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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Prod.instT0Space [TopologicalSpace Y] [T0Space X] [T0Space Y] : T0Space (X × Y) | ⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩ | instance | Prod.instT0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace",
"continuous_fst",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.instT0Space {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, T0Space (X i)] :
T0Space (∀ i, X i) | ⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩ | instance | Pi.instT0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace",
"continuous_apply"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ULift.instT0Space [T0Space X] : T0Space (ULift X) | IsEmbedding.uliftDown.t0Space | instance | ULift.instT0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) :
T0Space X | by
refine ⟨fun x y hxy => ?_⟩
rcases h x y hxy with ⟨s, hxs, hys, hs⟩
lift x to s using hxs; lift y to s using hys
rw [← subtype_inseparable_iff] at hxy
exact congr_arg Subtype.val hxy.eq | theorem | T0Space.of_cover | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"Set",
"T0Space",
"subtype_inseparable_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T0Space.of_open_cover (h : ∀ x, ∃ s : Set X, x ∈ s ∧ IsOpen s ∧ T0Space s) : T0Space X | T0Space.of_cover fun x _ hxy =>
let ⟨s, hxs, hso, hs⟩ := h x
⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, hs⟩ | theorem | T0Space.of_open_cover | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Set",
"T0Space",
"T0Space.of_cover"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R0Space (X : Type u) [TopologicalSpace X] : Prop where
/-- In an R₀ space, the `Specializes` relation is symmetric. -/
specializes_symmetric : Symmetric (Specializes : X → X → Prop) | class | R0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Specializes",
"Symmetric",
"TopologicalSpace"
] | A topological space is called an R₀ space, if `Specializes` relation is symmetric.
In other words, given two points `x y : X`,
if every neighborhood of `y` contains `x`, then every neighborhood of `x` contains `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Specializes.symm (h : x ⤳ y) : y ⤳ x | specializes_symmetric h | theorem | Specializes.symm | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [] | In an R₀ space, the `Specializes` relation is symmetric, dot notation version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
specializes_comm : x ⤳ y ↔ y ⤳ x | ⟨Specializes.symm, Specializes.symm⟩ | theorem | specializes_comm | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [] | In an R₀ space, the `Specializes` relation is symmetric, `Iff` version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y | ⟨fun h ↦ h.antisymm h.symm, Inseparable.specializes⟩
/-- In an R₀ space, `Specializes` implies `Inseparable`. -/
alias ⟨Specializes.inseparable, _⟩ := specializes_iff_inseparable | theorem | specializes_iff_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable"
] | In an R₀ space, `Specializes` is equivalent to `Inseparable`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Topology.IsInducing.r0Space [TopologicalSpace Y] {f : Y → X} (hf : IsInducing f) :
R0Space Y | where
specializes_symmetric a b := by
simpa only [← hf.specializes_iff] using Specializes.symm | theorem | Topology.IsInducing.r0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R0Space",
"Specializes.symm",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
R0Space.closure_singleton (x : X) : closure {x} = (𝓝 x).ker | by
ext; simp [ker_nhds_eq_specializes, ← specializes_iff_mem_closure, specializes_comm] | lemma | R0Space.closure_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"closure",
"ker_nhds_eq_specializes",
"specializes_comm",
"specializes_iff_mem_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCompact_closure_singleton : IsCompact (closure {x}) | by
refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi | theorem | isCompact_closure_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsCompact",
"closure",
"isCompact_of_finite_subcover",
"specializes_comm",
"specializes_iff_mem_closure",
"subset_closure"
] | In an R₀ space, the closure of a singleton is a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.coclosedCompact_le_cofinite : coclosedCompact X ≤ cofinite | le_cofinite_iff_compl_singleton_mem.2 fun _ ↦
compl_mem_coclosedCompact.2 isCompact_closure_singleton | theorem | Filter.coclosedCompact_le_cofinite | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"isCompact_closure_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bornology.relativelyCompact : Bornology X | where
cobounded := Filter.coclosedCompact X
le_cofinite := Filter.coclosedCompact_le_cofinite | def | Bornology.relativelyCompact | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Bornology",
"Filter.coclosedCompact",
"Filter.coclosedCompact_le_cofinite"
] | In an R₀ space, relatively compact sets form a bornology.
Its cobounded filter is `Filter.coclosedCompact`.
