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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upperHemicontinuousWithinAt_iff_frequently :
UpperHemicontinuousWithinAt f s x ↔
∀ t, IsClosed t → (∃ᶠ x' in 𝓝[s] x, ((f x') ∩ t).Nonempty) → ((f x) ∩ t).Nonempty | by
rw [UpperHemicontinuousWithinAt, semicontinuousWithinAt_iff_frequently, compl_surjective.forall]
simp [← subset_interior_iff_mem_nhdsSet, not_subset, forall_isClosed_iff, inter_nonempty]
alias ⟨UpperHemicontinuousWithinAt.frequently, UpperHemicontinuousWithinAt.of_frequently⟩ :=
upperHemicontinuousWithinAt_if... | lemma | upperHemicontinuousWithinAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuousWithinAt",
"forall_isClosed_iff",
"semicontinuousWithinAt_iff_frequently",
"subset_interior_iff_mem_nhdsSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_iff_frequently :
UpperHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ t, IsClosed t →
(∃ᶠ x' in 𝓝[s] x, ((f x') ∩ t).Nonempty) → ((f x) ∩ t).Nonempty | by
simp_rw [upperHemicontinuousOn_iff, upperHemicontinuousWithinAt_iff_frequently]
alias ⟨UpperHemicontinuousOn.frequently, UpperHemicontinuousOn.of_frequently⟩ :=
upperHemicontinuousOn_iff_frequently | lemma | upperHemicontinuousOn_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuousOn",
"upperHemicontinuousOn_iff",
"upperHemicontinuousWithinAt_iff_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousAt_iff_frequently :
UpperHemicontinuousAt f x ↔
∀ t, IsClosed t → (∃ᶠ x' in 𝓝 x, ((f x') ∩ t).Nonempty) → ((f x) ∩ t).Nonempty | by
rw [← upperHemicontinuousWithinAt_univ_iff, upperHemicontinuousWithinAt_iff_frequently]
simp
alias ⟨UpperHemicontinuousAt.frequently, UpperHemicontinuousAt.of_frequently⟩ :=
upperHemicontinuousAt_iff_frequently | lemma | upperHemicontinuousAt_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuousAt",
"upperHemicontinuousWithinAt_iff_frequently",
"upperHemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_iff_frequently :
UpperHemicontinuous f ↔
∀ x t, IsClosed t → (∃ᶠ x' in 𝓝 x, ((f x') ∩ t).Nonempty) → ((f x) ∩ t).Nonempty | by
simp_rw [upperHemicontinuous_iff, upperHemicontinuousAt_iff_frequently]
alias ⟨UpperHemicontinuous.frequently, UpperHemicontinuous.of_frequently⟩ :=
upperHemicontinuous_iff_frequently | lemma | upperHemicontinuous_iff_frequently | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuous",
"upperHemicontinuousAt_iff_frequently",
"upperHemicontinuous_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousWithinAt.const : UpperHemicontinuousWithinAt (fun _x => z) s x | SemicontinuousWithinAt.const | theorem | UpperHemicontinuousWithinAt.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousWithinAt.const",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousAt.const : UpperHemicontinuousAt (fun _x => z) x | SemicontinuousAt.const | theorem | UpperHemicontinuousAt.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousAt.const",
"UpperHemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousOn.const : UpperHemicontinuousOn (fun _x => z) s | SemicontinuousOn.const | theorem | UpperHemicontinuousOn.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn.const",
"UpperHemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuous.const : UpperHemicontinuous fun _x : α => z | Semicontinuous.const | theorem | UpperHemicontinuous.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous.const",
"UpperHemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousWithinAt.comp
(hf : UpperHemicontinuousWithinAt f s (g c)) (hg : ContinuousWithinAt g t c)
(hg' : MapsTo g t s) :
UpperHemicontinuousWithinAt (f ∘ g) t c | -- the elaboration aid is necessary.
SemicontinuousWithinAt.comp (r := (fun x t ↦ t ∈ 𝓝ˢ (f x))) hf hg hg' | theorem | UpperHemicontinuousWithinAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousWithinAt",
"SemicontinuousWithinAt.comp",
"UpperHemicontinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousAt.comp
(hf : UpperHemicontinuousAt f (g c)) (hg : ContinuousAt g c) :
UpperHemicontinuousAt (f ∘ g) c | SemicontinuousAt.comp (r := (fun x t ↦ t ∈ 𝓝ˢ (f x))) hf hg | theorem | UpperHemicontinuousAt.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousAt",
"SemicontinuousAt.comp",
"UpperHemicontinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousOn.comp
(hf : UpperHemicontinuousOn f s) (hg : ContinuousOn g t) (hg' : MapsTo g t s) :
UpperHemicontinuousOn (f ∘ g) t | SemicontinuousOn.comp (r := (fun x t ↦ t ∈ 𝓝ˢ (f x))) hf hg hg' | theorem | UpperHemicontinuousOn.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"SemicontinuousOn.comp",
"UpperHemicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuous.comp
(hf : UpperHemicontinuous f) (hg : Continuous g) : UpperHemicontinuous (f ∘ g) | Semicontinuous.comp (r := (fun x t ↦ t ∈ 𝓝ˢ (f x))) hf hg | theorem | UpperHemicontinuous.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"Semicontinuous.comp",
"UpperHemicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSectionsOn (f : α → Set β) (s : Set α) | SemicontinuousOn (fun x b ↦ b ∈ f x) s | abbrev | HasOpenLowerSectionsOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"SemicontinuousOn",
"Set"
] | A function `f : α → Set β` has open lower sections on `s` if it has open lower sections within
`s` at every `x ∈ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
HasOpenLowerSections (f : α → Set β) | Semicontinuous (fun x b ↦ b ∈ f x) | abbrev | HasOpenLowerSections | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Semicontinuous",
"Set"
] | A function `f : α → Set β` has open lower sections if, for every `b`, the set `{x | b ∈ f x}`
is open. Equivalently, whenever `b ∈ f x`, then `b ∈ f x'` for all `x'` sufficiently close to
`x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hasOpenLowerSections_iff_isOpen : HasOpenLowerSections f ↔ ∀ b, IsOpen {x | b ∈ f x} | by
