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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setSeq_mono ⦃m n : ℕ⦄ (h : m ≤ n) : setSeq hf U_mem n ⊆ setSeq hf U_mem m | biInter_subset_biInter_left <| Iic_subset_Iic.2 h | theorem | SequentiallyComplete.setSeq_mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
setSeq_sub_aux (n : ℕ) : setSeq hf U_mem n ⊆ setSeqAux hf U_mem n | biInter_subset_of_mem self_mem_Iic | theorem | SequentiallyComplete.setSeq_sub_aux | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
setSeq_prod_subset {N m n} (hm : N ≤ m) (hn : N ≤ n) :
setSeq hf U_mem m ×ˢ setSeq hf U_mem n ⊆ U N | fun p hp => by
refine (setSeqAux hf U_mem N).2.2 ⟨?_, ?_⟩ <;> apply setSeq_sub_aux
· exact setSeq_mono hf U_mem hm hp.1
· exact setSeq_mono hf U_mem hn hp.2 | theorem | SequentiallyComplete.setSeq_prod_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
seq (n : ℕ) : α | (hf.1.nonempty_of_mem (setSeq_mem hf U_mem n)).choose | def | SequentiallyComplete.seq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | A sequence of points such that `seq n ∈ setSeq n`. Here `setSeq` is an antitone
sequence of sets `setSeq n ∈ f` with diameters controlled by a given sequence
of entourages. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
seq_mem (n : ℕ) : seq hf U_mem n ∈ setSeq hf U_mem n | (hf.1.nonempty_of_mem (setSeq_mem hf U_mem n)).choose_spec | theorem | SequentiallyComplete.seq_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
seq_pair_mem ⦃N m n : ℕ⦄ (hm : N ≤ m) (hn : N ≤ n) :
(seq hf U_mem m, seq hf U_mem n) ∈ U N | setSeq_prod_subset hf U_mem hm hn ⟨seq_mem hf U_mem m, seq_mem hf U_mem n⟩ | theorem | SequentiallyComplete.seq_pair_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
seq_is_cauchySeq (U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) : CauchySeq <| seq hf U_mem | cauchySeq_of_controlled U U_le <| seq_pair_mem hf U_mem | theorem | SequentiallyComplete.seq_is_cauchySeq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"cauchySeq_of_controlled"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_nhds_of_seq_tendsto_nhds (U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s)
⦃a : α⦄ (ha : Tendsto (seq hf U_mem) atTop (𝓝 a)) : f ≤ 𝓝 a | le_nhds_of_cauchy_adhp_aux
(fun s hs => by
rcases U_le s hs with ⟨m, hm⟩
rcases tendsto_atTop'.1 ha _ (mem_nhds_left a (U_mem m)) with ⟨n, hn⟩
refine
⟨setSeq hf U_mem (max m n), setSeq_mem hf U_mem _, ?_, seq hf U_mem (max m n), ?_,
seq_mem hf U_mem _⟩
· have := le_max_left... | theorem | SequentiallyComplete.le_nhds_of_seq_tendsto_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set.Subset.trans",
"le_nhds_of_cauchy_adhp_aux",
"mem_nhds_left"
] | If the sequence `SequentiallyComplete.seq` converges to `a`, then `f ≤ 𝓝 a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
complete_of_convergent_controlled_sequences (U : ℕ → SetRel α α) (U_mem : ∀ n, U n ∈ 𝓤 α)
(HU : ∀ u : ℕ → α, (∀ N m n, N ≤ m → N ≤ n → (u m, u n) ∈ U N) → ∃ a, Tendsto u atTop (𝓝 a)) :
CompleteSpace α | by
obtain ⟨U', -, hU'⟩ := (𝓤 α).exists_antitone_seq
have Hmem : ∀ n, U n ∩ U' n ∈ 𝓤 α := fun n => inter_mem (U_mem n) (hU'.2 ⟨n, Subset.refl _⟩)
refine ⟨fun hf => (HU (seq hf Hmem) fun N m n hm hn => ?_).imp <|
le_nhds_of_seq_tendsto_nhds _ _ fun s hs => ?_⟩
· exact inter_subset_left (seq_pair_mem hf Hmem... | theorem | UniformSpace.complete_of_convergent_controlled_sequences | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CompleteSpace",
"SetRel"
