statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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ofSet_univ_eq_refl : ofSet univ isOpen_univ = OpenPartialHomeomorph.refl X | by
ext <;> simp | theorem | OpenPartialHomeomorph.ofSet_univ_eq_refl | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph.refl",
"isOpen_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EqOnSource (e e' : OpenPartialHomeomorph X Y) : Prop | e.source = e'.source ∧ EqOn e e' e.source | def | OpenPartialHomeomorph.EqOnSource | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | `EqOnSource e e'` means that `e` and `e'` have the same source, and coincide there. They
should really be considered the same partial equivalence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eqOnSource_iff (e e' : OpenPartialHomeomorph X Y) :
EqOnSource e e' ↔ PartialEquiv.EqOnSource e.toPartialEquiv e'.toPartialEquiv | Iff.rfl | theorem | OpenPartialHomeomorph.eqOnSource_iff | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.EqOnSource"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eqOnSourceSetoid : Setoid (OpenPartialHomeomorph X Y) | { PartialEquiv.eqOnSourceSetoid.comap toPartialEquiv with r := EqOnSource } | instance | OpenPartialHomeomorph.eqOnSourceSetoid | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | `EqOnSource` is an equivalence relation. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eqOnSource_refl : e ≈ e | Setoid.refl _ | theorem | OpenPartialHomeomorph.eqOnSource_refl | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EqOnSource.symm' {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : e.symm ≈ e'.symm | PartialEquiv.EqOnSource.symm' h | theorem | OpenPartialHomeomorph.EqOnSource.symm' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.EqOnSource.symm'"
] | If two open partial homeomorphisms are equivalent, so are their inverses. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EqOnSource.source_eq {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') :
e.source = e'.source | h.1 | theorem | OpenPartialHomeomorph.EqOnSource.source_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | Two equivalent open partial homeomorphisms have the same source. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EqOnSource.target_eq {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') :
e.target = e'.target | h.symm'.1 | theorem | OpenPartialHomeomorph.EqOnSource.target_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | Two equivalent open partial homeomorphisms have the same target. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EqOnSource.eqOn {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : EqOn e e' e.source | h.2 | theorem | OpenPartialHomeomorph.EqOnSource.eqOn | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | Two equivalent open partial homeomorphisms have coinciding `toFun` on the source | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EqOnSource.symm_eqOn_target {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') :
EqOn e.symm e'.symm e.target | h.symm'.2 | theorem | OpenPartialHomeomorph.EqOnSource.symm_eqOn_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | Two equivalent open partial homeomorphisms have coinciding `invFun` on the target | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EqOnSource.restr {e e' : OpenPartialHomeomorph X Y} (he : e ≈ e') (s : Set X) :
e.restr s ≈ e'.restr s | PartialEquiv.EqOnSource.restr he _ | theorem | OpenPartialHomeomorph.EqOnSource.restr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.EqOnSource.restr",
"Set"
] | Restriction of open partial homeomorphisms respects equivalence | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EqOn.restr_eqOn_source {e e' : OpenPartialHomeomorph X Y}
(h : EqOn e e' (e.source ∩ e'.source)) : e.restr e'.source ≈ e'.restr e.source | by
constructor
· rw [e'.restr_source' _ e.open_source]
rw [e.restr_source' _ e'.open_source]
exact Set.inter_comm _ _
· rw [e.restr_source' _ e'.open_source]
refine (EqOn.trans ?_ h).trans ?_ <;> simp only [mfld_simps, eqOn_refl] | theorem | OpenPartialHomeomorph.Set.EqOn.restr_eqOn_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"Set.inter_comm",
"trans"
] | Two equivalent open partial homeomorphisms are equal when the source and target are `univ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_of_eqOnSource_univ {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') (s : e.source = univ)
(t : e.target = univ) : e = e' | toPartialEquiv_injective <| PartialEquiv.eq_of_eqOnSource_univ _ _ h s t | theorem | OpenPartialHomeomorph.eq_of_eqOnSource_univ | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.eq_of_eqOnSource_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_eqOnSource_restr {s' : Set X}
