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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of_isCompact_of_forall_exists_isCompactOpenCovered [TopologicalSpace S] {U : Set S}
(hU : IsCompact U) (H : ∀ x ∈ U, ∃ t ⊆ U, x ∈ t ∧ IsOpen t ∧ IsCompactOpenCovered f t) :
IsCompactOpenCovered f U | by
choose Us hU' hUx hUo hU'' using H
refine of_biUnion_eq_of_isCompact hU { Us x h | (x : S) (h : x ∈ U) } ?_ ?_
· refine subset_antisymm (fun x ↦ ?_) fun x hx ↦ ?_
· simp [Opens.forall]
grind
· simpa using ⟨⟨Us x hx, hUo _ _⟩, ⟨x, by simpa⟩, hUx _ _⟩
· grind | lemma | IsCompactOpenCovered.of_isCompact_of_forall_exists_isCompactOpenCovered | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompact",
"IsCompactOpenCovered",
"IsOpen",
"Set",
"TopologicalSpace",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image {i : ι} (V : Opens (X i)) (hV : IsCompact (X := X i) V) :
IsCompactOpenCovered f (f i '' V) | by
refine ⟨{i}, Set.finite_singleton i, fun j hj ↦ hj ▸ V, by rintro i rfl; simpa, by simp⟩ | lemma | IsCompactOpenCovered.image | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"IsCompact",
"IsCompactOpenCovered",
"Set.finite_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_finite {U : Set S} {κ : Type*} [Finite κ] (a : κ → ι) (V : ∀ k, Opens (X (a k)))
(hV : ∀ k, IsCompact (V k).1) (hU : ⋃ k, f (a k) '' V k = U) :
IsCompactOpenCovered f U | of_iUnion_eq_of_finite _ hU (fun _ ↦ .image _ (hV _)) | lemma | IsCompactOpenCovered.of_finite | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"Finite",
"IsCompact",
"IsCompactOpenCovered",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_mem_of_isBasis {B : ∀ i, Set (Opens (X i))} (hB : ∀ i, IsBasis (B i))
(hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.1)
{U : Set S} (hU : IsCompactOpenCovered f U) :
∃ (n : ℕ) (a : Fin n → ι) (V : ∀ i, Opens (X (a i))),
(∀ i, V i ∈ B (a i)) ∧ ⋃ i, f (a i) '' V i = U | by
suffices h : ∃ (κ : Type _) (_ : Finite κ) (a : κ → ι) (V : ∀ i, Opens (X (a i))),
(∀ i, V i ∈ B (a i)) ∧ (∀ i, IsCompact (V i).1) ∧ ⋃ i, f (a i) '' V i = U by
obtain ⟨κ, _, a, V, hB, hc, hU⟩ := h
cases nonempty_fintype κ
refine ⟨Fintype.card κ, a ∘ (Fintype.equivFin κ).symm, fun i ↦ V _, fun i ↦... | lemma | IsCompactOpenCovered.exists_mem_of_isBasis | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"Finite",
"Fintype.equivFin",
"IsCompact",
"IsCompactOpenCovered",
"Set",
"Set.mem_iUnion",
"SetLike.mem_coe",
"nonempty_fintype",
"symm"
] | If `U` is compact-open covered and the `X i` have a basis of compact opens,
`U` can be written as the union of images of elements of the basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
of_finite_of_isSpectralMap [Finite ι] [TopologicalSpace S]
(hf : ∀ i, IsSpectralMap (f i)) {U : Set S} (hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i))
(hU : IsOpen U) (hc : IsCompact U) :
IsCompactOpenCovered f U | by
refine ⟨.univ, Set.finite_univ, fun i _ ↦ ⟨f i ⁻¹' U, hU.preimage (hf i).1⟩,
fun i _ ↦ hc.preimage_of_isOpen (hf i) hU, subset_antisymm (by simp) fun x hx ↦ ?_⟩
obtain ⟨i, y, rfl⟩ := hs x hx
simpa using ⟨i, y, hx, rfl⟩ | lemma | IsCompactOpenCovered.of_finite_of_isSpectralMap | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"Finite",
"IsCompact",
"IsCompactOpenCovered",
"IsOpen",
"IsSpectralMap",
"Set",
"Set.finite_univ",
"Set.range",
"TopologicalSpace",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_isOpenMap [TopologicalSpace S] [∀ i, PrespectralSpace (X i)]
(hfc : ∀ i, Continuous (f i)) (h : ∀ i, IsOpenMap (f i))
{U : Set S} (hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)) (hU : IsOpen U) (hc : IsCompact U) :
IsCompactOpenCovered f U | by
rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
refine (isOpenMap_sigma.mpr h).exists_opens_image_eq_of_prespectralSpace
(continuous_sigma_iff.mpr hfc) (fun x hx ↦ ?_) hU hc
simpa using hs x hx | lemma | IsCompactOpenCovered.of_isOpenMap | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"Continuous",
"IsCompact",
"IsCompactOpenCovered",
"IsOpen",
"IsOpenMap",
"PrespectralSpace",
"Set",