] | A uniform space is complete provided that (a) its uniformity filter has a countable basis;
(b) any sequence satisfying a "controlled" version of the Cauchy condition converges. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
complete_of_cauchySeq_tendsto (H' : ∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) :
CompleteSpace α | let ⟨U', _, hU'⟩ := (𝓤 α).exists_antitone_seq
complete_of_convergent_controlled_sequences U' (fun n => hU'.2 ⟨n, Subset.refl _⟩) fun u hu =>
H' u <| cauchySeq_of_controlled U' (fun _ hs => hU'.1 hs) hu | theorem | UniformSpace.complete_of_cauchySeq_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"CauchySeq",
"CompleteSpace",
"cauchySeq_of_controlled"
] | A sequentially complete uniform space with a countable basis of the uniformity filter is
complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
secondCountable_of_separable [SeparableSpace α] : SecondCountableTopology α | by
rcases exists_countable_dense α with ⟨s, hsc, hsd⟩
obtain
⟨t : ℕ → SetRel α α, hto : ∀ i : ℕ, t i ∈ (𝓤 α).sets ∧ IsOpen (t i) ∧ (t i).IsSymm,
h_basis : (𝓤 α).HasAntitoneBasis t⟩ :=
(@uniformity_hasBasis_open_symmetric α _).exists_antitone_subbasis
choose ht_mem hto hts using hto
refine ⟨⟨⋃ x ... | instance | UniformSpace.secondCountable_of_separable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"IsOpen",
"IsOpen.mem_nhds",
"IsSymm",
"SecondCountableTopology",
"SetRel",
"UniformSpace.mem_ball_self",
"comp_symm_of_uniformity",
"symm",
"uniformity_hasBasis_open_symmetric"
] | A separable uniform space with countably generated uniformity filter is second countable:
one obtains a countable basis by taking the balls centered at points in a dense subset,
and with rational "radii" from a countable open symmetric antitone basis of `𝓤 α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
subset_countable_closure_of_almost_dense_set (s : Set α)
(hs : ∀ U ∈ 𝓤 α, ∃ t : Set α, t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U) :
∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t | by
obtain ⟨B, hB, _⟩ := has_seq_basis α
replace hs (n : ℕ) := hs (B n) (hB.mem n)
choose t tC ht using hs
have := fun n => (tC n).to_subtype
choose o hox hos using fun (n : ℕ) (x : t n) (hx : (ball x.1 (B n) ∩ s).Nonempty) => hx
refine ⟨⋃ (n) (x), range (o n x), iUnion₂_subset fun _ _ => range_subset_iff.2 ... | theorem | UniformSpace.subset_countable_closure_of_almost_dense_set | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"SetRel.comp_subset_comp",
"closure",
"comp_mem_uniformity_sets",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
secondCountable_of_almost_dense_set
(hs : ∀ U ∈ 𝓤 α, ∃ t : Set α, t.Countable ∧ ⋃ x ∈ t, ball x U = univ) :
SecondCountableTopology α | by
suffices SeparableSpace α from UniformSpace.secondCountable_of_separable α
have : ∀ U ∈ 𝓤 α, ∃ t : Set α, Set.Countable t ∧ univ ⊆ ⋃ x ∈ t, ball x U := by
simpa only [univ_subset_iff] using hs
rcases subset_countable_closure_of_almost_dense_set (univ : Set α) this with ⟨t, -, htc, ht⟩
exact ⟨⟨t, htc, fu... | theorem | UniformSpace.secondCountable_of_almost_dense_set | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"SecondCountableTopology",
"Set",
"Set.Countable",
"UniformSpace.secondCountable_of_separable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.TotallyBounded.isSeparable {s : Set α} (h : TotallyBounded s) :
TopologicalSpace.IsSeparable s | by
obtain ⟨t, -, htc, hts⟩ := subset_countable_closure_of_almost_dense_set s fun U hU => by
obtain ⟨t, ht, hst⟩ := h (SetRel.inv U)