(hss' : e.source ∩ interior s = e.source ∩ interior s') :
e.restr s ≈ e.restr s' | by
constructor
· simpa [e.restr_source]
· simp [Set.eqOn_refl] | lemma | OpenPartialHomeomorph.restr_eqOnSource_restr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Set",
"Set.eqOn_refl",
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_inter_source : e.restr (e.source ∩ s) ≈ e.restr s | e.restr_eqOnSource_restr (by simp [interior_eq_iff_isOpen.mpr e.open_source]) | lemma | OpenPartialHomeomorph.restr_inter_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl_toOpenPartialHomeomorph :
(Homeomorph.refl X).toOpenPartialHomeomorph = OpenPartialHomeomorph.refl X | rfl | theorem | Homeomorph.refl_toOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Homeomorph.refl",
"OpenPartialHomeomorph.refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_toOpenPartialHomeomorph :
e.symm.toOpenPartialHomeomorph = e.toOpenPartialHomeomorph.symm | rfl | theorem | Homeomorph.symm_toOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_coe_nhdsLT_eq_atTop_iff :
comap ((↑) : s → X) (𝓝[<] b) = atTop ↔
s ⊆ Iio b ∧ (s.Nonempty → ∀ a < b, (s ∩ Ioo a b).Nonempty) | by
rcases s.eq_empty_or_nonempty with rfl | hsne
· simp [eq_iff_true_of_subsingleton]
have := hsne.to_subtype
simp only [hsne, true_imp_iff]
by_cases hsub : s ⊆ Iio b
· simp only [hsub, true_and]
constructor
· intro h a ha
have := preimage_mem_comap (m := ((↑) : s → X)) (Ioo_mem_nhdsLT ha)
... | theorem | comap_coe_nhdsLT_eq_atTop_iff | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"Filter",
"Filter.nonempty_of_mem",
"Ioo_mem_nhdsLT",
"by_cases",
"comap",
"eventually_mem_nhdsWithin",
"le_of_lt",
"lt_irrefl",
"nhdsLT_basis_of_exists_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_coe_nhdsGT_eq_atBot_iff :
comap ((↑) : s → X) (𝓝[>] b) = atBot ↔
s ⊆ Ioi b ∧ (s.Nonempty → ∀ a > b, (s ∩ Ioo b a).Nonempty) | by
refine comap_coe_nhdsLT_eq_atTop_iff (s := OrderDual.ofDual ⁻¹' s) (b := OrderDual.toDual b)
|>.trans ?_
simp [← preimage_inter, ofDual.surjective] | theorem | comap_coe_nhdsGT_eq_atBot_iff | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"OrderDual.ofDual",
"OrderDual.toDual",
"comap",
"comap_coe_nhdsLT_eq_atTop_iff",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_coe_nhdsLT_of_Ioo_subset (hsb : s ⊆ Iio b) (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s)
(hb : IsSuccPrelimit b := by exact .of_dense _) :
comap ((↑) : s → X) (𝓝[<] b) = atTop | by
rw [comap_coe_nhdsLT_eq_atTop_iff]
refine ⟨hsb, fun hsne a ha ↦ ?_⟩
rcases hs hsne with ⟨c, hcb, hcs⟩
rcases hb.lt_iff_exists_lt.mp (max_lt ha hcb) with ⟨x, hxb, hacx⟩
rw [max_lt_iff] at hacx
exact ⟨x, hcs ⟨hacx.2, hxb⟩, hacx.1, hxb⟩ | theorem | comap_coe_nhdsLT_of_Ioo_subset | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap",
"comap_coe_nhdsLT_eq_atTop_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_coe_nhdsGT_of_Ioo_subset (hsa : s ⊆ Ioi a) (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s)
(ha : IsPredPrelimit a := by exact .of_dense _) :
comap ((↑) : s → X) (𝓝[>] a) = atBot | by
refine comap_coe_nhdsLT_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from hsa) ?_ ha.dual
simpa only [OrderDual.exists, Ioo_toDual] | theorem | comap_coe_nhdsGT_of_Ioo_subset | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap",
"comap_coe_nhdsLT_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe_atTop_of_Ioo_subset (hsb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s)
(hb : IsSuccPrelimit b := by exact .of_dense _) :
map ((↑) : s → X) atTop = 𝓝[<] b | by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hsb x.2)⟩
rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]
· rw [← comap_coe_nhdsLT_of_Ioo_subset hsb (fun _ => hs a ha) hb, map_comap_of_mem]
rw [Subtype.range_val]
exact ... | theorem | map_coe_atTop_of_Ioo_subset | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"Filter.map_bot",
"IsEmpty",
"Subtype.range_val",
"comap_coe_nhdsLT_of_Ioo_subset",
"mem_nhdsLT_iff_exists_Ioo_subset'",
"nhdsWithin_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe_atBot_of_Ioo_subset (hsa : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s)
(ha : IsPredPrelimit a := by exact .of_dense _) :
map ((↑) : s → X) atBot = 𝓝[>] a | by
refine map_coe_atTop_of_Ioo_subset (s := ofDual ⁻¹' s) (b := toDual a) hsa ?_ ha.dual
intro b' hb'
simpa [OrderDual.exists] using hs (ofDual b') hb' | theorem | map_coe_atBot_of_Ioo_subset | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_atTop_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_coe_Ioo_nhdsLT (a b : X) (hb : IsSuccPrelimit b := by exact .of_dense _) :
comap ((↑) : Ioo a b → X) (𝓝[<] b) = atTop | comap_coe_nhdsLT_of_Ioo_subset Ioo_subset_Iio_self