"Set.range",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_comp [∀ i, PrespectralSpace (X i)] [TopologicalSpace S]
{σ : Type*} {Y : σ → Type*} [∀ i, TopologicalSpace (Y i)]
(g : ∀ i, Y i → S) {a : σ → ι} (t : ∀ i, Y i → X (a i)) (ht : ∀ i, Continuous (t i))
(hge : ∀ i, g i = f (a i) ∘ t i)
(hf : ∀ i, Continuous (f i)) {U : Set S} (ho : IsOpen U) (hU : IsComp... | by
rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique] at hU ⊢
let p : (Σ i, Y i) → (Σ i, X i) := Sigma.map a t
have hcomp : (fun x ↦ f x.1 x.2) ∘ p = fun x ↦ g x.1 x.2 := by
ext
simp [hge, p, Sigma.map]
have hp : Continuous p := Continuous.sigma_map ht
have hf : Continuous (fun p : Σ i, X i ↦ f p.1 ... | lemma | IsCompactOpenCovered.of_comp | Topology.Sets | Mathlib/Topology/Sets/CompactOpenCovered.lean | [] | [
"Continuous",
"Continuous.sigma_map",
"IsCompactOpenCovered",
"IsOpen",
"PrespectralSpace",
"PrespectralSpace.exists_isCompact_and_isOpen_between",
"Set",
"Set.image_comp",
"Set.image_mono",
"Set.preimage_comp",
"Set.subset_preimage_image",
"Sigma.map",
"TopologicalSpace",
"subset_antisymm... | Being compact open covered descends along refinements if the spaces are prespectral. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Compacts (α : Type*) [TopologicalSpace α] where
/-- the carrier set, i.e. the points in this set -/
carrier : Set α
isCompact' : IsCompact carrier | structure | TopologicalSpace.Compacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsCompact",
"Set",
"TopologicalSpace"
] | The type of compact sets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : Compacts α) : Set α | s
initialize_simps_projections Compacts (carrier → coe, as_prefix coe) | def | TopologicalSpace.Compacts.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact (s : Compacts α) : IsCompact (s : Set α) | s.isCompact' | theorem | TopologicalSpace.Compacts.isCompact | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCloseds [T2Space α] (s : Compacts α) : Closeds α | ⟨s, s.isCompact.isClosed⟩ | def | TopologicalSpace.Compacts.toCloseds | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | Reinterpret a compact as a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_toCloseds [T2Space α] {x : α} {s : Compacts α} :
x ∈ s.toCloseds ↔ x ∈ s | Iff.rfl | theorem | TopologicalSpace.Compacts.mem_toCloseds | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCloseds_injective [T2Space α] : Function.Injective (toCloseds (α := α)) | .of_comp (f := SetLike.coe) SetLike.coe_injective | theorem | TopologicalSpace.Compacts.toCloseds_injective | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"SetLike.coe_injective",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {s t : Compacts α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.Compacts.ext | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
carrier_eq_coe (s : Compacts α) : s.carrier = s | rfl | theorem | TopologicalSpace.Compacts.carrier_eq_coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | rfl | theorem | TopologicalSpace.Compacts.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t | rfl | theorem | TopologicalSpace.Compacts.coe_inf | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top [CompactSpace α] : (↑(⊤ : Compacts α) : Set α) = univ | rfl | theorem | TopologicalSpace.Compacts.coe_top | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"CompactSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_bot : (↑(⊥ : Compacts α) : Set α) = ∅ | rfl | theorem | TopologicalSpace.Compacts.coe_bot | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_eq_empty {s : Compacts α} : (s : Set α) = ∅ ↔ s = ⊥ | SetLike.coe_injective.eq_iff' rfl | theorem | TopologicalSpace.Compacts.coe_eq_empty | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_nonempty {s : Compacts α} : (s : Set α).Nonempty ↔ s ≠ ⊥ | nonempty_iff_ne_empty.trans coe_eq_empty.not | theorem | TopologicalSpace.Compacts.coe_nonempty | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :
(↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) | by
refine Finset.cons_induction_on s rfl fun a s _ h => ?_
simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]
