fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Icc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Icc b c) :=
inter_mem (Ici_mem_nhdsSet_Icc h) (Iic_mem_nhdsSet_Icc h')
/-! | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Icc_mem_nhdsSet_Icc | null |
Ici_mem_nhdsSet_Ico (h : a < b) : Ici a ∈ 𝓝ˢ (Ico b c) :=
nhdsSet_mono Ico_subset_Icc_self <| Ici_mem_nhdsSet_Icc h | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ici_mem_nhdsSet_Ico | null |
Ioi_mem_nhdsSet_Ico (h : a < b) : Ioi a ∈ 𝓝ˢ (Ico b c) :=
nhdsSet_mono Ico_subset_Icc_self <| Ioi_mem_nhdsSet_Icc h | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioi_mem_nhdsSet_Ico | null |
Iio_mem_nhdsSet_Ico (h : b ≤ c) : Iio c ∈ 𝓝ˢ (Ico a b) :=
nhdsSet_mono Ico_subset_Iio_self <| by simpa | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iio_mem_nhdsSet_Ico | null |
Iic_mem_nhdsSet_Ico (h : b ≤ c) : Iic c ∈ 𝓝ˢ (Ico a b) :=
mem_of_superset (Iio_mem_nhdsSet_Ico h) Iio_subset_Iic_self | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iic_mem_nhdsSet_Ico | null |
Ioo_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioo a d ∈ 𝓝ˢ (Ico b c) :=
inter_mem (Ioi_mem_nhdsSet_Ico h) (Iio_mem_nhdsSet_Ico h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioo_mem_nhdsSet_Ico | null |
Icc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Icc a d ∈ 𝓝ˢ (Ico b c) :=
inter_mem (Ici_mem_nhdsSet_Ico h) (Iic_mem_nhdsSet_Ico h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Icc_mem_nhdsSet_Ico | null |
Ioc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioc a d ∈ 𝓝ˢ (Ico b c) :=
inter_mem (Ioi_mem_nhdsSet_Ico h) (Iic_mem_nhdsSet_Ico h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioc_mem_nhdsSet_Ico | null |
Ico_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ico a d ∈ 𝓝ˢ (Ico b c) :=
inter_mem (Ici_mem_nhdsSet_Ico h) (Iio_mem_nhdsSet_Ico h')
/-! | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ico_mem_nhdsSet_Ico | null |
Ioi_mem_nhdsSet_Ioc (h : a ≤ b) : Ioi a ∈ 𝓝ˢ (Ioc b c) :=
nhdsSet_mono Ioc_subset_Ioi_self <| by simpa | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioi_mem_nhdsSet_Ioc | null |
Iio_mem_nhdsSet_Ioc (h : b < c) : Iio c ∈ 𝓝ˢ (Ioc a b) :=
nhdsSet_mono Ioc_subset_Icc_self <| Iio_mem_nhdsSet_Icc h | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iio_mem_nhdsSet_Ioc | null |
Ici_mem_nhdsSet_Ioc (h : a ≤ b) : Ici a ∈ 𝓝ˢ (Ioc b c) :=
mem_of_superset (Ioi_mem_nhdsSet_Ioc h) Ioi_subset_Ici_self | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ici_mem_nhdsSet_Ioc | null |
Iic_mem_nhdsSet_Ioc (h : b < c) : Iic c ∈ 𝓝ˢ (Ioc a b) :=
nhdsSet_mono Ioc_subset_Icc_self <| Iic_mem_nhdsSet_Icc h | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iic_mem_nhdsSet_Ioc | null |
Ioo_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Ioc b c) :=
inter_mem (Ioi_mem_nhdsSet_Ioc h) (Iio_mem_nhdsSet_Ioc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioo_mem_nhdsSet_Ioc | null |
Icc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Ioc b c) :=
inter_mem (Ici_mem_nhdsSet_Ioc h) (Iic_mem_nhdsSet_Ioc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Icc_mem_nhdsSet_Ioc | null |
Ioc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Ioc b c) :=
inter_mem (Ioi_mem_nhdsSet_Ioc h) (Iic_mem_nhdsSet_Ioc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ioc_mem_nhdsSet_Ioc | null |