(mem_of_superset (symmetrize_mem_uniformity hU) SetRel.symmetrize_subset_inv)
exact ⟨t, ht.countable, hst⟩
exact ⟨t, htc, hts⟩ | lemma | TotallyBounded.isSeparable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Cauchy.lean | [] | [
"Set",
"SetRel.inv",
"SetRel.symmetrize_subset_inv",
"TopologicalSpace.IsSeparable",
"TotallyBounded",
"symmetrize_mem_uniformity"
] | A totally bounded set is separable in countably generated uniform spaces. This can be obtained
from the more general `UniformSpace.subset_countable_closure_of_almost_dense_set`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hausdorffEntourage (U : SetRel α α) : SetRel (Set α) (Set α) | {x | x.1 ⊆ U.preimage x.2 ∧ x.2 ⊆ U.image x.1} | def | hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"SetRel"
] | The set of pairs of sets contained in each other's thickening with respect to an entourage. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_hausdorffEntourage (U : SetRel α α) (s t : Set α) :
(s, t) ∈ hausdorffEntourage U ↔ s ⊆ U.preimage t ∧ t ⊆ U.image s | Iff.rfl | theorem | mem_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"SetRel",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hausdorffEntourage_mono {U V : SetRel α α} (h : U ⊆ V) :
hausdorffEntourage U ⊆ hausdorffEntourage V | by
unfold hausdorffEntourage
gcongr | theorem | hausdorffEntourage_mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"SetRel",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
monotone_hausdorffEntourage : Monotone (hausdorffEntourage (α := α)) | fun _ _ => hausdorffEntourage_mono | theorem | monotone_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Monotone",
"hausdorffEntourage",
"hausdorffEntourage_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hausdorffEntourage_id : hausdorffEntourage (.id : SetRel α α) = .id | by
simp_rw [hausdorffEntourage, preimage_id, image_id, ← subset_antisymm_iff, SetRel.id] | theorem | hausdorffEntourage_id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"SetRel",
"SetRel.id",
"hausdorffEntourage",
"subset_antisymm_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isRefl_hausdorffEntourage (U : SetRel α α) [U.IsRefl] :
(hausdorffEntourage U).IsRefl | ⟨fun _ => ⟨U.self_subset_preimage _, U.self_subset_image _⟩⟩ | instance | isRefl_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsRefl",
"SetRel",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inv_hausdorffEntourage (U : SetRel α α) :
(hausdorffEntourage U).inv = hausdorffEntourage U.inv | Set.ext fun _ => And.comm | theorem | inv_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set.ext",
"SetRel",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSymm_hausdorffEntourage (U : SetRel α α) [U.IsSymm] :
(hausdorffEntourage U).IsSymm | by
rw [← inv_eq_self_iff, inv_hausdorffEntourage, inv_eq_self] | instance | isSymm_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsSymm",
"SetRel",
"hausdorffEntourage",
"inv_eq_self",
"inv_hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hausdorffEntourage_comp (U V : SetRel α α) :
hausdorffEntourage (U ○ V) = hausdorffEntourage U ○ hausdorffEntourage V | by
apply subset_antisymm
· intro ⟨s, t⟩ ⟨hst, hts⟩
simp only [mem_comp, mem_hausdorffEntourage] at *
refine ⟨U.image s ∩ V.preimage t, ⟨?_, Set.inter_subset_left⟩, ⟨Set.inter_subset_right, ?_⟩⟩
· intro x hx
obtain ⟨z, hz, y, hxy, hyz⟩ := hst hx
exact ⟨y, ⟨⟨x, hx, hxy⟩, ⟨z, hz, hyz⟩⟩, hxy⟩
... | theorem | hausdorffEntourage_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"SetRel",