congr | theorem | TopologicalSpace.Compacts.coe_finset_sup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Finset",
"Finset.cons_induction_on",
"Finset.sup_cons",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_singleton (x y : α) : x ∈ ({y} : Compacts α) ↔ x = y | Iff.rfl | theorem | TopologicalSpace.Compacts.mem_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCloseds_singleton [T2Space α] (x : α) : toCloseds {x} = {x} | rfl | theorem | TopologicalSpace.Compacts.toCloseds_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_injective : Function.Injective ({·} : α → Compacts α) | .of_comp (f := SetLike.coe) Set.singleton_injective | theorem | TopologicalSpace.Compacts.singleton_injective | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.singleton_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_inj {x y : α} : ({x} : Compacts α) = {y} ↔ x = y | singleton_injective.eq_iff | theorem | TopologicalSpace.Compacts.singleton_inj | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subsingleton_iff : Subsingleton (Compacts α) ↔ IsEmpty α | by
refine ⟨fun h => ?_, fun _ => inferInstance⟩
contrapose! h
infer_instance | theorem | TopologicalSpace.Compacts.subsingleton_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsEmpty",
"subsingleton_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nontrivial_iff : Nontrivial (Compacts α) ↔ Nonempty α | by
rw [← not_subsingleton_iff_nontrivial, subsingleton_iff, not_isEmpty_iff] | theorem | TopologicalSpace.Compacts.nontrivial_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Nontrivial",
"nontrivial_iff",
"not_isEmpty_iff",
"not_subsingleton_iff_nontrivial",
"subsingleton_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α → β) (hf : Continuous f) (K : Compacts α) : Compacts β | ⟨f '' K.1, K.2.image hf⟩ | def | TopologicalSpace.Compacts.map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous"
] | The image of a compact set under a continuous function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_map {f : α → β} (hf : Continuous f) (s : Compacts α) : (s.map f hf : Set β) = f '' s | rfl | theorem | TopologicalSpace.Compacts.coe_map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_id (K : Compacts α) : K.map id continuous_id = K | Compacts.ext <| Set.image_id _ | theorem | TopologicalSpace.Compacts.map_id | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.image_id",
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (K : Compacts α) :
K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf | Compacts.ext <| Set.image_comp _ _ _ | theorem | TopologicalSpace.Compacts.map_comp | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"Set.image_comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_injective {f : α → β} (hf : Continuous f) (hf' : Function.Injective f) :
Function.Injective (Compacts.map f hf) | .of_comp (f := SetLike.coe) <| hf'.image_injective.comp SetLike.coe_injective | theorem | TopologicalSpace.Compacts.map_injective | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_singleton {f : α → β} (hf : Continuous f) (x : α) : Compacts.map f hf {x} = {f x} | Compacts.ext Set.image_singleton | theorem | TopologicalSpace.Compacts.map_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"Set.image_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_injective_iff {f : α → β} (hf : Continuous f) :
Function.Injective (Compacts.map f hf) ↔ Function.Injective f | by
refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩
simp_rw [Function.comp_def, ← map_singleton hf]
exact h.comp singleton_injective | theorem | TopologicalSpace.Compacts.map_injective_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_map {f : α → β} (hf : Topology.IsInducing f) :
range (Compacts.map f hf.continuous) = {K : Compacts β | ↑K ⊆ range f} | subset_antisymm
(range_subset_iff.mpr fun _ => image_subset_range _ _)
(fun L hL => ⟨
{ carrier := f ⁻¹' L
isCompact' := hf.isCompact_preimage' L.isCompact hL },
Compacts.ext (image_preimage_eq_of_subset hL)⟩) | theorem | TopologicalSpace.Compacts.range_map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Topology.IsInducing",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equiv (f : α ≃ₜ β) : Compacts α ≃ Compacts β | where