Ico_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Ioc b c) :=
inter_mem (Ici_mem_nhdsSet_Ioc h) (Iio_mem_nhdsSet_Ioc h') | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ico_mem_nhdsSet_Ioc | null |
hasBasis_nhdsSet_Iic_Iio (a : α) [h : Nonempty (Ioi a)] :
HasBasis (𝓝ˢ (Iic a)) (a < ·) Iio := by
refine ⟨fun s ↦ ⟨fun hs ↦ ?_, fun ⟨b, hab, hb⟩ ↦ mem_of_superset (Iio_mem_nhdsSet_Iic hab) hb⟩⟩
rw [nhdsSet_Iic, mem_sup, mem_principal] at hs
rcases exists_Ico_subset_of_mem_nhds hs.1 (Set.nonempty_coe_sort.1 h) with ⟨b, hab, hbs⟩
exact ⟨b, hab, Iio_subset_Iio_union_Ico.trans (union_subset hs.2 hbs)⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | hasBasis_nhdsSet_Iic_Iio | null |
hasBasis_nhdsSet_Iic_Iic (a : α) [NeBot (𝓝[>] a)] :
HasBasis (𝓝ˢ (Iic a)) (a < ·) Iic := by
have : Nonempty (Ioi a) :=
(Filter.nonempty_of_mem (self_mem_nhdsWithin : Ioi a ∈ 𝓝[>] a)).to_subtype
refine (hasBasis_nhdsSet_Iic_Iio _).to_hasBasis
(fun c hc ↦ ?_) (fun _ h ↦ ⟨_, h, Iio_subset_Iic_self⟩)
simpa only [Iic_subset_Iio] using Filter.nonempty_of_mem (Ioo_mem_nhdsGT hc)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | hasBasis_nhdsSet_Iic_Iic | null |
Iic_mem_nhdsSet_Iic_iff {a b : α} [NeBot (𝓝[>] b)] : Iic a ∈ 𝓝ˢ (Iic b) ↔ b < a :=
(hasBasis_nhdsSet_Iic_Iic b).mem_iff.trans
⟨fun ⟨_c, hbc, hca⟩ ↦ hbc.trans_le (Iic_subset_Iic.1 hca), fun h ↦ ⟨_, h, Subset.rfl⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Iic_mem_nhdsSet_Iic_iff | null |
hasBasis_nhdsSet_Ici_Ioi (a : α) [Nonempty (Iio a)] :
HasBasis (𝓝ˢ (Ici a)) (· < a) Ioi :=
have : Nonempty (Ioi (toDual a)) := ‹_›; hasBasis_nhdsSet_Iic_Iio (toDual a) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | hasBasis_nhdsSet_Ici_Ioi | null |
hasBasis_nhdsSet_Ici_Ici (a : α) [NeBot (𝓝[<] a)] :
HasBasis (𝓝ˢ (Ici a)) (· < a) Ici :=
have : NeBot (𝓝[>] (toDual a)) := ‹_›; hasBasis_nhdsSet_Iic_Iic (toDual a)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | hasBasis_nhdsSet_Ici_Ici | null |
Ici_mem_nhdsSet_Ici_iff {a b : α} [NeBot (𝓝[<] b)] : Ici a ∈ 𝓝ˢ (Ici b) ↔ a < b :=
have : NeBot (𝓝[>] (toDual b)) := ‹_›; Iic_mem_nhdsSet_Iic_iff (a := toDual a) (b := toDual b) | theorem | Topology | [
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/NhdsSet.lean | Ici_mem_nhdsSet_Ici_iff | null |
ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- For any `a`, the set `(-∞, a]` is closed. -/
isClosed_Iic (a : α) : IsClosed (Iic a) | class | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | ClosedIicTopology | If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is
closed for all `a : α`. |
ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- For any `a`, the set `[a, +∞)` is closed. -/
isClosed_Ici (a : α) : IsClosed (Ici a) | class | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | ClosedIciTopology | If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is
closed for all `a : α`. |
OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- The set `{ (x, y) | x ≤ y }` is a closed set. -/
isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 } | class | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | OrderClosedTopology | A topology on a set which is both a topological space and a preorder is _order-closed_ if the
set of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.