"hausdorffEntourage",
"mem_hausdorffEntourage",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isTrans_hausdorffEntourage (U : SetRel α α) [U.IsTrans] :
(hausdorffEntourage U).IsTrans | by
grw [isTrans_iff_comp_subset_self, ← hausdorffEntourage_comp, comp_subset_self] | instance | isTrans_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsTrans",
"SetRel",
"hausdorffEntourage",
"hausdorffEntourage_comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
singleton_mem_hausdorffEntourage (U : SetRel α α) (x y : α) :
({x}, {y}) ∈ hausdorffEntourage U ↔ (x, y) ∈ U | by
simp [hausdorffEntourage] | theorem | singleton_mem_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"SetRel",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
union_mem_hausdorffEntourage (U : SetRel α α) {s₁ s₂ t₁ t₂ : Set α}
(h₁ : (s₁, t₁) ∈ hausdorffEntourage U) (h₂ : (s₂, t₂) ∈ hausdorffEntourage U) :
(s₁ ∪ s₂, t₁ ∪ t₂) ∈ hausdorffEntourage U | by
grind [mem_hausdorffEntourage, preimage_union, image_union] | theorem | union_mem_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"SetRel",
"hausdorffEntourage",
"mem_hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TotallyBounded.exists_prodMk_finset_mem_hausdorffEntourage [UniformSpace α]
{s : Set α} (hs : TotallyBounded s) {U : SetRel α α} (hU : U ∈ 𝓤 α) :
∃ t : Finset α, (↑t, s) ∈ hausdorffEntourage U | by
obtain ⟨t, ht₁, ht₂⟩ := hs _ (symm_le_uniformity hU)
lift t to Finset α using ht₁
classical
refine ⟨{x ∈ t | ∃ y ∈ s, (x, y) ∈ U}, ?_⟩
rw [Finset.coe_filter]
refine ⟨fun _ h => h.2, fun x hx => ?_⟩
obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht₂ hx)
exact ⟨y, ⟨hy, x, hx, hxy⟩, hxy⟩ | theorem | TotallyBounded.exists_prodMk_finset_mem_hausdorffEntourage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Finset",
"Set",
"SetRel",
"TotallyBounded",
"UniformSpace",
"hausdorffEntourage",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prod_mem_hausdorffEntourage_entourageProd
(U₁ : SetRel α α) (U₂ : SetRel β β) {s₁ t₁ : Set α} {s₂ t₂ : Set β}
(h₁ : (s₁, t₁) ∈ hausdorffEntourage U₁) (h₂ : (s₂, t₂) ∈ hausdorffEntourage U₂) :
(s₁ ×ˢ s₂, t₁ ×ˢ t₂) ∈ hausdorffEntourage (entourageProd U₁ U₂) | by
simp only [mem_hausdorffEntourage] at *
grind [preimage_entourageProd_prod, image_entourageProd_prod] | theorem | prod_mem_hausdorffEntourage_entourageProd | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"SetRel",
"entourageProd",
"hausdorffEntourage",
"image_entourageProd_prod",
"mem_hausdorffEntourage",
"preimage_entourageProd_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.hausdorff : UniformSpace (Set α) | .ofCore
{ uniformity := (𝓤 α).lift' hausdorffEntourage
refl := by
simp_rw [Filter.principal_le_lift', SetRel.id_subset_iff]
intro (U : SetRel α α) hU
have := isRefl_of_mem_uniformity hU
exact isRefl_hausdorffEntourage U
symm :=
Filter.tendsto_lift'.mpr fun U hU => Filter.mem_of_... | abbrev | UniformSpace.hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.le_lift'",
"Filter.mem_lift'",
"Filter.mem_of_superset",
"Filter.principal_le_lift'",
"Set",
"SetRel",
"SetRel.id_subset_iff",
"UniformSpace",
"comp_mem_uniformity_sets",
"hausdorffEntourage",
"hausdorffEntourage_comp",
"inv_hausdorffEntourage",
"isRefl_hausdorffEntourage",
"isRefl... | The Hausdorff uniformity on the powerset of a uniform space. Used for defining the uniformities
on `Closeds`, `Compacts` and `NonemptyCompacts`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.HasBasis.uniformity_hausdorff
{ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