toFun := Compacts.map f f.continuous
invFun := Compacts.map _ f.symm.continuous
left_inv s := by
ext1
simp only [coe_map, ← image_comp, f.symm_comp_self, image_id]
right_inv s := by
ext1
simp only [coe_map, ← image_comp, f.self_comp_symm, image_id] | def | TopologicalSpace.Compacts.equiv | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | A homeomorphism induces an equivalence on compact sets, by taking the image. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equiv_refl : Compacts.equiv (Homeomorph.refl α) = Equiv.refl _ | Equiv.ext map_id | theorem | TopologicalSpace.Compacts.equiv_refl | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Equiv.ext",
"Equiv.refl",
"Homeomorph.refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equiv_trans (f : α ≃ₜ β) (g : β ≃ₜ γ) :
Compacts.equiv (f.trans g) = (Compacts.equiv f).trans (Compacts.equiv g) | Equiv.ext <| map_comp g f g.continuous f.continuous | theorem | TopologicalSpace.Compacts.equiv_trans | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Equiv.ext",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equiv_symm (f : α ≃ₜ β) : Compacts.equiv f.symm = (Compacts.equiv f).symm | rfl | theorem | TopologicalSpace.Compacts.equiv_symm | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_equiv_apply_eq_preimage (f : α ≃ₜ β) (K : Compacts α) :
(Compacts.equiv f K : Set β) = f.symm ⁻¹' (K : Set α) | f.toEquiv.image_eq_preimage_symm K | theorem | TopologicalSpace.Compacts.coe_equiv_apply_eq_preimage | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | The image of a compact set under a homeomorphism can also be expressed as a preimage. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_prod (K : Compacts α) (L : Compacts β) :
(K ×ˢ L : Compacts (α × β)) = (K : Set α) ×ˢ (L : Set β) | rfl | theorem | TopologicalSpace.Compacts.coe_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCloseds_prod [T2Space α] [T2Space β] (K : Compacts α) (L : Compacts β) :
(K ×ˢ L).toCloseds = K.toCloseds ×ˢ L.toCloseds | by
rfl | theorem | TopologicalSpace.Compacts.toCloseds_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_prod_singleton (x : α) (y : β) :
({x} ×ˢ {y} : Compacts (α × β)) = {(x, y)} | Compacts.ext Set.singleton_prod_singleton | theorem | TopologicalSpace.Compacts.singleton_prod_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.singleton_prod_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
NonemptyCompacts (α : Type*) [TopologicalSpace α] extends Compacts α where
nonempty' : carrier.Nonempty | structure | TopologicalSpace.NonemptyCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"TopologicalSpace"
] | The type of nonempty compact sets of a topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : NonemptyCompacts α) : Set α | s
initialize_simps_projections NonemptyCompacts (carrier → coe, as_prefix coe, as_prefix toCompacts) | def | TopologicalSpace.NonemptyCompacts.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact (s : NonemptyCompacts α) : IsCompact (s : Set α) | s.isCompact' | theorem | TopologicalSpace.NonemptyCompacts.isCompact | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty (s : NonemptyCompacts α) : (s : Set α).Nonempty | s.nonempty' | theorem | TopologicalSpace.NonemptyCompacts.nonempty | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCloseds [T2Space α] (s : NonemptyCompacts α) : Closeds α | ⟨s, s.isCompact.isClosed⟩ | def | TopologicalSpace.NonemptyCompacts.toCloseds | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | Reinterpret a nonempty compact as a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toCloseds_toCompacts [T2Space α] (s : NonemptyCompacts α) :
s.toCompacts.toCloseds = s.toCloseds | rfl | theorem | TopologicalSpace.NonemptyCompacts.toCloseds_toCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_toCloseds [T2Space α] {x : α} {s : NonemptyCompacts α} :