This property is satisfied for the order topology on a linear order, but it can be satisfied more
generally, and suffices to derive many interesting properties relating order and topology. |
Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :
Dense (OrderDual.ofDual ⁻¹' s) :=
hs | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.orderDual | null |
protected BddAbove.of_closure : BddAbove (closure s) → BddAbove s :=
BddAbove.mono subset_closure | lemma | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | BddAbove.of_closure | null |
protected BddBelow.of_closure : BddBelow (closure s) → BddBelow s :=
BddBelow.mono subset_closure | lemma | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | BddBelow.of_closure | null |
isClosed_Iic : IsClosed (Iic a) :=
ClosedIicTopology.isClosed_Iic a | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_Iic | null |
@[simp]
closure_Iic (a : α) : closure (Iic a) = Iic a :=
isClosed_Iic.closure_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | closure_Iic | null |
le_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a))
(h : ∃ᶠ c in x, f c ≤ b) : a ≤ b :=
isClosed_Iic.mem_of_frequently_of_tendsto h lim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_of_tendsto_of_frequently | null |
le_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
(h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=
isClosed_Iic.mem_of_tendsto lim h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_of_tendsto | null |
le_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
(h : ∀ c, f c ≤ b) : a ≤ b :=
le_of_tendsto lim (Eventually.of_forall h)
@[simp] lemma upperBounds_closure (s : Set α) : upperBounds (closure s : Set α) = upperBounds s :=
ext fun a ↦ by simp_rw [mem_upperBounds_iff_subset_Iic, isClosed_Iic.closure_subset_iff]
@[simp] lemma bddAbove_closure : BddAbove (closure s) ↔ BddAbove s := by
simp_rw [BddAbove, upperBounds_closure]
protected alias ⟨_, BddAbove.closure⟩ := bddAbove_closure
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | le_of_tendsto' | null |
disjoint_nhds_atBot_iff : Disjoint (𝓝 a) atBot ↔ ¬IsBot a := by
constructor
· intro hd hbot
rw [hbot.atBot_eq, disjoint_principal_right] at hd
exact mem_of_mem_nhds hd le_rfl
· simp only [IsBot, not_forall]
rintro ⟨b, hb⟩
refine disjoint_of_disjoint_of_mem disjoint_compl_left ?_ (Iic_mem_atBot b)
exact isClosed_Iic.isOpen_compl.mem_nhds hb | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | disjoint_nhds_atBot_iff | null |
IsLUB.range_of_tendsto {F : Filter β} [F.NeBot] (hle : ∀ i, f i ≤ a)
(hlim : Tendsto f F (𝓝 a)) : IsLUB (range f) a :=
⟨forall_mem_range.mpr hle, fun _c hc ↦ le_of_tendsto' hlim fun i ↦ hc <| mem_range_self i⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | IsLUB.range_of_tendsto | null |
disjoint_nhds_atBot (a : α) : Disjoint (𝓝 a) atBot := by simp
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | disjoint_nhds_atBot | null |
inf_nhds_atBot (a : α) : 𝓝 a ⊓ atBot = ⊥ := (disjoint_nhds_atBot a).eq_bot | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | inf_nhds_atBot | null |
not_tendsto_nhds_of_tendsto_atBot (hf : Tendsto f l atBot) (a : α) : ¬Tendsto f l (𝓝 a) :=
hf.not_tendsto (disjoint_nhds_atBot a).symm | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | not_tendsto_nhds_of_tendsto_atBot | null |
not_tendsto_atBot_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atBot :=
hf.not_tendsto (disjoint_nhds_atBot a) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | not_tendsto_atBot_of_tendsto_nhds | null |
iSup_eq_of_forall_le_of_tendsto {ι : Type*} {F : Filter ι} [Filter.NeBot F]
[ConditionallyCompleteLattice α] [TopologicalSpace α] [ClosedIicTopology α]
{a : α} {f : ι → α} (hle : ∀ i, f i ≤ a) (hlim : Filter.Tendsto f F (𝓝 a)) :
⨆ i, f i = a :=
have := F.nonempty_of_neBot
(IsLUB.range_of_tendsto hle hlim).ciSup_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | iSup_eq_of_forall_le_of_tendsto | null |
iUnion_Iic_eq_Iio_of_lt_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α]
{a : α} {f : ι → α} (hlt : ∀ i, f i < a) (hlim : Tendsto f F (𝓝 a)) :