(𝓤 (Set α)).HasBasis p (hausdorffEntourage ∘ s) | h.lift' monotone_hausdorffEntourage | theorem | Filter.HasBasis.uniformity_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"hausdorffEntourage",
"monotone_hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_inter_nonempty_of_isOpen {U : Set α} (hU : IsOpen U) :
IsOpen {s | (s ∩ U).Nonempty} | by
rw [isOpen_iff_mem_nhds]
intro s ⟨x, hx₁, hx₂⟩
rw [← hU.mem_nhds_iff, mem_nhds_iff] at hx₂
obtain ⟨V, hV, hVU⟩ := hx₂
rw [mem_nhds_iff]
refine ⟨_, Filter.mem_lift' hV, ?_⟩
rintro s' ⟨hs', -⟩
obtain ⟨y, hy, hxy⟩ := hs' hx₁
exact ⟨y, hy, hVU hxy⟩ | theorem | UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpen | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.mem_lift'",
"IsOpen",
"Set",
"isOpen_iff_mem_nhds",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsClosed.powerset_hausdorff {F : Set α} (hF : IsClosed F) :
IsClosed F.powerset | by
simp_rw [Set.powerset, ← isOpen_compl_iff, Set.compl_setOf, ← Set.inter_compl_nonempty_iff]
exact isOpen_inter_nonempty_of_isOpen hF.isOpen_compl
@[deprecated (since := "2025-11-23")] alias isClosed_powerset := IsClosed.powerset_hausdorff | theorem | IsClosed.powerset_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsClosed",
"Set",
"Set.compl_setOf",
"Set.inter_compl_nonempty_iff",
"Set.powerset",
"isOpen_compl_iff"
] | In the Hausdorff uniformity, the powerset of a closed set is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClopen_singleton_empty : IsClopen {(∅ : Set α)} | by
constructor
· rw [← Set.powerset_empty]
exact isClosed_empty.powerset_hausdorff
· simp_rw [isOpen_iff_mem_nhds, Set.mem_singleton_iff, forall_eq, nhds_eq_uniformity]
filter_upwards [Filter.mem_lift' <| Filter.mem_lift' Filter.univ_mem] with F ⟨_, hF⟩
simpa using hF | theorem | UniformSpace.hausdorff.isClopen_singleton_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.mem_lift'",
"Filter.univ_mem",
"IsClopen",
"Set",
"Set.mem_singleton_iff",
"Set.powerset_empty",
"isOpen_iff_mem_nhds",
"nhds_eq_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_singleton : IsUniformEmbedding ({·} : α → Set α) | where
injective := Set.singleton_injective
comap_uniformity := by
change Filter.comap _ (Filter.lift' _ _) = _
simp_rw [Filter.comap_lift'_eq, Function.comp_def, Set.preimage,
singleton_mem_hausdorffEntourage]
exact Filter.lift'_id | theorem | UniformSpace.hausdorff.isUniformEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.comap",
"Filter.comap_lift'_eq",
"Filter.lift'",
"Filter.lift'_id",
"IsUniformEmbedding",
"Set",
"Set.preimage",
"Set.singleton_injective",
"singleton_mem_hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedEmbedding_singleton [T0Space α] :
Topology.IsClosedEmbedding ({·} : α → Set α) | where
__ := isUniformEmbedding_singleton.isEmbedding
isClosed_range :=
TopologicalSpace.isClosed_range_singleton
isClopen_singleton_empty.isOpen
isOpen_inter_nonempty_of_isOpen | theorem | UniformSpace.hausdorff.isClosedEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"T0Space",
"TopologicalSpace.isClosed_range_singleton",
"Topology.IsClosedEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_union : UniformContinuous (fun x : Set α × Set α => x.1 ∪ x.2) | by
refine Filter.tendsto_lift'.mpr fun U hU => ?_
filter_upwards [entourageProd_mem_uniformity (Filter.mem_lift' hU) (Filter.mem_lift' hU)]
with _ ⟨h₁, h₂⟩ using union_mem_hausdorffEntourage U h₁ h₂ | theorem | UniformSpace.hausdorff.uniformContinuous_union | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.mem_lift'",