(fun h => ⟨a, h.elim fun _x hx ↦ hx.1.trans hx.2, Subset.rfl⟩) hb | theorem | comap_coe_Ioo_nhdsLT | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap",
"comap_coe_nhdsLT_of_Ioo_subset"
] | The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at
the right endpoint in the ambient order. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comap_coe_Ioo_nhdsGT (a b : X) (ha : IsPredPrelimit a := by exact .of_dense _) :
comap ((↑) : Ioo a b → X) (𝓝[>] a) = atBot | comap_coe_nhdsGT_of_Ioo_subset Ioo_subset_Ioi_self
(fun h => ⟨b, h.elim fun _x hx ↦ hx.1.trans hx.2, Subset.rfl⟩) ha | theorem | comap_coe_Ioo_nhdsGT | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap",
"comap_coe_nhdsGT_of_Ioo_subset"
] | The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at
the left endpoint in the ambient order. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comap_coe_Ioi_nhdsGT (a : X) (ha : IsPredPrelimit a := by exact .of_dense _) :
comap ((↑) : Ioi a → X) (𝓝[>] a) = atBot | comap_coe_nhdsGT_of_Ioo_subset Subset.rfl (fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩) ha | theorem | comap_coe_Ioi_nhdsGT | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap",
"comap_coe_nhdsGT_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_coe_Iio_nhdsLT (a : X) (ha : IsSuccPrelimit a := by exact .of_dense _) :
comap ((↑) : Iio a → X) (𝓝[<] a) = atTop | comap_coe_Ioi_nhdsGT (toDual a) ha.dual | theorem | comap_coe_Iio_nhdsLT | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap",
"comap_coe_Ioi_nhdsGT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe_Ioo_atTop (h : a < b) (hb : IsSuccPrelimit b := by exact .of_dense _) :
map ((↑) : Ioo a b → X) atTop = 𝓝[<] b | map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self (fun _ _ => ⟨_, h, Subset.rfl⟩) hb | theorem | map_coe_Ioo_atTop | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_atTop_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe_Ioo_atBot (h : a < b) (ha : IsPredPrelimit a := by exact .of_dense _) :
map ((↑) : Ioo a b → X) atBot = 𝓝[>] a | map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self (fun _ _ => ⟨_, h, Subset.rfl⟩) ha | theorem | map_coe_Ioo_atBot | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_atBot_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe_Ioi_atBot (a : X) (ha : IsPredPrelimit a := by exact .of_dense _) :
map ((↑) : Ioi a → X) atBot = 𝓝[>] a | map_coe_atBot_of_Ioo_subset Subset.rfl (fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩) ha | theorem | map_coe_Ioi_atBot | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_atBot_of_Ioo_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_coe_Iio_atTop (a : X) (ha : IsSuccPrelimit a := by exact .of_dense _) :
map ((↑) : Iio a → X) atTop = 𝓝[<] a | map_coe_Ioi_atBot (toDual a) ha.dual | theorem | map_coe_Iio_atTop | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_Ioi_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_comp_coe_Ioo_atTop (h : a < b) (hb : IsSuccPrelimit b := by exact .of_dense _) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l | by
rw [← map_coe_Ioo_atTop h hb, tendsto_map'_iff, Function.comp_def] | theorem | tendsto_comp_coe_Ioo_atTop | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_Ioo_atTop"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_comp_coe_Ioo_atBot (h : a < b) (ha : IsPredPrelimit a := by exact .of_dense _) :
Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l | by
rw [← map_coe_Ioo_atBot h ha, tendsto_map'_iff, Function.comp_def] | theorem | tendsto_comp_coe_Ioo_atBot | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_Ioo_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_comp_coe_Ioi_atBot (ha : IsPredPrelimit a := by exact .of_dense _) :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l | by
rw [← map_coe_Ioi_atBot a ha, tendsto_map'_iff, Function.comp_def] | theorem | tendsto_comp_coe_Ioi_atBot | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_Ioi_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_comp_coe_Iio_atTop (ha : IsSuccPrelimit a := by exact .of_dense _) :
Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l | by
rw [← map_coe_Iio_atTop a ha, tendsto_map'_iff, Function.comp_def] | theorem | tendsto_comp_coe_Iio_atTop | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"map_coe_Iio_atTop"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_Ioo_atTop {f : α → Ioo a b} (hb : IsSuccPrelimit b := by exact .of_dense _) :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : X)) l (𝓝[<] b) | by