simp [semicontinuous_iff_isOpen] | theorem | hasOpenLowerSections_iff_isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"IsOpen",
"semicontinuous_iff_isOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSectionsOn.mono (h : HasOpenLowerSectionsOn f s) (hst : t ⊆ s) :
HasOpenLowerSectionsOn f t | SemicontinuousOn.mono h hst | theorem | HasOpenLowerSectionsOn.mono | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSectionsOn",
"SemicontinuousOn.mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasOpenLowerSectionsOn_univ_iff :
HasOpenLowerSectionsOn f univ ↔ HasOpenLowerSections f | semicontinuousOn_univ_iff | theorem | hasOpenLowerSectionsOn_univ_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"HasOpenLowerSectionsOn",
"semicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasOpenLowerSections_restrict_iff :
HasOpenLowerSections (s.restrict f) ↔ HasOpenLowerSectionsOn f s | semicontinuous_restrict_iff (r := (fun x b ↦ b ∈ f x)) | theorem | hasOpenLowerSections_restrict_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"HasOpenLowerSectionsOn",
"semicontinuous_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSections.hasOpenLowerSectionsOn (h : HasOpenLowerSections f) (s : Set α) :
HasOpenLowerSectionsOn f s | h.semicontinuousOn s | theorem | HasOpenLowerSections.hasOpenLowerSectionsOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"HasOpenLowerSectionsOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSectionsOn.const : HasOpenLowerSectionsOn (fun _x => z) s | SemicontinuousOn.const | theorem | HasOpenLowerSectionsOn.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSectionsOn",
"SemicontinuousOn.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSections.const : HasOpenLowerSections fun _x : α => z | Semicontinuous.const | theorem | HasOpenLowerSections.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"Semicontinuous.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSectionsOn.inter {f g : α → Set β} {s : Set α} (hf : HasOpenLowerSectionsOn f s)
(hg : HasOpenLowerSectionsOn g s) : HasOpenLowerSectionsOn (fun x ↦ f x ∩ g x) s | hf.inf hg | theorem | HasOpenLowerSectionsOn.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSectionsOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSectionsOn.union {f g : α → Set β} {s : Set α} (hf : HasOpenLowerSectionsOn f s)
(hg : HasOpenLowerSectionsOn g s) : HasOpenLowerSectionsOn (fun x ↦ f x ∪ g x) s | hf.sup hg | theorem | HasOpenLowerSectionsOn.union | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSectionsOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSections.inter {f g : α → Set β} (hf : HasOpenLowerSections f)
(hg : HasOpenLowerSections g) : HasOpenLowerSections (fun x ↦ f x ∩ g x) | hf.inf hg | theorem | HasOpenLowerSections.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSections.union {f g : α → Set β} (hf : HasOpenLowerSections f)
(hg : HasOpenLowerSections g) : HasOpenLowerSections (fun x ↦ f x ∪ g x) | hf.sup hg | theorem | HasOpenLowerSections.union | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSectionsOn.comp
(hf : HasOpenLowerSectionsOn f s) (hg : ContinuousOn g t) (hg' : MapsTo g t s) :
HasOpenLowerSectionsOn (f ∘ g) t | SemicontinuousOn.comp (r := (fun x b ↦ b ∈ f x)) hf hg hg' | theorem | HasOpenLowerSectionsOn.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"HasOpenLowerSectionsOn",
"SemicontinuousOn.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSections.comp
(hf : HasOpenLowerSections f) (hg : Continuous g) : HasOpenLowerSections (f ∘ g) | Semicontinuous.comp (r := (fun x b ↦ b ∈ f x)) hf hg | theorem | HasOpenLowerSections.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"HasOpenLowerSections",
"Semicontinuous.comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenCGraph (f : α → Set β) | IsOpen {x : α × β | x.2 ∈ f x.1} | abbrev | HasOpenCGraph | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"Set"
] | A function `f : α → Set β` has an open cgraph if the set of all `x` such that `x.2 ∈ f x.1`
is open in `α × β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
HasOpenCGraph.const {z : Set β} (hz : IsOpen z) : HasOpenCGraph (fun _x : α => z) | hz.preimage continuous_snd | theorem | HasOpenCGraph.const | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenCGraph",
"IsOpen",
"Set",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenCGraph.inter {f g : α → Set β} (hf : HasOpenCGraph f)
(hg : HasOpenCGraph g) : HasOpenCGraph (fun x ↦ f x ∩ g x) | by
have : {x : α × β | x.2 ∈ f x.1 ∩ g x.1} =
{x | x.2 ∈ f x.1} ∩ {x | x.2 ∈ g x.1} := by ext; simp
rw [HasOpenCGraph, this]
exact IsOpen.inter hf hg | theorem | HasOpenCGraph.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenCGraph",
"IsOpen.inter",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenCGraph.comp {f : α → Set β} (hf : HasOpenCGraph f) (hg : Continuous g') :
HasOpenCGraph (f ∘ g') | hf.preimage (hg.prodMap continuous_id) | theorem | HasOpenCGraph.comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"HasOpenCGraph",
"Set",
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenLowerSections.lowerHemicontinuous {f : α → Set β} (hf : HasOpenLowerSections f) :
LowerHemicontinuous f | fun x _ ⟨hopen, y, hyfx, hyt⟩ ↦
(hf x y hyfx).mono fun _ hy' ↦ ⟨hopen, y, hy', hyt⟩ | theorem | HasOpenLowerSections.lowerHemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"HasOpenLowerSections",
"LowerHemicontinuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasOpenCGraph.hasOpenLowerSections
{f : α → Set β} (h : HasOpenCGraph f) :
HasOpenLowerSections f | by
intro x b hb
have hopen : IsOpen {x' : α | b ∈ f x'} := by
simpa using h.preimage (continuous_id.prodMk continuous_const)
simpa [Filter.Eventually] using hopen.mem_nhds hb | theorem | HasOpenCGraph.hasOpenLowerSections | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Defs.lean | [
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually",