See also `Bornology.inCompact` the bornology of sets contained in a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Bornology.relativelyCompact.isBounded_iff {s : Set X} :
@Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) | compl_mem_coclosedCompact | theorem | Bornology.relativelyCompact.isBounded_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Bornology.IsBounded",
"Bornology.relativelyCompact",
"IsCompact",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.isCompact_closure {s : Set X} (hs : s.Finite) : IsCompact (closure s) | let _ : Bornology X := .relativelyCompact X
Bornology.relativelyCompact.isBounded_iff.1 hs.isBounded | theorem | Set.Finite.isCompact_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Bornology",
"IsCompact",
"Set",
"closure"
] | In an R₀ space, the closure of a finite set is a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
T1Space (X : Type u) [TopologicalSpace X] : Prop where
/-- A singleton in a T₁ space is a closed set. -/
t1 : ∀ x, IsClosed ({x} : Set X) | class | T1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"TopologicalSpace"
] | A T₁ space, also known as a Fréchet space, is a topological space
where every singleton set is closed. Equivalently, for every pair
`x ≠ y`, there is an open set containing `x` and not `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_singleton [T1Space X] {x : X} : IsClosed ({x} : Set X) | T1Space.t1 x | theorem | isClosed_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_compl_singleton [T1Space X] {x : X} : IsOpen ({x}ᶜ : Set X) | isClosed_singleton.isOpen_compl | theorem | isOpen_compl_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_ne [T1Space X] {x : X} : IsOpen { y | y ≠ x } | isOpen_compl_singleton | theorem | isOpen_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"T1Space",
"isOpen_compl_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.isOpen_mulSupport [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X}
(hf : Continuous f) : IsOpen (mulSupport f) | isOpen_ne.preimage hf | theorem | Continuous.isOpen_mulSupport | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Continuous",
"IsOpen",
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ne.nhdsWithin_compl_singleton [T1Space X] {x y : X} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x | isOpen_ne.nhdsWithin_eq h | theorem | Ne.nhdsWithin_compl_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ne.nhdsWithin_diff_singleton [T1Space X] {x y : X} (h : x ≠ y) (s : Set X) :
𝓝[s \ {y}] x = 𝓝[s] x | by
rw [diff_eq, inter_comm, nhdsWithin_inter_of_mem]
exact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_nhds h) | theorem | Ne.nhdsWithin_diff_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"mem_nhdsWithin_of_mem_nhds",
"nhdsWithin_inter_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_compl_singleton_le [T1Space X] (x y : X) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x | by
rcases eq_or_ne x y with rfl | hy
· exact Eq.le rfl
· rw [Ne.nhdsWithin_compl_singleton hy]
exact nhdsWithin_le_nhds | lemma | nhdsWithin_compl_singleton_le | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Ne.nhdsWithin_compl_singleton",
"T1Space",
"eq_or_ne",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_setOf_eventually_nhdsWithin [T1Space X] {p : X → Prop} :
IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } | by
refine isOpen_iff_mem_nhds.mpr fun a ha => ?_
filter_upwards [eventually_nhds_nhdsWithin.mpr ha] with b hb
rcases eq_or_ne a b with rfl | h
· exact hb
· rw [h.symm.nhdsWithin_compl_singleton] at hb
exact hb.filter_mono nhdsWithin_le_nhds | theorem | isOpen_setOf_eventually_nhdsWithin | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"T1Space",
"eq_or_ne",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.isClosed [T1Space X] {s : Set X} (hs : s.Finite) :
IsClosed s | by
rw [← biUnion_of_singleton s]
exact hs.isClosed_biUnion fun i _ => isClosed_singleton | lemma | Set.Finite.isClosed | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T1Space",
"isClosed_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space X] {b : Set (Set X)}
(hb : IsTopologicalBasis b) {x y : X} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a | by
rcases hb.isOpen_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩
exact ⟨a, ab, xa, fun h => ha h rfl⟩ | theorem | TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"isOpen_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Finset.isClosed [T1Space X] (s : Finset X) : IsClosed (s : Set X) | s.finite_toSet.isClosed | theorem | Finset.isClosed | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Finset",