rw [← comap_coe_Ioo_nhdsLT a b hb, tendsto_comap_iff, Function.comp_def] | theorem | tendsto_Ioo_atTop | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap_coe_Ioo_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_Ioo_atBot {f : α → Ioo a b} (ha : IsPredPrelimit a := by exact .of_dense _) :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : X)) l (𝓝[>] a) | by
rw [← comap_coe_Ioo_nhdsGT a b ha, tendsto_comap_iff, Function.comp_def] | theorem | tendsto_Ioo_atBot | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap_coe_Ioo_nhdsGT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_Ioi_atBot {f : α → Ioi a} (ha : IsPredPrelimit a := by exact .of_dense _) :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : X)) l (𝓝[>] a) | by
rw [← comap_coe_Ioi_nhdsGT a ha, tendsto_comap_iff, Function.comp_def] | theorem | tendsto_Ioi_atBot | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap_coe_Ioi_nhdsGT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_Iio_atTop {f : α → Iio a} (ha : IsSuccPrelimit a := by exact .of_dense _) :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : X)) l (𝓝[<] a) | by
rw [← comap_coe_Iio_nhdsLT a ha, tendsto_comap_iff, Function.comp_def] | theorem | tendsto_Iio_atTop | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"comap_coe_Iio_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
locallyFinite_Icc_of_tendsto {f g : α → X}
(hl : Tendsto f atTop atTop) (hu : Tendsto g atBot atBot) :
LocallyFinite (fun n => Set.Icc (f n) (g n)) | by
intro x
cases isEmpty_or_nonempty α
· use univ
simp [Subsingleton.elim _ (∅ : Set α)]
obtain ⟨x_L, hx_L⟩ := exists_lt x
obtain ⟨x_R, hx_R⟩ := exists_gt x
obtain ⟨a_L, ha_L : ∀ a ≤ a_L, g a ≤ x_L⟩ :=
hu.eventually_le_atBot x_L |>.exists_forall_of_atBot
obtain ⟨a_R, ha_R : ∀ a ≥ a_R, x_R ≤ f a⟩ :... | theorem | locallyFinite_Icc_of_tendsto | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"Ioo_mem_nhds",
"LocallyFinite",
"Set",
"Set.Icc",
"isEmpty_or_nonempty",
"trans"
] | A family of closed intervals bounded by diverging limits is locally finite. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
locallyFinite_Ico_of_tendsto {l u : α → X}
(hl : Tendsto l atTop atTop) (hu : Tendsto u atBot atBot) :
LocallyFinite (fun n => Set.Ico (l n) (u n)) | locallyFinite_Icc_of_tendsto hl hu |>.subset fun _ => Set.Ico_subset_Icc_self | theorem | locallyFinite_Ico_of_tendsto | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"LocallyFinite",
"Set.Ico",
"locallyFinite_Icc_of_tendsto"
] | A family of half-open intervals bounded by diverging limits is locally finite. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
locallyFinite_Ioc_of_tendsto {l u : α → X}
(hl : Tendsto l atTop atTop) (hu : Tendsto u atBot atBot) :
LocallyFinite (fun n => Set.Ioc (l n) (u n)) | locallyFinite_Icc_of_tendsto hl hu |>.subset fun _ => Set.Ioc_subset_Icc_self | theorem | locallyFinite_Ioc_of_tendsto | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"LocallyFinite",
"Set.Ioc",
"Set.Ioc_subset_Icc_self",
"locallyFinite_Icc_of_tendsto"
] | A family of half-open intervals bounded by diverging limits is locally finite. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
locallyFinite_Ioo_of_tendsto {l u : α → X}
(hl : Tendsto l atTop atTop) (hu : Tendsto u atBot atBot) :
LocallyFinite (fun n => Set.Ioo (l n) (u n)) | locallyFinite_Icc_of_tendsto hl hu |>.subset fun _ => Set.Ioo_subset_Icc_self | theorem | locallyFinite_Ioo_of_tendsto | Topology.Order | Mathlib/Topology/Order/AtTopBotIxx.lean | [
"Mathlib.Topology.Order.LeftRightNhds"
] | [
"LocallyFinite",
"Set.Ioo",
"Set.Ioo_subset_Icc_self",
"locallyFinite_Icc_of_tendsto"
] | A family of open intervals bounded by diverging limits is locally finite. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Preorder.topology (α : Type*) [Preorder α] : TopologicalSpace α | generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }
to_dual_insert_cast Preorder.topology := by simp_rw [or_comm] | def | Preorder.topology | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Preorder",
"TopologicalSpace"
] | (Order) topology on a partial order `α` generated by the subbase of open intervals
`(a, ∞) = { x ∣ a < x }, (-∞, b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an
instance as many ordered sets are already endowed with the same topology, most often in a non-defeq
way though. Register as a local instan... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
OrderTopology (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
/-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/
topology_eq_generate_intervals : t = Preorder.topology α | class | OrderTopology | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Preorder",
"Preorder.topology",
"TopologicalSpace"
] | The order topology on an ordered type is the topology generated by open intervals. We register
it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.