⋃ i : ι, Iic (f i) = Iio a := by
have obs : a ∉ range f := by
rw [mem_range]
rintro ⟨i, rfl⟩
exact (hlt i).false
rw [← biUnion_range, (IsLUB.range_of_tendsto (le_of_lt <| hlt ·) hlim).biUnion_Iic_eq_Iio obs] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | iUnion_Iic_eq_Iio_of_lt_of_tendsto | null |
isOpen_Ioi : IsOpen (Ioi a) := by
rw [← compl_Iic]
exact isClosed_Iic.isOpen_compl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isOpen_Ioi | null |
interior_Ioi : interior (Ioi a) = Ioi a :=
isOpen_Ioi.interior_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | interior_Ioi | null |
Ioi_mem_nhds (h : a < b) : Ioi a ∈ 𝓝 b := IsOpen.mem_nhds isOpen_Ioi h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioi_mem_nhds | null |
eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | eventually_gt_nhds | null |
Ici_mem_nhds (h : a < b) : Ici a ∈ 𝓝 b :=
mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ici_mem_nhds | null |
eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | eventually_ge_nhds | null |
Filter.Tendsto.eventually_const_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
(h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=
h.eventually <| eventually_gt_nhds hv | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.eventually_const_lt | null |
Filter.Tendsto.eventually_const_le {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
(h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=
h.eventually <| eventually_ge_nhds hv | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.eventually_const_le | null |
protected Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
∃ y ∈ s, x < y :=
hs.exists_mem_open isOpen_Ioi (exists_gt x) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_gt | null |
protected Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
∃ y ∈ s, x ≤ y :=
(hs.exists_gt x).imp fun _ h ↦ ⟨h.1, h.2.le⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_ge | null |
Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :
∃ y ∈ s, x ≤ y := by
by_cases hx : IsTop x
· exact ⟨x, htop x hx, le_rfl⟩
· simp only [IsTop, not_forall, not_le] at hx
rcases hs.exists_mem_open isOpen_Ioi hx with ⟨y, hys, hy : x < y⟩
exact ⟨y, hys, hy.le⟩
/-! | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Dense.exists_ge' | null |
Ioo_mem_nhdsLT (H : a < b) : Ioo a b ∈ 𝓝[<] b := by
simpa only [← Iio_inter_Ioi] using inter_mem_nhdsWithin _ (Ioi_mem_nhds H) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhdsLT | null |
Ioo_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b :=
mem_of_superset (Ioo_mem_nhdsLT H.1) <| Ioo_subset_Ioo_right H.2 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhdsLT_of_mem | null |
protected CovBy.nhdsLT (h : a ⋖ b) : 𝓝[<] b = ⊥ :=
empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsLT h.1 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | CovBy.nhdsLT | null |
protected PredOrder.nhdsLT [PredOrder α] : 𝓝[<] a = ⊥ := by
if h : IsMin a then simp [h.Iio_eq]
else exact (Order.pred_covBy_of_not_isMin h).nhdsLT | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | PredOrder.nhdsLT | null |
PredOrder.nhdsGT_eq_nhdsNE [PredOrder α] (a : α) : 𝓝[>] a = 𝓝[≠] a := by
rw [← nhdsLT_sup_nhdsGT, PredOrder.nhdsLT, bot_sup_eq] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | PredOrder.nhdsGT_eq_nhdsNE | null |
PredOrder.nhdsGE_eq_nhds [PredOrder α] (a : α) : 𝓝[≥] a = 𝓝 a := by
rw [← nhdsLT_sup_nhdsGE, PredOrder.nhdsLT, bot_sup_eq] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | PredOrder.nhdsGE_eq_nhds | null |
Ico_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b :=
mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Ico_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsLT_of_mem | null |
Ico_mem_nhdsLT (H : a < b) : Ico a b ∈ 𝓝[<] b := Ico_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsLT | null |
Ioc_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b :=
mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Ioc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsLT_of_mem | null |