"HasOpenCGraph",
"HasOpenLowerSections",
"IsOpen",
"Set",
"continuous_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousWithinAt_iff_forall_isOpen :
UpperHemicontinuousWithinAt f s x ↔ ∀ u, IsOpen u → f x ⊆ u → ∀ᶠ x' in 𝓝[s] x, f x' ⊆ u | by
rw [upperHemicontinuousWithinAt_iff, hasBasis_nhdsSet _ |>.forall_iff ?mono]
case mono => exact fun t₁ t₂ ht h ↦ h.mp <| .of_forall fun x' ↦ by gcongr
simp only [and_imp]
apply forall₂_congr
simp +contextual [← subset_interior_iff_mem_nhdsSet, IsOpen.interior_eq]
alias ⟨UpperHemicontinuousWithinAt.forall_... | lemma | upperHemicontinuousWithinAt_iff_forall_isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"IsOpen.interior_eq",
"UpperHemicontinuousWithinAt",
"hasBasis_nhdsSet",
"subset_interior_iff_mem_nhdsSet",
"upperHemicontinuousWithinAt_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_iff_forall_isOpen :
UpperHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ u, IsOpen u → f x ⊆ u → ∀ᶠ x' in 𝓝[s] x, f x' ⊆ u | by
simp [upperHemicontinuousOn_iff, upperHemicontinuousWithinAt_iff_forall_isOpen]
alias ⟨UpperHemicontinuousOn.forall_isOpen, UpperHemicontinuousOn.of_forall_isOpen⟩ :=
upperHemicontinuousOn_iff_forall_isOpen | lemma | upperHemicontinuousOn_iff_forall_isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"UpperHemicontinuousOn",
"upperHemicontinuousOn_iff",
"upperHemicontinuousWithinAt_iff_forall_isOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousAt_iff_forall_isOpen :
UpperHemicontinuousAt f x ↔ ∀ u, IsOpen u → f x ⊆ u → ∀ᶠ x' in 𝓝 x, f x' ⊆ u | by
simpa [upperHemicontinuousWithinAt_univ_iff] using
upperHemicontinuousWithinAt_iff_forall_isOpen (s := Set.univ)
alias ⟨UpperHemicontinuousAt.forall_isOpen, UpperHemicontinuousAt.of_forall_isOpen⟩ :=
upperHemicontinuousAt_iff_forall_isOpen | lemma | upperHemicontinuousAt_iff_forall_isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"Set.univ",
"UpperHemicontinuousAt",
"upperHemicontinuousWithinAt_iff_forall_isOpen",
"upperHemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_iff_forall_isOpen :
UpperHemicontinuous f ↔ ∀ x u, IsOpen u → f x ⊆ u → ∀ᶠ x' in 𝓝 x, f x' ⊆ u | by
simp [upperHemicontinuous_iff, upperHemicontinuousAt_iff_forall_isOpen]
alias ⟨UpperHemicontinuous.forall_isOpen, UpperHemicontinuous.of_forall_isOpen⟩ :=
upperHemicontinuous_iff_forall_isOpen | lemma | upperHemicontinuous_iff_forall_isOpen | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"UpperHemicontinuous",
"upperHemicontinuousAt_iff_forall_isOpen",
"upperHemicontinuous_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousWithinAt_iff_preimage_Iic :
UpperHemicontinuousWithinAt f s x ↔ ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' (Iic u) ∈ 𝓝[s] x | by
simp_rw [upperHemicontinuousWithinAt_iff]
rw [hasBasis_nhdsSet (f x) |>.forall_iff ?h₁, hasBasis_nhdsSet (f x) |>.forall_iff ?h₂]
case h₂ =>
intro s t hst
gcongr
exact hst
case h₁ =>
intro s t hst
gcongr
refine forall₂_congr fun u ⟨hu, hfu⟩ ↦ ?_
simp [hu.mem_nhdsSet, eventually_iff, I... | lemma | upperHemicontinuousWithinAt_iff_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousWithinAt",
"hasBasis_nhdsSet",
"upperHemicontinuousWithinAt_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousAt_iff_preimage_Iic :
UpperHemicontinuousAt f x ↔ ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' (Iic u) ∈ 𝓝 x | by
simpa [upperHemicontinuousWithinAt_univ_iff] using
upperHemicontinuousWithinAt_iff_preimage_Iic (s := univ) | lemma | upperHemicontinuousAt_iff_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousAt",
"upperHemicontinuousWithinAt_iff_preimage_Iic",
"upperHemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_iff_preimage_Iic :
UpperHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' (Iic u) ∈ 𝓝[s] x | by
simp [upperHemicontinuousOn_iff, upperHemicontinuousWithinAt_iff_preimage_Iic] | lemma | upperHemicontinuousOn_iff_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousOn",
"upperHemicontinuousOn_iff",
"upperHemicontinuousWithinAt_iff_preimage_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_iff_preimage_Iic :
UpperHemicontinuous f ↔ ∀ x, ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' (Iic u) ∈ 𝓝 x | by
simp [upperHemicontinuous_iff, upperHemicontinuousAt_iff_preimage_Iic] | lemma | upperHemicontinuous_iff_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuous",
"upperHemicontinuousAt_iff_preimage_Iic",
"upperHemicontinuous_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_iff_isOpen_preimage_Iic :
UpperHemicontinuous f ↔ ∀ u, IsOpen u → IsOpen (f ⁻¹' (Iic u)) | by
simp_rw [upperHemicontinuous_iff_preimage_Iic, isOpen_iff_mem_nhds (s := f ⁻¹' (Iic _))]
conv =>
enter [1, x]
rw [hasBasis_nhdsSet (f x) |>.forall_iff <|
fun s t hst ↦ by gcongr; exact hst]
simp [forall_comm (α := α)] | lemma | upperHemicontinuous_iff_isOpen_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpen",
"UpperHemicontinuous",
"hasBasis_nhdsSet",
"isOpen_iff_mem_nhds",
"upperHemicontinuous_iff_preimage_Iic"
] | A correspondence `f : α → Set β` is upper hemicontinuous if and only if its *upper inverse*
(i.e., `u : Set β ↦ f ⁻¹' (Iic u)`, note that `f ⁻¹' (Iic u) = {x | f x ⊆ u}`) sends open sets
to open sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upperHemicontinuous_iff_isClosed_compl_preimage_Iic_compl :
UpperHemicontinuous f ↔ ∀ u, IsClosed u → IsClosed (f ⁻¹' (Iic uᶜ))ᶜ | by
conv_rhs =>
rw [compl_surjective.forall]
simp [← isOpen_compl_iff]
exact upperHemicontinuous_iff_isOpen_preimage_Iic | lemma | upperHemicontinuous_iff_isClosed_compl_preimage_Iic_compl | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuous",
"isOpen_compl_iff",
"upperHemicontinuous_iff_isOpen_preimage_Iic"
] | A correspondence `f : α → Set β` is upper hemicontinuous if and only if its *lower inverse*
(i.e., `u : Set β ↦ (f ⁻¹' (Iic uᶜ))ᶜ`, note that `f ⁻¹' (Iic u) = {x | (f x ∩ u).Nonempty}`)