"Set",
"UniformContinuous",
"entourageProd_mem_uniformity",
"union_mem_hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_prod : UniformContinuous (fun x : Set α × Set β => x.1 ×ˢ x.2) | by
refine (𝓤 α).basis_sets.uniformity_prod (𝓤 β).basis_sets |>.lift' monotone_hausdorffEntourage
|>.tendsto_right_iff.mpr fun ⟨U, V⟩ ⟨hU, hV⟩ => ?_
filter_upwards [entourageProd_mem_uniformity (Filter.mem_lift' hU) (Filter.mem_lift' hV)]
with ⟨⟨s₁, s₂⟩, ⟨t₁, t₂⟩⟩ ⟨h₁, h₂⟩ using prod_mem_hausdorffEntourage... | theorem | UniformSpace.hausdorff.uniformContinuous_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.mem_lift'",
"Set",
"UniformContinuous",
"entourageProd_mem_uniformity",
"monotone_hausdorffEntourage",
"prod_mem_hausdorffEntourage_entourageProd"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_closure : UniformContinuous (closure (X := α)) | by
simp_rw [UniformContinuous, (𝓤 α).basis_sets.uniformity_hausdorff.tendsto_iff
(𝓤 α).basis_sets.uniformity_hausdorff, Function.comp_id, mem_hausdorffEntourage]
intro U hU
obtain ⟨V : SetRel α α, hV, hVU⟩ := comp_mem_uniformity_sets hU
refine ⟨V, hV, fun ⟨s, t⟩ ⟨hst, hts⟩ => ?_⟩
simp only at *
constr... | theorem | UniformSpace.hausdorff.uniformContinuous_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"SetRel",
"SetRel.image_comp",
"SetRel.preimage_comp",
"UniformContinuous",
"closure",
"comp_mem_uniformity_sets",
"mem_hausdorffEntourage",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_closure : Continuous (closure (X := α)) | uniformContinuous_closure.continuous | theorem | UniformSpace.hausdorff.continuous_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Continuous",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing_closure : IsUniformInducing (closure (X := α)) | by
refine ⟨le_antisymm ?_ <| Filter.map_le_iff_le_comap.mp uniformContinuous_closure⟩
rw [(𝓤 α).basis_sets.uniformity_hausdorff.comap _ |>.le_basis_iff
(𝓤 α).basis_sets.uniformity_hausdorff, Function.comp_id]
intro U hU
obtain ⟨V : SetRel α α, hV, hVU⟩ := comp_mem_uniformity_sets hU
refine ⟨V, hV, fun ⟨... | theorem | UniformSpace.hausdorff.isUniformInducing_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformInducing",
"SetRel",
"SetRel.image_comp",
"SetRel.preimage_comp",
"closure",
"comp_mem_uniformity_sets",
"mem_hausdorffEntourage",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_closure (s : Set α) : 𝓝 (closure s) = 𝓝 s | by
simp_rw +singlePass [isUniformInducing_closure.isInducing.nhds_eq_comap, closure_closure] | theorem | UniformSpace.hausdorff.nhds_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"closure",
"closure_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_setOf_totallyBounded : IsClosed {s : Set α | TotallyBounded s} | by
simp_rw [isClosed_iff_frequently, nhds_eq_comap_uniformity]
intro s hs U hU
obtain ⟨V : SetRel α α, hV, hVU⟩ := comp_mem_uniformity_sets hU
rw [(𝓤 α).basis_sets.uniformity_hausdorff.comap _ |>.frequently_iff] at hs
obtain ⟨t, ⟨hst : s ⊆ V.preimage t, -⟩, ht⟩ := hs V hV
obtain ⟨u, hu, htu⟩ := ht V hV
r... | theorem | UniformSpace.hausdorff.isClosed_setOf_totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsClosed",
"Set",
"Set.subset_def",
"SetRel",
"TotallyBounded",
"comp_mem_uniformity_sets",
"isClosed_iff_frequently",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.image_hausdorff {f : α → β} (hf : UniformContinuous f) :
UniformContinuous (f '' ·) | by
refine Filter.tendsto_lift'.mpr fun U hU => ?_