Ioc_mem_nhdsLT (H : a < b) : Ioc a b ∈ 𝓝[<] b := Ioc_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsLT | null |
Icc_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b :=
mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Icc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsLT_of_mem | null |
Icc_mem_nhdsLT (H : a < b) : Icc a b ∈ 𝓝[<] b := Icc_mem_nhdsLT_of_mem ⟨H, le_rfl⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsLT | null |
nhdsWithin_Ico_eq_nhdsLT (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b :=
nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ici_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Ico_eq_nhdsLT | null |
nhdsWithin_Ioo_eq_nhdsLT (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b :=
nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ioi_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Ioo_eq_nhdsLT | null |
continuousWithinAt_Ico_iff_Iio (h : a < b) :
ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by
simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsLT h]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Ico_iff_Iio | null |
continuousWithinAt_Ioo_iff_Iio (h : a < b) :
ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by
simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsLT h]
/-! | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Ioo_iff_Iio | null |
protected CovBy.nhdsLE (H : a ⋖ b) : 𝓝[≤] b = pure b := by
rw [← Iio_insert, nhdsWithin_insert, H.nhdsLT, sup_bot_eq] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | CovBy.nhdsLE | null |
protected PredOrder.nhdsLE [PredOrder α] : 𝓝[≤] b = pure b := by
rw [← Iio_insert, nhdsWithin_insert, PredOrder.nhdsLT, sup_bot_eq] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | PredOrder.nhdsLE | null |
Ioc_mem_nhdsLE (H : a < b) : Ioc a b ∈ 𝓝[≤] b :=
inter_mem (nhdsWithin_le_nhds <| Ioi_mem_nhds H) self_mem_nhdsWithin | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsLE | null |
Ioo_mem_nhdsLE_of_mem (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b :=
mem_of_superset (Ioc_mem_nhdsLE H.1) <| Ioc_subset_Ioo_right H.2 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioo_mem_nhdsLE_of_mem | null |
Ico_mem_nhdsLE_of_mem (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b :=
mem_of_superset (Ioo_mem_nhdsLE_of_mem H) Ioo_subset_Ico_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ico_mem_nhdsLE_of_mem | null |
Ioc_mem_nhdsLE_of_mem (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b :=
mem_of_superset (Ioc_mem_nhdsLE H.1) <| Ioc_subset_Ioc_right H.2 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Ioc_mem_nhdsLE_of_mem | null |
Icc_mem_nhdsLE_of_mem (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b :=
mem_of_superset (Ioc_mem_nhdsLE_of_mem H) Ioc_subset_Icc_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsLE_of_mem | null |
Icc_mem_nhdsLE (H : a < b) : Icc a b ∈ 𝓝[≤] b := Icc_mem_nhdsLE_of_mem ⟨H, le_rfl⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Icc_mem_nhdsLE | null |
nhdsWithin_Icc_eq_nhdsLE (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b :=
nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ici_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Icc_eq_nhdsLE | null |
nhdsWithin_Ioc_eq_nhdsLE (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b :=
nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ioi_mem_nhds h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | nhdsWithin_Ioc_eq_nhdsLE | null |
continuousWithinAt_Icc_iff_Iic (h : a < b) :
ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by
simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsLE h]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Icc_iff_Iic | null |
continuousWithinAt_Ioc_iff_Iic (h : a < b) :
ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by
simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | continuousWithinAt_Ioc_iff_Iic | null |
isClosed_Ici {a : α} : IsClosed (Ici a) :=
ClosedIciTopology.isClosed_Ici a | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isClosed_Ici | null |
@[simp]