sends closed sets to closed sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosedMap_iff_upperHemicontinuous {f : α → β} :
IsClosedMap f ↔ UpperHemicontinuous (f ⁻¹' {·}) | by
rw [isClosedMap_iff_kernImage, upperHemicontinuous_iff_isOpen_preimage_Iic]
aesop | lemma | isClosedMap_iff_upperHemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosedMap",
"UpperHemicontinuous",
"upperHemicontinuous_iff_isOpen_preimage_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl :
LowerHemicontinuous f ↔ ∀ u, IsOpen u → IsOpen (f ⁻¹' (Iic uᶜ))ᶜ | by
have (u : Set β) : (f ⁻¹' (Iic uᶜ))ᶜ = {x | (f x ∩ u).Nonempty} := by
simp [Set.ext_iff, Iic, Set.mem_compl_iff, Set.not_subset, Set.Nonempty]
simp_rw [lowerHemicontinuous_iff, lowerHemicontinuousAt_iff, this, isOpen_iff_mem_nhds,
forall_comm (α := α), mem_setOf, Filter.Eventually] | lemma | lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"Filter.Eventually",
"IsOpen",
"LowerHemicontinuous",
"Set",
"Set.Nonempty",
"Set.mem_compl_iff",
"Set.not_subset",
"isOpen_iff_mem_nhds",
"lowerHemicontinuousAt_iff",
"lowerHemicontinuous_iff"
] | A correspondence `f : α → Set β` is lower hemicontinuous if and only if its *lower inverse*
(i.e., `u : Set β ↦ (f ⁻¹' (Iic uᶜ))ᶜ`, note that `f ⁻¹' (Iic u) = {x | (f x ∩ u).Nonempty}`)
sends open sets to open sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lowerHemicontinuous_iff_isClosed_preimage_Iic :
LowerHemicontinuous f ↔ ∀ u, IsClosed u → IsClosed (f ⁻¹' (Iic u)) | by
conv_rhs =>
rw [compl_surjective.forall]
simp [← isOpen_compl_iff]
exact lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl | lemma | lowerHemicontinuous_iff_isClosed_preimage_Iic | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"LowerHemicontinuous",
"isOpen_compl_iff",
"lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl"
] | A correspondence `f : α → Set β` is lower hemicontinuous if and only if its *upper inverse*
(i.e., `u : Set β ↦ f ⁻¹' (Iic u)`, note that `f ⁻¹' (Iic u) = {x | f x ⊆ u}`) sends closed sets
to closed sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpenMap_iff_lowerHemicontinuous {f : α → β} :
IsOpenMap f ↔ LowerHemicontinuous (f ⁻¹' {·}) | by
rw [isOpenMap_iff_kernImage, lowerHemicontinuous_iff_isClosed_preimage_Iic]
aesop | lemma | isOpenMap_iff_lowerHemicontinuous | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsOpenMap",
"LowerHemicontinuous",
"lowerHemicontinuous_iff_isClosed_preimage_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_singleton_id : UpperHemicontinuous ({·} : α → Set α) | by
simp [upperHemicontinuous_iff, upperHemicontinuousAt_iff] | lemma | upperHemicontinuous_singleton_id | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"Set",
"UpperHemicontinuous",
"upperHemicontinuousAt_iff",
"upperHemicontinuous_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousWithinAt_singleton_iff {f : α → β} {s : Set α} {x : α} :
UpperHemicontinuousWithinAt ({f ·}) s x ↔ ContinuousWithinAt f s x | by
refine ⟨?_, fun hf ↦ upperHemicontinuous_singleton_id.upperHemicontinuousWithinAt _ _ |>.comp hf
(mapsTo_image _ _)⟩
simp only [upperHemicontinuousWithinAt_iff, nhdsSet_singleton, ContinuousWithinAt,
tendsto_iff_forall_eventually_mem]
intro h t ht
filter_upwards [h t ht] with x
exact mem_of_mem_nhd... | lemma | upperHemicontinuousWithinAt_singleton_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousWithinAt",
"Set",
"UpperHemicontinuousWithinAt",
"mem_of_mem_nhds",
"nhdsSet_singleton",
"upperHemicontinuousWithinAt_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousAt_singleton_iff {f : α → β} {x : α} :
UpperHemicontinuousAt ({f ·}) x ↔ ContinuousAt f x | by
simp [← upperHemicontinuousWithinAt_univ_iff, continuousWithinAt_univ] | lemma | upperHemicontinuousAt_singleton_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousAt",
"UpperHemicontinuousAt",
"continuousWithinAt_univ",
"upperHemicontinuousWithinAt_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuousOn_singleton_iff {f : α → β} {s : Set α} :
UpperHemicontinuousOn ({f ·}) s ↔ ContinuousOn f s | forall₂_congr <| fun _ _ ↦ upperHemicontinuousWithinAt_singleton_iff | lemma | upperHemicontinuousOn_singleton_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"ContinuousOn",
"Set",
"UpperHemicontinuousOn",
"upperHemicontinuousWithinAt_singleton_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperHemicontinuous_singleton_iff {f : α → β} :
UpperHemicontinuous ({f ·}) ↔ Continuous f | by
simp [← upperHemicontinuousOn_univ_iff] | lemma | upperHemicontinuous_singleton_iff | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"Continuous",
"UpperHemicontinuous",
"upperHemicontinuousOn_univ_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UpperHemicontinuousWithinAt.union (hf : UpperHemicontinuousWithinAt f s x)
(hg : UpperHemicontinuousWithinAt g s x) :
UpperHemicontinuousWithinAt (fun x ↦ f x ∪ g x) s x | by
rw [upperHemicontinuousWithinAt_iff] at hf hg ⊢
aesop | lemma | UpperHemicontinuousWithinAt.union | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousWithinAt",
"upperHemicontinuousWithinAt_iff"
] | Pointwise unions of upper hemicontinuous maps are upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousOn.union (hf : UpperHemicontinuousOn f s)
(hg : UpperHemicontinuousOn g s) : UpperHemicontinuousOn (fun x ↦ f x ∪ g x) s | by
rw [upperHemicontinuousOn_iff] at hf hg ⊢
exact fun x hx ↦ (hf x hx).union (hg x hx) | lemma | UpperHemicontinuousOn.union | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousOn",
"upperHemicontinuousOn_iff"
] | Pointwise unions of upper hemicontinuous maps are upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousAt.union (hf : UpperHemicontinuousAt f x)
(hg : UpperHemicontinuousAt g x) :
UpperHemicontinuousAt (fun x ↦ f x ∪ g x) x | by
rw [← upperHemicontinuousWithinAt_univ_iff] at hf hg ⊢