We define it as a mixin. If you want to introduce the order topology on a preorder, use
`Preorder.topology`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderTopology.continuous_iff [OrderTopology α] [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ a, IsOpen (f ⁻¹' Ioi a) ∧ IsOpen (f ⁻¹' Iio a) | by
simp_rw +instances [OrderTopology.topology_eq_generate_intervals, continuous_generateFrom_iff]
aesop | theorem | OrderTopology.continuous_iff | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Continuous",
"IsOpen",
"OrderTopology",
"TopologicalSpace",
"continuous_generateFrom_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_generate_intervals [t : OrderTopology α] {s : Set α} :
IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s | by
rw [t.topology_eq_generate_intervals]; rfl | theorem | isOpen_iff_generate_intervals | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsOpen",
"OrderTopology",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_lt' [OrderTopology α] (a : α) : IsOpen { b : α | a < b } | isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩ | theorem | isOpen_lt' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsOpen",
"OrderTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_Ioi' [OrderTopology α] (a : α) : IsOpen (Ioi a) | isOpen_lt' a | theorem | isOpen_Ioi' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsOpen",
"OrderTopology",
"isOpen_lt'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lt_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x | (isOpen_lt' _).mem_nhds h | theorem | lt_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"isOpen_lt'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x | (lt_mem_nhds h).mono fun _ => le_of_lt | theorem | le_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"le_of_lt",
"lt_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_order [OrderTopology α] (a : α) :
𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) | by
rw [OrderTopology.topology_eq_generate_intervals (α := α), nhds_generateFrom]
simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and,
iInf_exists, iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio] | theorem | nhds_eq_order | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_order [OrderTopology α] {f : β → α} {a : α} {x : Filter β} :
Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' | by
simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl | theorem | tendsto_order | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"OrderTopology",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoIccClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) | by
simp only [nhds_eq_order, iInf_subtype']
refine
((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass
fun s _ => ?_
refine ((ordConnected_biInter ?_).inter (ordConnected_biInter ?_)).out <;> intro _ _
exacts [ordConnected_Ioi, ordConnected_Iio] | instance | tendstoIccClassNhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoIcoClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) | tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self | instance | tendstoIcoClassNhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoIocClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) | tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self | instance | tendstoIocClassNhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoIooClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) | tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self | instance | tendstoIooClassNhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_countable_generateFrom_Ioi_Iio
[OrderTopology α] [SecondCountableTopology α] :
∃ (c : Set α), c.Countable ∧ ts = generateFrom { s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a } | by
rcases isEmpty_or_nonempty α with hα | hα
· exact ⟨∅, by simp, Subsingleton.elim _ _⟩
obtain ⟨t, t_subs, t_count, ht⟩ : ∃ t ⊆ { s | ∃ a, s = Ioi a ∨ s = Iio a },
t.Countable ∧ ts = generateFrom t :=
exists_countable_of_generateFrom OrderTopology.topology_eq_generate_intervals
have A : ∀ s ∈ t, ∃ a,... | lemma | exists_countable_generateFrom_Ioi_Iio | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"SecondCountableTopology",
"Set",
"exists_exists_and_eq_and",
"isEmpty_or_nonempty",
"isOpen_Ioi'",
"le_antisymm"
] | In a second countable topological space with the order topology, the topology is generated
by half-infinite open intervals with endpoints in a countable set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isTopologicalBasis_biInter_Ioi_Iio_of_generateFrom (c : Set α)
(h : ts = generateFrom { s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a }) :
IsTopologicalBasis {s | ∃ (f g : Set α), f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite
∧ s = (⋂ a ∈ f, Ioi a) ∩ (⋂ a ∈ g, Iio a)} | by
refine IsTopologicalBasis.of_isOpen_of_subset ?_ (isTopologicalBasis_of_subbasis h) ?_
· rintro u ⟨f, g, hfc, hgc, hf_fin, hg_fin, rfl⟩
apply IsOpen.inter
· apply hf_fin.isOpen_biInter (fun i hi ↦ ?_)
rw [h]
exact isOpen_generateFrom_of_mem ⟨i, hfc hi, Or.inl rfl⟩
· apply hg_fin.isOpen_bi... | lemma | isTopologicalBasis_biInter_Ioi_Iio_of_generateFrom | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Finite",
"IsOpen.inter",
"Set"
] | If a topology is generated by half-open intervals with endpoints in a set `c`, then the sets
formed by intersecting finitely many of these intervals form a topological basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_of_tendsto_of_tendsto_of_le_of_le' [OrderTopology α] {f g h : β → α} {b : Filter β}
{a : α} (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b)
(hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) | (hg.Icc hh).of_smallSets <| hgf.and hfh
@[to_dual self (reorder := g h, hg hh, hgf hfh)]
alias Filter.Tendsto.squeeze' := tendsto_of_tendsto_of_tendsto_of_le_of_le' | theorem | tendsto_of_tendsto_of_tendsto_of_le_of_le' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"OrderTopology"
] | **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities
hold eventually for the filter. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_of_tendsto_of_tendsto_of_le_of_le [OrderTopology α] {f g h : β → α} {b : Filter β}
{a : α} (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :
Tendsto f b (𝓝 a) | tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (Eventually.of_forall hgf)
(Eventually.of_forall hfh)
@[to_dual self (reorder := g h, hg hh, hgf hfh)]