x ∈ s.toCloseds ↔ x ∈ s | Iff.rfl | theorem | TopologicalSpace.NonemptyCompacts.mem_toCloseds | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {s t : NonemptyCompacts α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.NonemptyCompacts.ext | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk (s : Compacts α) (h) : (mk s h : Set α) = s | rfl | theorem | TopologicalSpace.NonemptyCompacts.coe_mk | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
carrier_eq_coe (s : NonemptyCompacts α) : s.carrier = s | rfl | theorem | TopologicalSpace.NonemptyCompacts.carrier_eq_coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_toCompacts (s : NonemptyCompacts α) : (s.toCompacts : Set α) = s | rfl | theorem | TopologicalSpace.NonemptyCompacts.coe_toCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_toCompacts {x : α} {s : NonemptyCompacts α} :
x ∈ s.toCompacts ↔ x ∈ s | Iff.rfl | theorem | TopologicalSpace.NonemptyCompacts.mem_toCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCompacts_injective : Function.Injective (toCompacts (α := α)) | .of_comp (f := SetLike.coe) SetLike.coe_injective | theorem | TopologicalSpace.NonemptyCompacts.toCompacts_injective | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_toCompacts : range (toCompacts (α := α)) = {⊥}ᶜ | by
ext K
rw [mem_compl_singleton_iff, ← Compacts.coe_nonempty]
refine ⟨?_, fun h => ⟨⟨K, h⟩, rfl⟩⟩
rintro ⟨K, rfl⟩
exact K.nonempty | theorem | TopologicalSpace.NonemptyCompacts.range_toCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : NonemptyCompacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | rfl | theorem | TopologicalSpace.NonemptyCompacts.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCompacts_sup (s t : NonemptyCompacts α) :
(s ⊔ t).toCompacts = s.toCompacts ⊔ t.toCompacts | rfl | theorem | TopologicalSpace.NonemptyCompacts.toCompacts_sup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top [CompactSpace α] [Nonempty α] : (↑(⊤ : NonemptyCompacts α) : Set α) = univ | rfl | theorem | TopologicalSpace.NonemptyCompacts.coe_top | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"CompactSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_singleton (x y : α) : x ∈ ({y} : NonemptyCompacts α) ↔ x = y | Iff.rfl | theorem | TopologicalSpace.NonemptyCompacts.mem_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_injective : Function.Injective ({·} : α → NonemptyCompacts α) | .of_comp (f := SetLike.coe) Set.singleton_injective | theorem | TopologicalSpace.NonemptyCompacts.singleton_injective | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.singleton_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_inj {x y : α} : ({x} : NonemptyCompacts α) = {y} ↔ x = y | singleton_injective.eq_iff | theorem | TopologicalSpace.NonemptyCompacts.singleton_inj | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmpty_iff : IsEmpty (NonemptyCompacts α) ↔ IsEmpty α | ⟨fun _ => Function.isEmpty ({·} : α → NonemptyCompacts α), fun _ => inferInstance⟩ | theorem | TopologicalSpace.NonemptyCompacts.isEmpty_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Function.isEmpty",
"IsEmpty",
"isEmpty_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_iff : Nonempty (NonemptyCompacts α) ↔ Nonempty α | by
simp_rw [← not_isEmpty_iff, isEmpty_iff] | theorem | TopologicalSpace.NonemptyCompacts.nonempty_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"isEmpty_iff",
"not_isEmpty_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subsingleton_iff : Subsingleton (NonemptyCompacts α) ↔ Subsingleton α | ⟨fun _ => singleton_injective.subsingleton, fun _ => inferInstance⟩ | theorem | TopologicalSpace.NonemptyCompacts.subsingleton_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"subsingleton_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nontrivial_iff : Nontrivial (NonemptyCompacts α) ↔ Nontrivial α | by
simp_rw [← not_subsingleton_iff_nontrivial, subsingleton_iff] | theorem | TopologicalSpace.NonemptyCompacts.nontrivial_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Nontrivial",
"nontrivial_iff",
"not_subsingleton_iff_nontrivial",
"subsingleton_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α → β) (hf : Continuous f) (K : NonemptyCompacts α) : NonemptyCompacts β | ⟨K.toCompacts.map f hf, K.nonempty.image f⟩ | def | TopologicalSpace.NonemptyCompacts.map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous"