exact hf.union hg | lemma | UpperHemicontinuousAt.union | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuousAt",
"upperHemicontinuousWithinAt_univ_iff"
] | Pointwise unions of upper hemicontinuous maps are upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuous.union (hf : UpperHemicontinuous f) (hg : UpperHemicontinuous g) :
UpperHemicontinuous (fun x ↦ f x ∪ g x) | by
rw [upperHemicontinuous_iff] at hf hg ⊢
exact fun x ↦ (hf x).union (hg x) | lemma | UpperHemicontinuous.union | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"UpperHemicontinuous",
"upperHemicontinuous_iff"
] | Pointwise unions of upper hemicontinuous maps are upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousWithinAt.inter (hf : UpperHemicontinuousWithinAt f s x)
{u : Set β} (hu : IsClosed u) :
UpperHemicontinuousWithinAt (fun x ↦ f x ∩ u) s x | by
rw [upperHemicontinuousWithinAt_iff_forall_isOpen] at hf ⊢
intro t ht_open ht
specialize hf (t ∪ uᶜ) (ht_open.union hu.isOpen_compl) (by grind)
grind | lemma | UpperHemicontinuousWithinAt.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"Set",
"UpperHemicontinuousWithinAt",
"upperHemicontinuousWithinAt_iff_forall_isOpen"
] | The pointwise intersection of an upper hemicontinuous function with a fixed closed set is
upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousOn.inter (hf : UpperHemicontinuousOn f s) {u : Set β} (hu : IsClosed u) :
UpperHemicontinuousOn (fun x ↦ f x ∩ u) s | by
rw [upperHemicontinuousOn_iff] at hf ⊢
exact (hf · · |>.inter hu) | lemma | UpperHemicontinuousOn.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"Set",
"UpperHemicontinuousOn",
"upperHemicontinuousOn_iff"
] | The pointwise intersection of an upper hemicontinuous function with a fixed closed set is
upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousAt.inter (hf : UpperHemicontinuousAt f x) {u : Set β} (hu : IsClosed u) :
UpperHemicontinuousAt (fun x ↦ f x ∩ u) x | by
rw [← upperHemicontinuousWithinAt_univ_iff] at hf ⊢
exact hf.inter hu | lemma | UpperHemicontinuousAt.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"Set",
"UpperHemicontinuousAt",
"upperHemicontinuousWithinAt_univ_iff"
] | The pointwise intersection of an upper hemicontinuous function with a fixed closed set is
upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuous.inter (hf : UpperHemicontinuous f) {u : Set β} (hu : IsClosed u) :
UpperHemicontinuous (fun x ↦ f x ∩ u) | by
rw [upperHemicontinuous_iff] at hf ⊢
exact fun x ↦ (hf x).inter hu | lemma | UpperHemicontinuous.inter | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"Set",
"UpperHemicontinuous",
"upperHemicontinuous_iff"
] | The pointwise intersection of an upper hemicontinuous function with a fixed closed set is
upper hemicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousWithinAt.isInducing_comp (hf : UpperHemicontinuousWithinAt f s x)
(hi : IsInducing i) (h_cl : IsClosed (range i)) :
UpperHemicontinuousWithinAt (fun x ↦ i ⁻¹' (f x)) s x | by
refine .of_forall_isOpen fun u hu hifu ↦ ?_
obtain ⟨v, hv, rfl⟩ := hi.isOpen_iff.mp hu
simp_rw [← preimage_inter_range (s := f _), preimage_subset_preimage_iff inter_subset_right]
at hifu ⊢
exact hf.inter h_cl |>.forall_isOpen v hv hifu | lemma | UpperHemicontinuousWithinAt.isInducing_comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuousWithinAt"
] | Post-composition with the preimage of an inducing function whose range is closed preserves
upper hemicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousOn.isInducing_comp (hf : UpperHemicontinuousOn f s)
(hi : IsInducing i) (h_cl : IsClosed (range i)) :
UpperHemicontinuousOn (fun x ↦ i ⁻¹' (f x)) s | by
rw [upperHemicontinuousOn_iff] at hf ⊢
exact fun x hx ↦ (hf x hx).isInducing_comp hi h_cl | lemma | UpperHemicontinuousOn.isInducing_comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuousOn",
"upperHemicontinuousOn_iff"
] | Post-composition with the preimage of an inducing function whose range is closed preserves
upper hemicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousAt.isInducing_comp (hf : UpperHemicontinuousAt f x)
(hi : IsInducing i) (h_cl : IsClosed (range i)) :
UpperHemicontinuousAt (fun x ↦ i ⁻¹' (f x)) x | by
simpa [upperHemicontinuousWithinAt_univ_iff] using
hf.upperHemicontinuousWithinAt (s := Set.univ) |>.isInducing_comp hi h_cl | lemma | UpperHemicontinuousAt.isInducing_comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"Set.univ",
"UpperHemicontinuousAt",
"upperHemicontinuousWithinAt_univ_iff"
] | Post-composition with the preimage of an inducing function whose range is closed preserves
upper hemicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuous.isInducing_comp (hf : UpperHemicontinuous f)
(hi : IsInducing i) (h_cl : IsClosed (range i)) :
UpperHemicontinuous (fun x ↦ i ⁻¹' (f x)) | by
rw [upperHemicontinuous_iff] at hf ⊢
exact fun x ↦ (hf x).isInducing_comp hi h_cl | lemma | UpperHemicontinuous.isInducing_comp | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"UpperHemicontinuous",
"upperHemicontinuous_iff"
] | Post-composition with the preimage of an inducing function whose range is closed preserves
upper hemicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuous.isClosed_domain {α β : Type*} [TopologicalSpace α]
[TopologicalSpace β] {f : α → Set β} (hf : UpperHemicontinuous f) :
IsClosed {x | (f x).Nonempty} | by
simp only [← isOpen_compl_iff, compl_setOf, not_nonempty_iff_eq_empty, isOpen_iff_mem_nhds]
intro x (hx : f x = ∅)
simp_rw [upperHemicontinuous_iff, upperHemicontinuousAt_iff] at hf
simpa [hx, empty_mem_iff_bot, nhdsSet_eq_bot_iff] using! hf x ∅ | lemma | UpperHemicontinuous.isClosed_domain | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsClosed",
"Set",
"TopologicalSpace",
"UpperHemicontinuous",
"isOpen_compl_iff",
"isOpen_iff_mem_nhds",
"nhdsSet_eq_bot_iff",
"upperHemicontinuousAt_iff",
"upperHemicontinuous_iff"
] | Upper hemicontinuous functions always have closed domain.