"IsClosed",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_TFAE (X : Type u) [TopologicalSpace X] :
List.TFAE [T1Space X,
∀ x, IsClosed ({ x } : Set X),
∀ x, IsOpen ({ x }ᶜ : Set X),
Continuous (@CofiniteTopology.of X),
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsO... | by
tfae_have 1 ↔ 2 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
tfae_have 2 ↔ 3 := by
simp only [isOpen_compl_iff]
tfae_have 5 ↔ 3 := by
refine forall_comm.trans ?_
simp only [isOpen_iff_mem_nhds, mem_compl_iff, mem_singleton_iff]
tfae_have 5 ↔ 6 := by
simp only [← subset_compl_singleton_iff, exists_mem_subs... | theorem | t1Space_TFAE | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"CofiniteTopology.isOpen_iff'",
"CofiniteTopology.of",
"Continuous",
"Disjoint",
"Equiv.injective",
"IsClosed",
"IsOpen",
"List.TFAE",
"R0Space",
"Set",
"Set.Finite.isClosed",
"T0Space",
"T1Space",
"TopologicalSpace",
"closure_subset_iff_isClosed",
"compl_compl",
"continuous_def",
... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_iff_continuous_cofinite_of : T1Space X ↔ Continuous (@CofiniteTopology.of X) | (t1Space_TFAE X).out 0 3 | theorem | t1Space_iff_continuous_cofinite_of | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"CofiniteTopology.of",
"Continuous",
"T1Space",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CofiniteTopology.continuous_of [T1Space X] : Continuous (@CofiniteTopology.of X) | t1Space_iff_continuous_cofinite_of.mp ‹_› | theorem | CofiniteTopology.continuous_of | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"CofiniteTopology.of",
"Continuous",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_iff_exists_open :
T1Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U | (t1Space_TFAE X).out 0 6 | theorem | t1Space_iff_exists_open | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Pairwise",
"Set",
"T1Space",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_iff_disjoint_pure_nhds : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y) | (t1Space_TFAE X).out 0 8 | theorem | t1Space_iff_disjoint_pure_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"T1Space",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_iff_disjoint_nhds_pure : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y) | (t1Space_TFAE X).out 0 7 | theorem | t1Space_iff_disjoint_nhds_pure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"T1Space",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_iff_specializes_imp_eq : T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y | (t1Space_TFAE X).out 0 9 | theorem | t1Space_iff_specializes_imp_eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_iff_t0Space_and_r0Space : T1Space X ↔ T0Space X ∧ R0Space X | (t1Space_TFAE X).out 0 10 | theorem | t1Space_iff_t0Space_and_r0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R0Space",
"T0Space",
"T1Space",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_pure_nhds [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) | t1Space_iff_disjoint_pure_nhds.mp ‹_› h | theorem | disjoint_pure_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_nhds_pure [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) | t1Space_iff_disjoint_nhds_pure.mp ‹_› h | theorem | disjoint_nhds_pure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Disjoint",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Specializes.eq [T1Space X] {x y : X} (h : x ⤳ y) : x = y | t1Space_iff_specializes_imp_eq.1 ‹_› h | theorem | Specializes.eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_iff_eq [T1Space X] {x y : X} : x ⤳ y ↔ x = y | ⟨Specializes.eq, fun h => h ▸ specializes_rfl⟩ | theorem | specializes_iff_eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializes_eq_eq [T1Space X] : (· ⤳ ·) = @Eq X | funext₂ fun _ _ => propext specializes_iff_eq | theorem | specializes_eq_eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"specializes_iff_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pure_le_nhds_iff [T1Space X] {a b : X} : pure a ≤ 𝓝 b ↔ a = b | specializes_iff_pure.symm.trans specializes_iff_eq | theorem | pure_le_nhds_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"specializes_iff_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_le_nhds_iff [T1Space X] {a b : X} : 𝓝 a ≤ 𝓝 b ↔ a = b | specializes_iff_eq | theorem | nhds_le_nhds_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"specializes_iff_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_antitone {X} : Antitone (@T1Space X) | fun a _ h _ =>