] | The image of a nonempty compact set under a continuous function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_map {f : α → β} (hf : Continuous f) (s : NonemptyCompacts α) :
(s.map f hf : Set β) = f '' s | rfl | theorem | TopologicalSpace.NonemptyCompacts.coe_map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_id (K : NonemptyCompacts α) : K.map id continuous_id = K | by
ext
simp | theorem | TopologicalSpace.NonemptyCompacts.map_id | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g)
(K : NonemptyCompacts α) : K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf | by
ext
simp | theorem | TopologicalSpace.NonemptyCompacts.map_comp | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_singleton {f : α → β} (hf : Continuous f) (x : α) :
NonemptyCompacts.map f hf {x} = {f x} | by
ext
simp | theorem | TopologicalSpace.NonemptyCompacts.map_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_injective {f : α → β} (hf : Continuous f) (hf' : Function.Injective f) :
Function.Injective (NonemptyCompacts.map f hf) | .of_comp (f := SetLike.coe) <| hf'.image_injective.comp SetLike.coe_injective | theorem | TopologicalSpace.NonemptyCompacts.map_injective | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_injective_iff {f : α → β} (hf : Continuous f) :
Function.Injective (NonemptyCompacts.map f hf) ↔ Function.Injective f | ⟨fun h => .of_comp (f := ({·} : β → NonemptyCompacts β)) fun _ _ _ ↦
singleton_injective (h (by simp_all)), map_injective hf⟩ | theorem | TopologicalSpace.NonemptyCompacts.map_injective_iff | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_map {f : α → β} (hf : Topology.IsInducing f) :
range (NonemptyCompacts.map f hf.continuous) = {K : NonemptyCompacts β | ↑K ⊆ range f} | subset_antisymm
(range_subset_iff.mpr fun _ => image_subset_range _ _)
(fun L hL => ⟨
{ carrier := f ⁻¹' L
isCompact' := hf.isCompact_preimage' L.isCompact hL
nonempty' := L.nonempty.preimage' hL },
NonemptyCompacts.ext (image_preimage_eq_of_subset hL)⟩) | theorem | TopologicalSpace.NonemptyCompacts.range_map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Topology.IsInducing",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCompactSpace {s : NonemptyCompacts α} : CompactSpace s | isCompact_iff_compactSpace.1 s.isCompact | instance | TopologicalSpace.NonemptyCompacts.toCompactSpace | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"CompactSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toNonempty {s : NonemptyCompacts α} : Nonempty s | s.nonempty.to_subtype | instance | TopologicalSpace.NonemptyCompacts.toNonempty | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_prod (K : NonemptyCompacts α) (L : NonemptyCompacts β) :
(K ×ˢ L : NonemptyCompacts (α × β)) = (K : Set α) ×ˢ (L : Set β) | rfl | theorem | TopologicalSpace.NonemptyCompacts.coe_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCompacts_prod (K : NonemptyCompacts α) (L : NonemptyCompacts β) :
(K ×ˢ L).toCompacts = K.toCompacts ×ˢ L.toCompacts | rfl | theorem | TopologicalSpace.NonemptyCompacts.toCompacts_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toCloseds_prod [T2Space α] [T2Space β] (K : NonemptyCompacts α) (L : NonemptyCompacts β) :
(K ×ˢ L).toCloseds = K.toCloseds ×ˢ L.toCloseds | by
rfl | theorem | TopologicalSpace.NonemptyCompacts.toCloseds_prod | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_prod_singleton (x : α) (y : β) :
({x} ×ˢ {y} : NonemptyCompacts (α × β)) = {(x, y)} | NonemptyCompacts.ext Set.singleton_prod_singleton | theorem | TopologicalSpace.NonemptyCompacts.singleton_prod_singleton | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.singleton_prod_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PositiveCompacts (α : Type*) [TopologicalSpace α] extends Compacts α where
interior_nonempty' : (interior carrier).Nonempty | structure | TopologicalSpace.PositiveCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"TopologicalSpace",
"interior"
] | The type of compact sets with nonempty interior of a topological space.