alias Filter.Tendsto.squeeze := tendsto_of_tendsto_of_tendsto_of_le_of_le | theorem | tendsto_of_tendsto_of_tendsto_of_le_of_le | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"OrderTopology",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | **Squeeze theorem** (also known as **sandwich theorem**). This version assumes that inequalities
hold everywhere. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_order_unbounded [OrderTopology α] {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :
𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) | by
simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl | theorem | nhds_order_unbounded | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_order_unbounded [OrderTopology α] {f : β → α} {a : α} {x : Filter β}
(hu : ∃ u, a < u) (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :
Tendsto f x (𝓝 a) | by
simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal]
exact fun l hl u => h l u hl | theorem | tendsto_order_unbounded | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"OrderTopology",
"nhds_order_unbounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoIxxNhdsWithin {α : Type*} [TopologicalSpace α] (a : α) {s t : Set α}
{Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] :
TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) | Filter.tendstoIxxClass_inf | instance | tendstoIxxNhdsWithin | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter.tendstoIxxClass_inf",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoIccClassNhdsPi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)]
[∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) :
TendstoIxxClass Icc (𝓝 f) (𝓝 f) | by
constructor
conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi]
simp only [smallSets_iInf, smallSets_comap_eq_comap_image, tendsto_iInf, tendsto_comap_iff]
intro i
have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f
refine (this.comp tendsto_fst).Icc (this.comp te... | instance | tendstoIccClassNhdsPi | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter.pi",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"continuous_apply",
"nhds_pi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
[OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) :
induced f ‹TopologicalSpace β› ≤ Preorder.topology α | by
let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩
refine le_of_nhds_le_nhds fun x => ?_
simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf]
refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_)
exacts [iInf₂_le (f a) ha, iInf₂_le (f a)... | theorem | induced_topology_le_preorder | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Preorder",
"Preorder.topology",
"TopologicalSpace",
"le_of_nhds_le_nhds",
"nhds_eq_order",
"nhds_induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
[OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)
(H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)
(H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :
induced f ‹TopologicalSpace β› = Preor... | by
let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩
refine le_antisymm (induced_topology_le_preorder hf) ?_
refine le_of_nhds_le_nhds fun a => ?_
simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]
refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)
... | theorem | induced_topology_eq_preorder | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Preorder",
"Preorder.topology",
"TopologicalSpace",
"induced_topology_le_preorder",
"le_antisymm",
"le_of_nhds_le_nhds",
"nhds_eq_order",
"nhds_induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
induced_orderTopology' {α β : Type*} [Preorder α] [ta : TopologicalSpace β]
[Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :
@OrderTopology _ (induced f ta) _ | let _ := induced f ta
⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩ | theorem | induced_orderTopology' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
induced_orderTopology {α β : Type*} [Preorder α] [ta : TopologicalSpace β]
[Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H : ∀ {x y}, x < y → ∃ a, f a ∈ Ioo x y) : @OrderTopology _ (induced f ta) _ | induced_orderTopology' f (hf)
(fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩)
fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩ | theorem | induced_orderTopology | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Preorder",
"TopologicalSpace",
"induced_orderTopology'",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
StrictMono.induced_topology_eq_preorder {α β : Type*} [LinearOrder α]
[LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β}
(hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α | by
refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_
· rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩
exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩
· rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩
exact... | theorem | StrictMono.induced_topology_eq_preorder | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"OrderTopology",
"Preorder.topology",
"StrictMono",
"TopologicalSpace",
"induced_topology_eq_preorder"
] | The topology induced by a strictly monotone function with order-connected range is the preorder
topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMono.isEmbedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]
[TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(hf : StrictMono f) (hc : OrdConnected (range f)) : IsEmbedding f | ⟨⟨h.1.trans <| Eq.symm <| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩ | theorem | StrictMono.isEmbedding_of_ordConnected | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"OrderTopology",
"StrictMono",
"TopologicalSpace"
] | A strictly monotone function between linear orders with order topology is a topological
embedding provided that the range of `f` is order-connected. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
OrderEmbedding.isEmbedding_of_ordConnected {α β : Type*} [LinearOrder α] [LinearOrder β]
[TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [OrderTopology β]
(f : α ↪o β) (hc : OrdConnected (range f)) : Topology.IsEmbedding f | f.strictMono.isEmbedding_of_ordConnected hc | lemma | OrderEmbedding.isEmbedding_of_ordConnected | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"OrderTopology",
"TopologicalSpace",
"Topology.IsEmbedding"
] | An `OrderEmbedding` is a topological embedding provided that the range of `f` is
order-connected | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α]
[OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t | ⟨(Subtype.strictMono_coe (· ∈ t)).induced_topology_eq_preorder <| by
rwa [← @Subtype.range_val _ t] at ht⟩ | instance | orderTopology_of_ordConnected | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"OrderTopology",
"Set",
"Subtype.range_val",
"Subtype.strictMono_coe",
"TopologicalSpace",
"induced_topology_eq_preorder"
] | On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the
order is the same as the restriction to the subset of the order topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsGE_eq_iInf_inf_principal [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) :
𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) | by
rw [nhdsWithin, nhds_eq_order]
refine le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf ?_ inf_le_left) inf_le_right)
exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl) | theorem | nhdsGE_eq_iInf_inf_principal | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Preorder",
"TopologicalSpace",
"le_antisymm",
"nhdsWithin",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGE_eq_iInf_principal [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}
(ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) | by
simp only [nhdsGE_eq_iInf_inf_principal, biInf_inf ha, inf_principal, Iio_inter_Ici] | theorem | nhdsGE_eq_iInf_principal | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTopology",
"Preorder",
"TopologicalSpace",
"nhdsGE_eq_iInf_inf_principal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGE_basis_of_exists_gt [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α}
(ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u | nhdsGE_eq_iInf_principal ha ▸ hasBasis_biInf_principal
(fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _),
Ico_subset_Ico_right (min_le_right _ _)⟩) ha | theorem | nhdsGE_basis_of_exists_gt | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"OrderTopology",
"TopologicalSpace",
"lt_min",
"min_le_left",
"min_le_right",
"nhdsGE_eq_iInf_principal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGE_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] (a : α) :
(𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u | nhdsGE_basis_of_exists_gt (exists_gt a) | theorem | nhdsGE_basis | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"NoMaxOrder",
"OrderTopology",
"TopologicalSpace",
"nhdsGE_basis_of_exists_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :
𝓝 (⊤ : α) = ⨅ (l) (_ : l < ⊤), 𝓟 (Ioi l) | by simp [nhds_eq_order (⊤ : α)] | theorem | nhds_top_order | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"OrderTop",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"nhds_eq_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]
[Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a | by
have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top
simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsLE_basis_of_exists_lt this | theorem | nhds_top_basis | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"LinearOrder",
"Nontrivial",
"OrderTop",
"OrderTopology",
"TopologicalSpace",
"exists_ne",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]
[Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici | nhds_top_basis.to_hasBasis
(fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩)
fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩ | theorem | nhds_top_basis_Ici | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"DenselyOrdered",
"LinearOrder",
"Nontrivial",
"OrderTop",
"OrderTopology",
"TopologicalSpace",
"exists_between"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]
{l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) | by
simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢
intro x hx
filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le | theorem | tendsto_nhds_top_mono | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"OrderTop",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"lt_of_lt_of_le",
"nhds_top_order"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]
{l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) | tendsto_nhds_top_mono hf (Eventually.of_forall hg) | theorem | tendsto_nhds_top_mono' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Filter",
"OrderTop",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"tendsto_nhds_top_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
order_separated {a₁ a₂ : α} (h : a₁ < a₂) :
∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ | let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi
⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩ | theorem | order_separated | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsOpen",
"Set",
"isOpen_lt'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_Ioc_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :
∃ l < a, Ioc l a ⊆ s | (nhdsLE_basis_of_exists_lt h).mem_iff.mp (nhdsWithin_le_nhds hs) | theorem | exists_Ioc_subset_of_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Set",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_Ioc_subset_of_mem_nhds' {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :
∃ l' ∈ Ico l a, Ioc l' a ⊆ s | let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩
⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩,
(Ioc_subset_Ioc_left <| le_max_right _ _).trans hl's⟩ | theorem | exists_Ioc_subset_of_mem_nhds' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Set",
"exists_Ioc_subset_of_mem_nhds",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_Icc_mem_subset_of_mem_nhdsGE {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) :
∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s | by
rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha | ha)
· use a
simpa [ha.Ici_eq] using hs
· rcases (nhdsGE_basis_of_exists_gt ha).mem_iff.mp hs with ⟨b, hab, hbs⟩
rcases eq_empty_or_nonempty (Ioo a b) with (H | ⟨c, hc⟩)
· have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab... | theorem | exists_Icc_mem_subset_of_mem_nhdsGE | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Set",
"le_rfl",
"nhdsGE_basis_of_exists_gt",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_Icc_mem_subset_of_mem_nhds {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :
∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s | by
rcases exists_Icc_mem_subset_of_mem_nhdsLE (nhdsWithin_le_nhds hs) with
⟨b, hba, hb_nhds, hbs⟩
rcases exists_Icc_mem_subset_of_mem_nhdsGE (nhdsWithin_le_nhds hs) with
⟨c, hac, hc_nhds, hcs⟩