@T1Space.mk _ a fun x => (T1Space.t1 x).mono h | theorem | t1Space_antitone | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Antitone",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
{s : Set X} {x x' : X} {y : Y} (hne : x' ≠ x) :
ContinuousWithinAt (Function.update f x y) s x' ↔ ContinuousWithinAt f s x' | EventuallyEq.congr_continuousWithinAt
(mem_nhdsWithin_of_mem_nhds <| mem_of_superset (isOpen_ne.mem_nhds hne) fun _y' hy' =>
Function.update_of_ne hy' _ _)
(Function.update_of_ne hne ..) | theorem | continuousWithinAt_update_of_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousWithinAt",
"Function.update",
"Function.update_of_ne",
"Set",
"T1Space",
"TopologicalSpace",
"mem_nhdsWithin_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y]
{f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) :
ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' | by
simp only [← continuousWithinAt_univ, continuousWithinAt_update_of_ne hne] | theorem | continuousAt_update_of_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousAt",
"Function.update",
"T1Space",
"TopologicalSpace",
"continuousWithinAt_univ",
"continuousWithinAt_update_of_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
{s : Set X} {x : X} {y : Y} :
ContinuousOn (Function.update f x y) s ↔
ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y)) | by
rw [ContinuousOn, ← and_forall_ne x, and_comm]
refine and_congr ⟨fun H z hz => ?_, fun H z hzx hzs => ?_⟩ (forall_congr' fun _ => ?_)
· specialize H z hz.2 hz.1
rw [continuousWithinAt_update_of_ne hz.2] at H
exact H.mono diff_subset
· rw [continuousWithinAt_update_of_ne hzx]
refine (H z ⟨hzs, hzx... | theorem | continuousOn_update_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousOn",
"Function.update",
"Set",
"T1Space",
"TopologicalSpace",
"and_forall_ne",
"continuousWithinAt_update_of_ne",
"continuousWithinAt_update_same",
"inter_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t1Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
(hf : Function.Injective f) (hf' : Continuous f) [T1Space Y] : T1Space X | t1Space_iff_specializes_imp_eq.2 fun _ _ h => hf (h.map hf').eq | theorem | t1Space_of_injective_of_continuous | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Continuous",
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsEmbedding.t1Space [TopologicalSpace Y] [T1Space Y] {f : X → Y}
(hf : IsEmbedding f) : T1Space X | t1Space_of_injective_of_continuous hf.injective hf.continuous | theorem | Topology.IsEmbedding.t1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"TopologicalSpace",
"t1Space_of_injective_of_continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Homeomorph.t1Space [TopologicalSpace Y] [T1Space X] (h : X ≃ₜ Y) : T1Space Y | h.symm.isEmbedding.t1Space | theorem | Homeomorph.t1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Subtype.t1Space {X : Type u} [TopologicalSpace X] [T1Space X] {p : X → Prop} :
T1Space (Subtype p) | IsEmbedding.subtypeVal.t1Space | instance | Subtype.t1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ULift.instT1Space [T1Space X] : T1Space (ULift X) | IsEmbedding.uliftDown.t1Space | instance | ULift.instT1Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_singleton_mem_nhds_iff [T1Space X] {x y : X} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x | isOpen_compl_singleton.mem_nhds_iff | theorem | compl_singleton_mem_nhds_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_singleton_mem_nhds [T1Space X] {x y : X} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y | compl_singleton_mem_nhds_iff.mpr h | theorem | compl_singleton_mem_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} | isClosed_singleton.closure_eq | theorem | closure_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Subsingleton.isClosed [T1Space X] {s : Set X} (hs : s.Subsingleton) : IsClosed s | by
rcases hs.eq_empty_or_singleton with rfl | ⟨x, rfl⟩
· exact isClosed_empty
· exact isClosed_singleton | lemma | Set.Subsingleton.isClosed | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T1Space",
"isClosed_empty",
"isClosed_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Subsingleton.closure_eq [T1Space X] {s : Set X} (hs : s.Subsingleton) :
closure s = s | hs.isClosed.closure_eq | theorem | Set.Subsingleton.closure_eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) :
(closure s).Subsingleton | by
rwa [hs.closure_eq] | theorem | Set.Subsingleton.closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subsingleton_closure [T1Space X] {s : Set X} : (closure s).Subsingleton ↔ s.Subsingleton | ⟨fun h => h.anti subset_closure, fun h => h.closure⟩ | theorem | subsingleton_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedMap_const {X Y} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y} :
