range_subset_closure_image_dense hs).of_closure | lemma | dense_range.dense_image | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"dense",
"dense_range"
] | The image of a dense set under a continuous map with dense range is a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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