The more general fact is that if `f` is upper hemicontinuous at `x₀` within `s`, and if
`x₀` is a cluster point of `s ∩ {x | (f x).Nonempty}`, then `(f x₀).Nonempty`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousAt.of_sequences {α β : Type*} [TopologicalSpace α]
[TopologicalSpace β] {f : α → Set β} {x₀ : α} [(𝓝 x₀).IsCountablyGenerated]
{K : Set β} (hK : IsSeqCompact K) (hf : ∀ᶠ x in 𝓝 x₀, f x ⊆ K)
(h : ∀ x : ℕ → α, Tendsto x atTop (𝓝 x₀) →
∀ y : ℕ → β, (∀ n, y n ∈ f (x n)) → ∀ y₀, Tends... | by
refine .of_frequently fun t ht hft ↦ ?_
obtain ⟨x, hx, hfx⟩ := exists_seq_forall_of_frequently hft
choose y hy using hfx
obtain ⟨y₀, hy₀, φ, hφ, hyφ⟩ := hK.subseq_of_frequently_in (x := y) <| by
refine Eventually.frequently ?_
filter_upwards [hx hf] with n hn
exact hn (hy n).1
specialize h (x ∘... | lemma | UpperHemicontinuousAt.of_sequences | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"IsSeqCompact",
"Set",
"TopologicalSpace",
"UpperHemicontinuousAt",
"mem_closure_of_tendsto"
] | **Sequential characterization of upper hemicontinuity**:
A set-valued function `f : α → Set β` is upper hemicontinuous at `x₀ : α` if for every pair
of sequences `x : ℕ → α` and `y : ℕ → β` such that `x` tends to `x₀` and `y n ∈ f (x n)` and
`y` tends to `y₀ : β`, then `y₀ ∈ f x₀`. This requires that there is some (seq... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UpperHemicontinuousAt.mem_of_tendsto {α β ι : Type*} [TopologicalSpace α]
[TopologicalSpace β] [RegularSpace β] {f : α → Set β} {x₀ : α} {l : Filter ι}
(hf : UpperHemicontinuousAt f x₀) (hf_closed : IsClosed (f x₀))
{x : ι → α} (hx : Tendsto x l (𝓝 x₀))
{y : ι → β} (hy : ∃ᶠ n in l, y n ∈ f (x n)) {y₀ :... | by
by_contra
obtain ⟨s, hs, t, ht, hst⟩ := Filter.disjoint_iff.mp <| RegularSpace.regular hf_closed this
suffices ∃ᶠ n in l, y n ∈ s by
apply this
filter_upwards [hy₀ ht] with n hn hyn
exact hst.notMem_of_mem_left hyn hn
apply hy.mp
filter_upwards [hx (hf s hs)] with n hn hyn
simp only [← subset... | lemma | UpperHemicontinuousAt.mem_of_tendsto | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Hemicontinuity.lean | [
"Mathlib.Topology.Sequences",
"Mathlib.Topology.ContinuousOn"
] | [
"Filter",
"IsClosed",
"RegularSpace",
"Set",
"TopologicalSpace",
"UpperHemicontinuousAt",
"interior_subset",
"subset_interior_iff_mem_nhdsSet"
] | **Sequential characterization of upper hemicontinuity**:
If `β` is a regular space and `f : α → Set β` is upper hemicontinuous at `x₀` and `f x₀` is
closed, then for any sequences `x` and `y` (in `α` and `β`, respectively) tending to `x₀` and `y₀`,
respectively, if `y n ∈ f (x n)` frequently, then `y₀ ∈ f x₀`.
This is... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_countable_upperSemicontinuous_isGLB {s : X → E} {𝓕 : Set (X → E)}
(h𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous f) (h𝓕 : IsGLB 𝓕 s) :
∃ 𝓕' ⊆ 𝓕, 𝓕'.Countable ∧ IsGLB 𝓕' s | by
simp_rw [isGLB_pi] at *
rcases exists_countable_dense E with ⟨D, D_count, D_dense⟩
let U (f : X → E) (d : E) : Set X := {x | f x < d}
have U_open {f} (hf : f ∈ 𝓕) (d : E) : IsOpen (U f d) := h𝓕_cont f hf |>.isOpen_preimage d
have (d : E) : {x | s x < d} = ⋃ f : 𝓕, U f d := by
ext x
simp [U, isGL... | theorem | exists_countable_upperSemicontinuous_isGLB | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Lindelof.lean | [] | [
"IsOpen",
"Set",
"UpperSemicontinuous",
"eq_open_union_countable",
"trans"
] | If a function `s : X → E` can be written as the infimum of a family `𝓕` of upper semicontinuous
functions then, assuming that `X` is hereditarily Lindelöf (for example, second countable),
`s` can in fact be written as the infimum of some *countable* subfamily `𝓕'`.