IsClosedMap (Function.const X y) | IsClosedMap.of_nonempty fun s _ h2s => by simp_rw [const, h2s.image_const, isClosed_singleton] | theorem | isClosedMap_const | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosedMap",
"T1Space",
"TopologicalSpace",
"isClosed_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedMap_prodMk_left [TopologicalSpace Y] [T1Space X] (x : X) :
IsClosedMap (fun y : Y ↦ Prod.mk x y) | fun _K hK ↦ Set.singleton_prod ▸ isClosed_singleton.prod hK | lemma | isClosedMap_prodMk_left | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosedMap",
"Set.singleton_prod",
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedMap_prodMk_right [TopologicalSpace Y] [T1Space Y] (y : Y) :
IsClosedMap (fun x : X ↦ Prod.mk x y) | fun _K hK ↦ Set.prod_singleton ▸ hK.prod isClosed_singleton | lemma | isClosedMap_prodMk_right | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosedMap",
"Set.prod_singleton",
"T1Space",
"TopologicalSpace",
"isClosed_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_insert_of_ne [T1Space X] {x y : X} {s : Set X} (hxy : x ≠ y) :
𝓝[insert y s] x = 𝓝[s] x | by
refine le_antisymm (Filter.le_def.2 fun t ht => ?_) (nhdsWithin_mono x <| subset_insert y s)
obtain ⟨o, ho, hxo, host⟩ := mem_nhdsWithin.mp ht
refine mem_nhdsWithin.mpr ⟨o \ {y}, ho.sdiff isClosed_singleton, ⟨hxo, hxy⟩, ?_⟩
rw [inter_insert_of_notMem <| notMem_diff_of_mem (mem_singleton y)]
exact (inter_su... | theorem | nhdsWithin_insert_of_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"isClosed_singleton",
"le_antisymm",
"nhdsWithin_mono",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
insert_mem_nhdsWithin_of_subset_insert [T1Space X] {x y : X} {s t : Set X}
(hu : t ⊆ insert y s) : insert x s ∈ 𝓝[t] x | by
rcases eq_or_ne x y with (rfl | h)
· exact mem_of_superset self_mem_nhdsWithin hu
refine nhdsWithin_mono x hu ?_
rw [nhdsWithin_insert_of_ne h]
exact mem_of_superset self_mem_nhdsWithin (subset_insert x s) | theorem | insert_mem_nhdsWithin_of_subset_insert | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"eq_or_ne",
"nhdsWithin_insert_of_ne",
"nhdsWithin_mono",
"self_mem_nhdsWithin"
] | If `t` is a subset of `s`, except for one point,
then `insert x s` is a neighborhood of `x` within `t`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eventuallyEq_insert [T1Space X] {s t : Set X} {x y : X} (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
(insert x s : Set X) =ᶠ[𝓝 x] (insert x t : Set X) | by
simp_rw [eventuallyEq_set] at h ⊢
simp_rw [← union_singleton, ← nhdsWithin_univ, ← compl_union_self {x},
nhdsWithin_union, eventually_sup, nhdsWithin_singleton,
eventually_pure, union_singleton, mem_insert_iff, true_or, and_true]
filter_upwards [nhdsWithin_compl_singleton_le x y h] with y using or_cong... | lemma | eventuallyEq_insert | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"nhdsWithin_compl_singleton_le",
"nhdsWithin_singleton",
"nhdsWithin_union",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ker_nhds [T1Space X] (x : X) : (𝓝 x).ker = {x} | by
simp [ker_nhds_eq_specializes] | theorem | ker_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"ker_nhds_eq_specializes"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
biInter_basis_nhds [T1Space X] {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {x : X}
(h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} | by
rw [← h.ker, ker_nhds] | theorem | biInter_basis_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"ker_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl_singleton_mem_nhdsSet_iff [T1Space X] {x : X} {s : Set X} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s | by
rw [isOpen_compl_singleton.mem_nhdsSet, subset_compl_singleton_iff] | theorem | compl_singleton_mem_nhdsSet_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_le_iff [T1Space X] {s t : Set X} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t | by
refine ⟨?_, fun h => monotone_nhdsSet h⟩
simp_rw [Filter.le_def]; intro h x hx
specialize h {x}ᶜ
simp_rw [compl_singleton_mem_nhdsSet_iff] at h
by_contra hxt
exact h hxt hx | theorem | nhdsSet_le_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter.le_def",
"Set",
"T1Space",
"compl_singleton_mem_nhdsSet_iff",
"monotone_nhdsSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_inj_iff [T1Space X] {s t : Set X} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t | by