closure_Ici (a : α) : closure (Ici a) = Ici a :=
isClosed_Ici.closure_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | closure_Ici | null |
ge_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a))
(h : ∃ᶠ c in x, b ≤ f c) : b ≤ a :=
isClosed_Ici.mem_of_frequently_of_tendsto h lim | lemma | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | ge_of_tendsto_of_frequently | null |
ge_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
(h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=
isClosed_Ici.mem_of_tendsto lim h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | ge_of_tendsto | null |
ge_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
(h : ∀ c, b ≤ f c) : b ≤ a :=
ge_of_tendsto lim (Eventually.of_forall h)
@[simp] lemma lowerBounds_closure (s : Set α) : lowerBounds (closure s : Set α) = lowerBounds s :=
ext fun a ↦ by simp_rw [mem_lowerBounds_iff_subset_Ici, isClosed_Ici.closure_subset_iff]
@[simp] lemma bddBelow_closure : BddBelow (closure s) ↔ BddBelow s := by
simp_rw [BddBelow, lowerBounds_closure]
protected alias ⟨_, BddBelow.closure⟩ := bddBelow_closure
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | ge_of_tendsto' | null |
disjoint_nhds_atTop_iff : Disjoint (𝓝 a) atTop ↔ ¬IsTop a :=
disjoint_nhds_atBot_iff (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | disjoint_nhds_atTop_iff | null |
IsGLB.range_of_tendsto {F : Filter β} [F.NeBot] (hle : ∀ i, a ≤ f i)
(hlim : Tendsto f F (𝓝 a)) : IsGLB (range f) a :=
IsLUB.range_of_tendsto (α := αᵒᵈ) hle hlim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | IsGLB.range_of_tendsto | null |
disjoint_nhds_atTop (a : α) : Disjoint (𝓝 a) atTop := disjoint_nhds_atBot (toDual a)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | disjoint_nhds_atTop | null |
inf_nhds_atTop (a : α) : 𝓝 a ⊓ atTop = ⊥ := (disjoint_nhds_atTop a).eq_bot | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | inf_nhds_atTop | null |
not_tendsto_nhds_of_tendsto_atTop (hf : Tendsto f l atTop) (a : α) : ¬Tendsto f l (𝓝 a) :=
hf.not_tendsto (disjoint_nhds_atTop a).symm | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | not_tendsto_nhds_of_tendsto_atTop | null |
not_tendsto_atTop_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atTop :=
hf.not_tendsto (disjoint_nhds_atTop a) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | not_tendsto_atTop_of_tendsto_nhds | null |
iInf_eq_of_forall_le_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLattice α] [TopologicalSpace α] [ClosedIciTopology α]
{a : α} {f : ι → α} (hle : ∀ i, a ≤ f i) (hlim : Tendsto f F (𝓝 a)) :
⨅ i, f i = a :=
iSup_eq_of_forall_le_of_tendsto (α := αᵒᵈ) hle hlim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | iInf_eq_of_forall_le_of_tendsto | null |
iUnion_Ici_eq_Ioi_of_lt_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIciTopology α]
{a : α} {f : ι → α} (hlt : ∀ i, a < f i) (hlim : Tendsto f F (𝓝 a)) :
⋃ i : ι, Ici (f i) = Ioi a :=
iUnion_Iic_eq_Iio_of_lt_of_tendsto (α := αᵒᵈ) hlt hlim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | iUnion_Ici_eq_Ioi_of_lt_of_tendsto | null |
isOpen_Iio : IsOpen (Iio a) := isOpen_Ioi (α := αᵒᵈ)
@[simp] theorem interior_Iio : interior (Iio a) = Iio a := isOpen_Iio.interior_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | isOpen_Iio | null |
Iio_mem_nhds (h : a < b) : Iio b ∈ 𝓝 a := isOpen_Iio.mem_nhds h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Iio_mem_nhds | null |
eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | eventually_lt_nhds | null |
Iic_mem_nhds (h : a < b) : Iic b ∈ 𝓝 a :=
mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Iic_mem_nhds | null |
eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | eventually_le_nhds | null |
Filter.Tendsto.eventually_lt_const {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
(h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=
h.eventually <| eventually_lt_nhds hv | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/Order/OrderClosed.lean | Filter.Tendsto.eventually_lt_const | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.