This is implication a) ⇒ b) in
[N. Bourbaki, *Topol... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_countable_lowerSemicontinuous_isLUB {s : X → E} {𝓕 : Set (X → E)}
(h𝓕_cont : ∀ f ∈ 𝓕, LowerSemicontinuous f) (h𝓕 : IsLUB 𝓕 s) :
∃ 𝓕' ⊆ 𝓕, 𝓕'.Countable ∧ IsLUB 𝓕' s | exists_countable_upperSemicontinuous_isGLB (E := Eᵒᵈ) h𝓕_cont h𝓕 | theorem | exists_countable_lowerSemicontinuous_isLUB | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Lindelof.lean | [] | [
"IsLUB",
"LowerSemicontinuous",
"Set",
"exists_countable_upperSemicontinuous_isGLB"
] | If a function `s : X → E` can be written as the supremum of a family `𝓕` of lower semicontinuous
functions then, assuming that `X` is hereditarily Lindelöf (for example, second countable),
`s` can in fact be written as the supremum of some *countable* subfamily `𝓕'`.
This is implication a) ⇒ b) in
[N. Bourbaki, *Top... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsKer_eq_of_t1Space [T1Space X] (s : Set X) : nhdsKer s = s | by
ext; simp [mem_nhdsKer_iff_specializes] | lemma | nhdsKer_eq_of_t1Space | Topology.Separation | Mathlib/Topology/Separation/AlexandrovDiscrete.lean | [] | [
"Set",
"T1Space",
"mem_nhdsKer_iff_specializes",
"nhdsKer"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
T0Space (X : Type u) [TopologicalSpace X] : Prop where
/-- Two inseparable points in a T₀ space are equal. -/
t0 : ∀ ⦃x y : X⦄, Inseparable x y → x = y | class | T0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"TopologicalSpace"
] | A T₀ space, also known as a Kolmogorov space, is a topological space such that for every pair
`x ≠ y`, there is an open set containing one but not the other. We formulate the definition in terms
of the `Inseparable` relation. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_inseparable (X : Type u) [TopologicalSpace X] :
T0Space X ↔ ∀ x y : X, Inseparable x y → x = y | ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩ | theorem | t0Space_iff_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"T0Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y | by
simp only [t0Space_iff_inseparable, Ne, not_imp_not, Pairwise] | theorem | t0Space_iff_not_inseparable | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"Pairwise",
"T0Space",
"TopologicalSpace",
"not_imp_not",
"t0Space_iff_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y | T0Space.t0 h | theorem | Inseparable.eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsInducing.injective [TopologicalSpace Y] [T0Space X] {f : X → Y}
(hf : IsInducing f) : Injective f | fun _ _ h =>
(hf.inseparable_iff.1 <| .of_eq h).eq | theorem | Topology.IsInducing.injective | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace",
"of_eq"
] | A topology inducing map from a T₀ space is injective. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Topology.IsInducing.isEmbedding [TopologicalSpace Y] [T0Space X] {f : X → Y}
(hf : IsInducing f) : IsEmbedding f | ⟨hf, hf.injective⟩ | theorem | Topology.IsInducing.isEmbedding | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace"
] | A topology inducing map from a T₀ space is a topological embedding. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isEmbedding_iff_isInducing [TopologicalSpace Y] [T0Space X] {f : X → Y} :
IsEmbedding f ↔ IsInducing f | ⟨IsEmbedding.isInducing, IsInducing.isEmbedding⟩ | lemma | isEmbedding_iff_isInducing | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_nhds_injective (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Injective (𝓝 : X → Filter X) | t0Space_iff_inseparable X | theorem | t0Space_iff_nhds_injective | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"T0Space",
"TopologicalSpace",
"t0Space_iff_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_injective [T0Space X] : Injective (𝓝 : X → Filter X) | (t0Space_iff_nhds_injective X).1 ‹_› | theorem | nhds_injective | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Filter",
"T0Space",
"t0Space_iff_nhds_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_iff_eq [T0Space X] {x y : X} : Inseparable x y ↔ x = y | nhds_injective.eq_iff | theorem | inseparable_iff_eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_nhds_iff [T0Space X] {a b : X} : 𝓝 a = 𝓝 b ↔ a = b | nhds_injective.eq_iff | theorem | nhds_eq_nhds_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_eq_eq [T0Space X] : Inseparable = @Eq X | funext₂ fun _ _ => propext inseparable_iff_eq | theorem | inseparable_eq_eq | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"T0Space",
"inseparable_iff_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)}
(hb : IsTopologicalBasis b) {x y : X} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) | ⟨fun h _ hs ↦ inseparable_iff_forall_isOpen.1 h _ (hb.isOpen hs),
fun h ↦ hb.nhds_hasBasis.eq_of_same_basis <| by
convert! hb.nhds_hasBasis using 2
exact and_congr_right (h _)⟩ | theorem | TopologicalSpace.IsTopologicalBasis.inseparable_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Inseparable",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space X] {b : Set (Set X)}
(hb : IsTopologicalBasis b) {x y : X} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) | inseparable_iff_eq.symm.trans hb.inseparable_iff | theorem | TopologicalSpace.IsTopologicalBasis.eq_iff | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Set",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_exists_isOpen_xor_mem (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ Xor (x ∈ U) (y ∈ U) | by
simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
inseparable_iff_forall_isOpen, Pairwise]
@[deprecated (since := "2026-04-04")]
alias t0Space_iff_exists_isOpen_xor'_mem := t0Space_iff_exists_isOpen_xor_mem | theorem | t0Space_iff_exists_isOpen_xor_mem | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Pairwise",
"Set",
"T0Space",
"TopologicalSpace",
"Xor",
"inseparable_iff_forall_isOpen",
"t0Space_iff_not_inseparable",
"xor_iff_not_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isOpen_xor_mem [T0Space X] {x y : X} (h : x ≠ y) :
∃ U : Set X, IsOpen U ∧ Xor (x ∈ U) (y ∈ U) | (t0Space_iff_exists_isOpen_xor_mem X).1 ‹_› h
@[deprecated (since := "2026-04-04")] alias exists_isOpen_xor'_mem := exists_isOpen_xor_mem | theorem | exists_isOpen_xor_mem | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Set",
"T0Space",
"Xor",
"t0Space_iff_exists_isOpen_xor_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