simp_rw [le_antisymm_iff]
exact and_congr nhdsSet_le_iff nhdsSet_le_iff | theorem | nhdsSet_inj_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"le_antisymm_iff",
"nhdsSet_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
injective_nhdsSet [T1Space X] : Function.Injective (𝓝ˢ : Set X → Filter X) | fun _ _ hst =>
nhdsSet_inj_iff.mp hst | theorem | injective_nhdsSet | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
strictMono_nhdsSet [T1Space X] : StrictMono (𝓝ˢ : Set X → Filter X) | monotone_nhdsSet.strictMono_of_injective injective_nhdsSet | theorem | strictMono_nhdsSet | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"Set",
"StrictMono",
"T1Space",
"injective_nhdsSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_le_nhdsSet_iff [T1Space X] {s : Set X} {x : X} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s | by
rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff] | theorem | nhds_le_nhdsSet_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T1Space",
"nhdsSet_le_iff",
"nhdsSet_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.diff_singleton [T1Space X] {s : Set X} (hs : Dense s) (x : X) [NeBot (𝓝[≠] x)] :
Dense (s \ {x}) | hs.inter_of_isOpen_right (dense_compl_singleton x) isOpen_compl_singleton | theorem | Dense.diff_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Dense",
"Set",
"T1Space",
"dense_compl_singleton",
"isOpen_compl_singleton"
] | Removing a non-isolated point from a dense set, one still obtains a dense set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.diff_finset [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
(t : Finset X) : Dense (s \ t) | by
classical
induction t using Finset.induction_on with
| empty => simpa using hs
| insert _ _ _ ih =>
rw [Finset.coe_insert, ← union_singleton, ← diff_diff]
exact ih.diff_singleton _ | theorem | Dense.diff_finset | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Dense",
"Finset",
"Finset.coe_insert",
"Finset.induction_on",
"Set",
"T1Space"
] | Removing a finset from a dense set in a space without isolated points, one still
obtains a dense set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.diff_finite [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
{t : Set X} (ht : t.Finite) : Dense (s \ t) | by
convert! hs.diff_finset ht.toFinset
exact (Finite.coe_toFinset _).symm | theorem | Dense.diff_finite | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Dense",
"Set",
"T1Space",
"symm"
] | Removing a finite set from a dense set in a space without isolated points, one still
obtains a dense set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
(h : Tendsto f (𝓝 x) (𝓝 y)) : f x = y | by_contra fun hfa : f x ≠ y =>
have fact₁ : {f x}ᶜ ∈ 𝓝 y := compl_singleton_mem_nhds hfa.symm
have fact₂ : Tendsto f (pure x) (𝓝 y) := h.comp (tendsto_id'.2 <| pure_le_nhds x)
fact₂ fact₁ (Eq.refl <| f x) | theorem | eq_of_tendsto_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"TopologicalSpace",
"compl_singleton_mem_nhds",
"pure_le_nhds"
] | If a function to a `T1Space` tends to some limit `y` at some point `x`, then necessarily
`y = f x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.eventually_ne {X} [TopologicalSpace Y] [T1Space Y] {g : X → Y}
{l : Filter X} {b₁ b₂ : Y} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ | hg.eventually (isOpen_compl_singleton.eventually_mem hb) | theorem | Filter.Tendsto.eventually_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y}
(hg1 : ContinuousAt g x) (hg2 : g x ≠ y) : ∀ᶠ z in 𝓝 x, g z ≠ y | hg1.tendsto.eventually_ne hg2 | theorem | ContinuousAt.eventually_ne | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousAt",
"T1Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_ne_nhds [T1Space X] {a b : X} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b | IsOpen.eventually_mem isOpen_ne h | theorem | eventually_ne_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen.eventually_mem",
"T1Space",
"isOpen_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_ne_nhdsWithin [T1Space X] {a b : X} {s : Set X} (h : a ≠ b) :
∀ᶠ x in 𝓝[s] a, x ≠ b | Filter.Eventually.filter_mono nhdsWithin_le_nhds <| eventually_ne_nhds h | theorem | eventually_ne_nhdsWithin | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter.Eventually.filter_mono",
"Set",
"T1Space",
"eventually_ne_nhds",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhdsWithin_eventually_nhds_iff_of_isOpen {s : Set X} {a : X} {p : X → Prop}
(hs : IsOpen s) : (∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x | by
nth_rw 2 [← eventually_eventually_nhdsWithin]
constructor
· intro h