See also `TopologicalSpace.Compacts` and `TopologicalSpace.NonemptyCompacts`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Simps.coe (s : PositiveCompacts α) : Set α | s
initialize_simps_projections PositiveCompacts (carrier → coe, as_prefix coe, as_prefix toCompacts) | def | TopologicalSpace.PositiveCompacts.Simps.coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isCompact (s : PositiveCompacts α) : IsCompact (s : Set α) | s.isCompact' | theorem | TopologicalSpace.PositiveCompacts.isCompact | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_nonempty (s : PositiveCompacts α) : (interior (s : Set α)).Nonempty | s.interior_nonempty' | theorem | TopologicalSpace.PositiveCompacts.interior_nonempty | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty (s : PositiveCompacts α) : (s : Set α).Nonempty | s.interior_nonempty.mono interior_subset | theorem | TopologicalSpace.PositiveCompacts.nonempty | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"interior_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toNonemptyCompacts (s : PositiveCompacts α) : NonemptyCompacts α | ⟨s.toCompacts, s.nonempty⟩ | def | TopologicalSpace.PositiveCompacts.toNonemptyCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | Reinterpret a positive compact as a nonempty compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ext {s t : PositiveCompacts α} (h : (s : Set α) = t) : s = t | SetLike.ext' h | theorem | TopologicalSpace.PositiveCompacts.ext | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set",
"SetLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
carrier_eq_coe (s : PositiveCompacts α) : s.carrier = s | rfl | theorem | TopologicalSpace.PositiveCompacts.carrier_eq_coe | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_toCompacts (s : PositiveCompacts α) : (s.toCompacts : Set α) = s | rfl | theorem | TopologicalSpace.PositiveCompacts.coe_toCompacts | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sup (s t : PositiveCompacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | rfl | theorem | TopologicalSpace.PositiveCompacts.coe_sup | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_top [CompactSpace α] [Nonempty α] : (↑(⊤ : PositiveCompacts α) : Set α) = univ | rfl | theorem | TopologicalSpace.PositiveCompacts.coe_top | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"CompactSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (K : PositiveCompacts α) :
PositiveCompacts β | { Compacts.map f hf K.toCompacts with
interior_nonempty' :=
(K.interior_nonempty'.image _).mono (hf'.image_interior_subset K.toCompacts) } | def | TopologicalSpace.PositiveCompacts.map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"IsOpenMap"
] | The image of a positive compact set under a continuous open map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_map {f : α → β} (hf : Continuous f) (hf' : IsOpenMap f) (s : PositiveCompacts α) :
(s.map f hf hf' : Set β) = f '' s | rfl | theorem | TopologicalSpace.PositiveCompacts.coe_map | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"IsOpenMap",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_id (K : PositiveCompacts α) : K.map id continuous_id IsOpenMap.id = K | PositiveCompacts.ext <| Set.image_id _ | theorem | TopologicalSpace.PositiveCompacts.map_id | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Set.image_id",
"continuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f)
(hg' : IsOpenMap g) (K : PositiveCompacts α) :
K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' | PositiveCompacts.ext <| Set.image_comp _ _ _ | theorem | TopologicalSpace.PositiveCompacts.map_comp | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"Continuous",
"IsOpenMap",
"Set.image_comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.exists_positiveCompacts_subset [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U)
(hn : U.Nonempty) : ∃ K : PositiveCompacts α, ↑K ⊆ U | let ⟨x, hx⟩ := hn
let ⟨K, hKc, hxK, hKU⟩ := exists_compact_subset ho hx
⟨⟨⟨K, hKc⟩, ⟨x, hxK⟩⟩, hKU⟩ | theorem | exists_positiveCompacts_subset | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsOpen",
"LocallyCompactSpace",
"Set",
"exists_compact_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsOpen.exists_positiveCompacts_closure_subset [R1Space α] [LocallyCompactSpace α]
{U : Set α} (ho : IsOpen U) (hn : U.Nonempty) : ∃ K : PositiveCompacts α, closure ↑K ⊆ U | let ⟨K, hKU⟩ := exists_positiveCompacts_subset ho hn
⟨K, K.isCompact.closure_subset_of_isOpen ho hKU⟩ | theorem | IsOpen.exists_positiveCompacts_closure_subset | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"IsOpen",
"LocallyCompactSpace",
"R1Space",
"Set",
"closure",
"exists_positiveCompacts_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty' [WeaklyLocallyCompactSpace α] [Nonempty α] : Nonempty (PositiveCompacts α) | by
inhabit α
rcases exists_compact_mem_nhds (default : α) with ⟨K, hKc, hK⟩
exact ⟨⟨K, hKc⟩, _, mem_interior_iff_mem_nhds.2 hK⟩ | instance | TopologicalSpace.PositiveCompacts.nonempty' | Topology.Sets | Mathlib/Topology/Sets/Compacts.lean | [] | [
"WeaklyLocallyCompactSpace"
] | In a nonempty locally compact space, there exists a compact set with nonempty interior. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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