specializationOrder (X) [TopologicalSpace X] [T0Space X] : PartialOrder X | { specializationPreorder X, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with } | def | specializationOrder | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"OrderDual.toDual",
"PartialOrder",
"PartialOrder.lift",
"T0Space",
"TopologicalSpace",
"nhds_injective",
"specializationPreorder"
] | Specialization forms a partial order on a t0 topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
SeparationQuotient.instT0Space : T0Space (SeparationQuotient X) | ⟨fun x y => Quotient.inductionOn₂' x y fun _ _ h =>
SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.isInducing_mk.inseparable_iff.1 h⟩ | instance | SeparationQuotient.instT0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Quotient.inductionOn₂'",
"SeparationQuotient",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
minimal_nonempty_closed_subsingleton [T0Space X] {s : Set X} (hs : IsClosed s)
(hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : s.Subsingleton | by
refine fun x hx y hy => of_not_not fun hxy => ?_
rcases exists_isOpen_xor_mem hxy with ⟨U, hUo, hU⟩
wlog h : x ∈ U ∧ y ∉ U
· refine this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h)
obtain ⟨hxU, hyU⟩ := h
have : s \ U = s := hmin (s \ U) diff_subset ⟨y, hy, hyU⟩ (hs.sdiff hUo)
exac... | theorem | minimal_nonempty_closed_subsingleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T0Space",
"exists_isOpen_xor_mem",
"of_not_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
minimal_nonempty_closed_eq_singleton [T0Space X] {s : Set X} (hs : IsClosed s)
(hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} | exists_eq_singleton_iff_nonempty_subsingleton.2
⟨hne, minimal_nonempty_closed_subsingleton hs hmin⟩ | theorem | minimal_nonempty_closed_eq_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsClosed",
"Set",
"T0Space",
"minimal_nonempty_closed_subsingleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.exists_closed_singleton [T0Space X] [CompactSpace X] {S : Set X}
(hS : IsClosed S) (hne : S.Nonempty) : ∃ x : X, x ∈ S ∧ IsClosed ({x} : Set X) | by
obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne
rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with ⟨x, rfl⟩
exact ⟨x, Vsub (mem_singleton x), Vcls⟩ | theorem | IsClosed.exists_closed_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"CompactSpace",
"IsClosed",
"Set",
"T0Space",
"minimal_nonempty_closed_eq_singleton"
] | Given a closed set `S` in a compact T₀ space, there is some `x ∈ S` such that `{x}` is
closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
minimal_nonempty_open_subsingleton [T0Space X] {s : Set X} (hs : IsOpen s)
(hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : s.Subsingleton | by
refine fun x hx y hy => of_not_not fun hxy => ?_
rcases exists_isOpen_xor_mem hxy with ⟨U, hUo, hU⟩
wlog h : x ∈ U ∧ y ∉ U
· exact this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h)
obtain ⟨hxU, hyU⟩ := h
have : s ∩ U = s := hmin (s ∩ U) inter_subset_left ⟨x, hx, hxU⟩ (hs.inter hUo)
... | theorem | minimal_nonempty_open_subsingleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Set",
"T0Space",
"exists_isOpen_xor_mem",
"of_not_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
minimal_nonempty_open_eq_singleton [T0Space X] {s : Set X} (hs : IsOpen s)
(hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} | exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_open_subsingleton hs hmin⟩ | theorem | minimal_nonempty_open_eq_singleton | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"IsOpen",
"Set",
"T0Space",
"minimal_nonempty_open_subsingleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_isOpen_singleton_of_isOpen_finite [T0Space X] {s : Set X} (hfin : s.Finite)
(hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set X) | by
lift s to Finset X using hfin
induction s using Finset.strongInductionOn
rename_i s ihs
rcases em (∃ t, t ⊂ s ∧ t.Nonempty ∧ IsOpen (t : Set X)) with (⟨t, hts, htne, hto⟩ | ht)
· rcases ihs t hts htne hto with ⟨x, hxt, hxo⟩
exact ⟨x, hts.1 hxt, hxo⟩
· -- Porting note: was `rcases minimal_nonempty_ope... | theorem | exists_isOpen_singleton_of_isOpen_finite | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Finset",
"Finset.strongInductionOn",
"IsOpen",
"Set",
"T0Space",
"minimal_nonempty_open_eq_singleton",
"of_not_not"
] | Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_open_singleton_of_finite [T0Space X] [Finite X] [Nonempty X] :
∃ x : X, IsOpen ({x} : Set X) | let ⟨x, _, h⟩ := exists_isOpen_singleton_of_isOpen_finite (Set.toFinite _)
univ_nonempty isOpen_univ
⟨x, h⟩ | theorem | exists_open_singleton_of_finite | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Finite",
"IsOpen",
"Set",
"Set.toFinite",
"T0Space",
"exists_isOpen_singleton_of_isOpen_finite",
"isOpen_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
(hf : Function.Injective f) (hf' : Continuous f) [T0Space Y] : T0Space X | ⟨fun _ _ h => hf <| (h.map hf').eq⟩ | theorem | t0Space_of_injective_of_continuous | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Continuous",
"T0Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsEmbedding.t0Space [TopologicalSpace Y] [T0Space Y] {f : X → Y}
(hf : IsEmbedding f) : T0Space X | t0Space_of_injective_of_continuous hf.injective hf.continuous | theorem | Topology.IsEmbedding.t0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace",
"t0Space_of_injective_of_continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Homeomorph.t0Space [TopologicalSpace Y] [T0Space X] (h : X ≃ₜ Y) : T0Space Y | h.symm.isEmbedding.t0Space | theorem | Homeomorph.t0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Subtype.t0Space [T0Space X] {p : X → Prop} : T0Space (Subtype p) | IsEmbedding.subtypeVal.t0Space | instance | Subtype.t0Space | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_or_notMem_closure (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun a b : X => a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) | by
simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or] | theorem | t0Space_iff_or_notMem_closure | Topology.Separation | Mathlib/Topology/Separation/Basic.lean | [] | [
"Pairwise",
"Set",
"T0Space",
"TopologicalSpace",
"closure",
"inseparable_iff_mem_closure",
"not_and_or",
"t0Space_iff_not_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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