filter_upwards [h] with _ hy
exact eventually_nhdsWithin_of_eventually_nhds hy
· intro h
filter_upwards [h, eventually_nhdsWithin_of_forall fun _ a ↦ a] with _ _ _
simp_all [IsOpen.nhdsWithin_eq] | theorem | eventually_nhdsWithin_eventually_nhds_iff_of_isOpen | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"IsOpen.nhdsWithin_eq",
"Set",
"eventually_eventually_nhdsWithin",
"eventually_nhdsWithin_of_eventually_nhds",
"eventually_nhdsWithin_of_forall"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhdsNE_eventually_nhds_iff [T1Space X] {a : X} {p : X → Prop} :
(∀ᶠ y in 𝓝[≠] a, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝[≠] a, p x | eventually_nhdsWithin_eventually_nhds_iff_of_isOpen isOpen_ne | theorem | eventually_nhdsNE_eventually_nhds_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T1Space",
"eventually_nhdsWithin_eventually_nhds_iff_of_isOpen",
"isOpen_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_insert [TopologicalSpace Y] [T1Space X]
{x y : X} {s : Set X} {f : X → Y} :
ContinuousWithinAt f (insert y s) x ↔ ContinuousWithinAt f s x | by
rcases eq_or_ne x y with (rfl | h)
· exact continuousWithinAt_insert_self
simp_rw [ContinuousWithinAt, nhdsWithin_insert_of_ne h]
alias ⟨ContinuousWithinAt.of_insert, ContinuousWithinAt.insert'⟩ := continuousWithinAt_insert | theorem | continuousWithinAt_insert | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousWithinAt",
"Set",
"T1Space",
"TopologicalSpace",
"continuousWithinAt_insert_self",
"eq_or_ne",
"nhdsWithin_insert_of_ne"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_diff_singleton [TopologicalSpace Y] [T1Space X]
{x y : X} {s : Set X} {f : X → Y} :
ContinuousWithinAt f (s \ {y}) x ↔ ContinuousWithinAt f s x | by
rw [← continuousWithinAt_insert, insert_diff_singleton, continuousWithinAt_insert] | theorem | continuousWithinAt_diff_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousWithinAt",
"Set",
"T1Space",
"TopologicalSpace",
"continuousWithinAt_insert"
] | See also `continuousWithinAt_diff_self` for the case `y = x` but not requiring `T1Space`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_congr_set' [TopologicalSpace Y] [T1Space X]
{x : X} {s t : Set X} {f : X → Y} (y : X) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x | by
rw [← continuousWithinAt_insert_self (s := s), ← continuousWithinAt_insert_self (s := t)]
exact continuousWithinAt_congr_set (eventuallyEq_insert h) | theorem | continuousWithinAt_congr_set' | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousWithinAt",
"Set",
"T1Space",
"TopologicalSpace",
"continuousWithinAt_congr_set",
"continuousWithinAt_insert_self",
"eventuallyEq_insert"
] | If two sets coincide locally around `x`, except maybe at `y`, then it is equivalent to be
continuous at `x` within one set or the other. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousWithinAt.eq_const_of_mem_closure [TopologicalSpace Y] [T1Space Y]
{f : X → Y} {s : Set X} {x : X} {c : Y} (h : ContinuousWithinAt f s x) (hx : x ∈ closure s)
(ht : ∀ y ∈ s, f y = c) : f x = c | by
rw [← Set.mem_singleton_iff, ← closure_singleton]
exact h.mem_closure hx ht | theorem | ContinuousWithinAt.eq_const_of_mem_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousWithinAt",
"Set",
"Set.mem_singleton_iff",
"T1Space",
"TopologicalSpace",
"closure",
"closure_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.eqOn_const_closure [TopologicalSpace Y] [T1Space Y]
{f : X → Y} {s : Set X} {c : Y} (h : ∀ x ∈ closure s, ContinuousWithinAt f s x)
(ht : s.EqOn f (fun _ ↦ c)) : (closure s).EqOn f (fun _ ↦ c) | by
intro x hx
apply ContinuousWithinAt.eq_const_of_mem_closure (h x hx) hx ht | theorem | ContinuousWithinAt.eqOn_const_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousWithinAt",
"ContinuousWithinAt.eq_const_of_mem_closure",
"Set",
"T1Space",
"TopologicalSpace",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
(h : Tendsto f (𝓝 x) (𝓝 y)) : ContinuousAt f x | by
rwa [ContinuousAt, eq_of_tendsto_nhds h] | theorem | continuousAt_of_tendsto_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousAt",
"T1Space",
"TopologicalSpace",
"eq_of_tendsto_nhds"
] | To prove a function to a `T1Space` is continuous at some point `x`, it suffices to prove that
`f` admits *some* limit at `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_const_nhds_iff [T1Space X] {l : Filter Y} [NeBot l] {c d : X} :
Tendsto (fun _ => c) l (𝓝 d) ↔ c = d | by simp_rw [Tendsto, Filter.map_const, pure_le_nhds_iff] | theorem | tendsto_const_nhds_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"Filter.map_const",