refine ⟨b, c, ⟨hba, hac⟩, ?_⟩
rw [← Icc_union_Icc_eq_Icc hba hac, ← nhdsLE_sup_nhdsGE]
exact ⟨union_mem_sup h... | theorem | exists_Icc_mem_subset_of_mem_nhds | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Set",
"exists_Icc_mem_subset_of_mem_nhdsGE",
"nhdsLE_sup_nhdsGE",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.exists_Ioo_subset [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) :
∃ a b, a < b ∧ Ioo a b ⊆ s | by
obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h
obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x
rcases lt_trichotomy x y with (H | rfl | H)
· obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s :=
exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩
exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩
· exact (hy rfl).elim
·... | theorem | IsOpen.exists_Ioo_subset | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsOpen",
"Nontrivial",
"Set",
"exists_Ioc_subset_of_mem_nhds",
"exists_ne",
"lt_trichotomy"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dense_of_exists_between [Nontrivial α] {s : Set α}
(h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, c ∈ Ioo a b) : Dense s | by
refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_
obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty
obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab
exact ⟨x, ⟨H hx, xs⟩⟩ | theorem | dense_of_exists_between | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Dense",
"Nontrivial",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUpperSet.isClosed [WellFoundedLT α] {s : Set α} (h : IsUpperSet s) : IsClosed s | by
obtain rfl | ⟨a, rfl⟩ := h.eq_empty_or_Ici
exacts [isClosed_empty, isClosed_Ici] | theorem | IsUpperSet.isClosed | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsClosed",
"IsUpperSet",
"Set",
"WellFoundedLT",
"isClosed_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLowerSet.isOpen [WellFoundedLT α] {s : Set α} (h : IsLowerSet s) : IsOpen s | by
simpa using h.compl.isClosed | theorem | IsLowerSet.isOpen | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"IsOpen",
"Set",
"WellFoundedLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dense_iff_exists_between [DenselyOrdered α] [Nontrivial α] {s : Set α} :
Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b | ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩ | theorem | dense_iff_exists_between | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Dense",
"DenselyOrdered",
"Nontrivial",
"Set"
] | A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only
if for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass
assumptions. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhds_iff_exists_Ioo_subset' {a : α} {s : Set α} (hl : ∃ l, l < a)
(hu : ∃ u, a < u) : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s | by
constructor
· intro h
rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩
rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩
exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩
· rintro ⟨l, u, ha, h⟩
apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h | theorem | mem_nhds_iff_exists_Ioo_subset' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Ioo_mem_nhds",
"Set",
"exists_Ioc_subset_of_mem_nhds"
] | A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,
provided `a` is neither a bottom element nor a top element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhds_iff_exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α}
{s : Set α} : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s | mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a) | theorem | mem_nhds_iff_exists_Ioo_subset | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"NoMaxOrder",
"NoMinOrder",
"Set",
"mem_nhds_iff_exists_Ioo_subset'"
] | A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_basis_Ioo' {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :
(𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 | ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <| by simp⟩ | theorem | nhds_basis_Ioo' | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"mem_nhds_iff_exists_Ioo_subset'",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_Ioo [NoMaxOrder α] [NoMinOrder α] (a : α) :
(𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 | nhds_basis_Ioo' (exists_lt a) (exists_gt a) | theorem | nhds_basis_Ioo | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"NoMaxOrder",
"NoMinOrder",
"nhds_basis_Ioo'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Eventually.exists_Ioo_subset [NoMaxOrder α] [NoMinOrder α] {a : α}
{p : α → Prop} (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x | p x } | mem_nhds_iff_exists_Ioo_subset.1 hp | theorem | Filter.Eventually.exists_Ioo_subset | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"NoMaxOrder",
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α}
(hs : Dense s) : ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) | by
refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_
refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)
· simp only [union_subset_iff, image_subset_iff]
exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩
· rintro _ ⟨a, rfl | rfl⟩
· rw [hs.Ioi_eq_biUnion]
let... | theorem | Dense.topology_eq_generateFrom | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Dense",
"DenselyOrdered",
"Set",
"isOpen_iUnion",
"le_antisymm",
"le_generateFrom",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SuccOrder.hasBasis_nhds_Ioc_of_exists_lt [SuccOrder α] {a : α} (ha : ∃ l, l < a) :
(𝓝 a).HasBasis (· < a) (Set.Ioc · a) | SuccOrder.nhdsLE_eq_nhds a ▸ nhdsLE_basis_of_exists_lt ha
@[deprecated (since := "2026-04-19")]
alias PredOrder.hasBasis_nhds_Ioc_of_exists_gt := PredOrder.hasBasis_nhds_Ico_of_exists_gt | theorem | SuccOrder.hasBasis_nhds_Ioc_of_exists_lt | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"Set.Ioc",
"SuccOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SuccOrder.hasBasis_nhds_Ioc [SuccOrder α] {a : α} [NoMinOrder α] :
(𝓝 a).HasBasis (· < a) (Set.Ioc · a) | SuccOrder.hasBasis_nhds_Ioc_of_exists_lt (exists_lt a)
@[deprecated (since := "2026-04-19")]
alias PredOrder.hasBasis_nhds_Ioc := PredOrder.hasBasis_nhds_Ico | theorem | SuccOrder.hasBasis_nhds_Ioc | Topology.Order | Mathlib/Topology/Order/Basic.lean | [] | [
"NoMinOrder",
"Set.Ioc",
"SuccOrder",
"SuccOrder.hasBasis_nhds_Ioc_of_exists_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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