"T1Space",
"pure_le_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_singleton_of_finite_mem_nhds [T1Space X] (x : X)
{s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set X) | by
have A : {x} ⊆ s := by simp only [singleton_subset_iff, mem_of_mem_nhds hs]
have B : IsClosed (s \ {x}) := (hsf.subset diff_subset).isClosed
have C : (s \ {x})ᶜ ∈ 𝓝 x := B.isOpen_compl.mem_nhds fun h => h.2 rfl
have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C
... | theorem | isOpen_singleton_of_finite_mem_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"IsOpen",
"Set",
"T1Space",
"mem_interior_iff_mem_nhds",
"mem_of_mem_nhds",
"subset_interior_iff_isOpen"
] | A point with a finite neighborhood has to be isolated. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
infinite_of_mem_nhds {X} [TopologicalSpace X] [T1Space X] (x : X) [hx : NeBot (𝓝[≠] x)]
{s : Set X} (hs : s ∈ 𝓝 x) : Set.Infinite s | by
refine fun hsf => hx.1 ?_
rw [← isOpen_singleton_iff_punctured_nhds]
exact isOpen_singleton_of_finite_mem_nhds x hs hsf | theorem | infinite_of_mem_nhds | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"Set.Infinite",
"T1Space",
"TopologicalSpace",
"isOpen_singleton_iff_punctured_nhds",
"isOpen_singleton_of_finite_mem_nhds"
] | If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is
infinite. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Finite.instDiscreteTopology [T1Space X] [Finite X] : DiscreteTopology X | discreteTopology_iff_forall_isClosed.mpr (·.toFinite.isClosed) | instance | Finite.instDiscreteTopology | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"DiscreteTopology",
"Finite",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.isDiscrete [T1Space X] {s : Set X} (hs : s.Finite) : IsDiscrete s | ⟨@Finite.instDiscreteTopology _ _ _ hs.to_subtype⟩ | lemma | Set.Finite.isDiscrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Finite.instDiscreteTopology",
"IsDiscrete",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.Finite.continuousOn [T1Space X] [TopologicalSpace Y] {s : Set X} (hs : s.Finite)
(f : X → Y) : ContinuousOn f s | by
rw [continuousOn_iff_continuous_restrict]
have : Finite s := hs
fun_prop | theorem | Set.Finite.continuousOn | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"ContinuousOn",
"Finite",
"Set",
"T1Space",
"TopologicalSpace",
"continuousOn_iff_continuous_restrict"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparationQuotient.t1Space_iff : T1Space (SeparationQuotient X) ↔ R0Space X | by
rw [r0Space_iff, ((t1Space_TFAE (SeparationQuotient X)).out 0 9 :)]
constructor
· intro h x y xspecy
rw [← IsInducing.specializes_iff isInducing_mk, h xspecy] at *
· -- TODO is there are better way to do this,
-- so the case split produces `SeparationQuotient.mk` directly, rather than `Quot.mk`?
... | theorem | SeparationQuotient.t1Space_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"R0Space",
"SeparationQuotient",
"T1Space",
"inseparable_iff_specializes_and",
"t1Space_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_inter_singleton [T1Space X] {A : Set X} {a : X} : IsClosed (A ∩ {a}) | Subsingleton.inter_singleton.isClosed | lemma | isClosed_inter_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_singleton_inter [T1Space X] {A : Set X} {a : X} : IsClosed ({a} ∩ A) | Subsingleton.singleton_inter.isClosed | lemma | isClosed_singleton_inter | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T1Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X}
(hx : x ∈ s) : {x} ∈ 𝓝[s] x | by
rw [isDiscrete_iff_discreteTopology] at hs
have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]
simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using
@image_mem_map _ _ _ ((↑) : s → X) _ this | theorem | singleton_mem_nhdsWithin_of_mem_discrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsDiscrete",
"Set",
"isDiscrete_iff_discreteTopology",
"nhdsWithin_eq_map_subtype_coe",
"nhds_discrete"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X} (hx : x ∈ s) :
𝓝[s] x = pure x | (le_pure_iff.2 <| singleton_mem_nhdsWithin_of_mem_discrete hs hx).antisymm (pure_le_nhdsWithin hx) | theorem | nhdsWithin_of_mem_discrete | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsDiscrete",
"Set",
"antisymm",
"pure_le_nhdsWithin",
"singleton_mem_nhdsWithin_of_mem_discrete"
] | The neighbourhoods filter of `x` within `s`, under the discrete topology, is equal to
the pure `x` filter (which is the principal filter at the singleton `{x}`.) | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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