filter_upwards [Filter.mem_lift' (hf hU)] with ⟨s, t⟩ ⟨h₁, h₂⟩
simp_rw [mem_hausdorffEntourage, Set.image_subset_iff]
exact ⟨h₁.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩,
h₂.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy,... | theorem | UniformContinuous.image_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.mem_lift'",
"Set.image_subset_iff",
"Set.mem_image_of_mem",
"UniformContinuous",
"mem_hausdorffEntourage"
] | When `Set` is equipped with the Hausdorff uniformity, taking the image under a uniformly
continuous map is uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.image_hausdorff {f : α → β} (hf : IsUniformInducing f) :
IsUniformInducing (f '' ·) | by
constructor
change Filter.comap _ (Filter.lift' _ _) = Filter.lift' _ _
rw [Filter.comap_lift'_eq, ← hf.comap_uniformity,
Filter.comap_lift'_eq2 monotone_hausdorffEntourage]
congr with U ⟨s, t⟩
simp only [Function.comp, hausdorffEntourage, SetRel.preimage, SetRel.image, Set.preimage,
Set.mem_setOf,... | theorem | IsUniformInducing.image_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.comap",
"Filter.comap_lift'_eq",
"Filter.comap_lift'_eq2",
"Filter.lift'",
"IsUniformInducing",
"Set.exists_mem_image",
"Set.image_subset_iff",
"Set.mem_setOf",
"Set.preimage",
"SetRel.image",
"SetRel.preimage",
"hausdorffEntourage",
"monotone_hausdorffEntourage"
] | When `Set` is equipped with the Hausdorff uniformity, taking the image under a uniform
inducing map is uniform inducing. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformEmbedding.image_hausdorff {f : α → β} (hf : IsUniformEmbedding f) :
IsUniformEmbedding (f '' ·) | where
__ := hf.isUniformInducing.image_hausdorff
injective := hf.injective.image_injective | theorem | IsUniformEmbedding.image_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding"
] | When `Set` is equipped with the Hausdorff uniformity, taking the image under a uniform
embedding is a uniform embedding. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TotallyBounded.powerset_hausdorff {t : Set α} (ht : TotallyBounded t) :
TotallyBounded t.powerset | by
simp_rw [(𝓤 α).basis_sets.uniformity_hausdorff.totallyBounded_iff, Function.comp_id,
Set.powerset, Set.setOf_subset, Set.mem_iUnion]
intro (U : SetRel α α) hU
obtain ⟨u, hu, ht⟩ := ht U hU
refine ⟨u.powerset, hu.powerset, fun s hs => ⟨u ∩ U.image s, by grind, fun x hx => ?_,
fun x ⟨_, hx⟩ => hx⟩⟩
... | theorem | TotallyBounded.powerset_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"Set.mem_iUnion",
"Set.powerset",
"Set.setOf_subset",
"SetRel",
"TotallyBounded"
] | In the Hausdorff uniformity, the powerset of a totally bounded set is totally bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TotallyBounded.nhds_vietoris_le_nhds_hausdorff {s : Set α} (hs : TotallyBounded s) :
@nhds _ (.vietoris α) s ≤ 𝓝 s | by | theorem | TotallyBounded.nhds_vietoris_le_nhds_hausdorff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"TotallyBounded"
] | The neighborhoods of a totally bounded set in the Hausdorff uniformity are neighborhoods in the
Vietoris topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.nhds_hausdorff_eq_nhds_vietoris {s : Set α} (hs : IsCompact s) :
𝓝 s = @nhds _ (.vietoris α) s | by
refine le_antisymm ?_ hs.totallyBounded.nhds_vietoris_le_nhds_hausdorff
simp_rw [TopologicalSpace.nhds_generateFrom, le_iInf₂_iff, Filter.le_principal_iff]
rintro _ ⟨hs', (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩)⟩
· obtain ⟨V : SetRel α α, hV₁, hV₂⟩ :=
hs.nhdsSet_basis_uniformity (𝓤 α).basis_sets |>.mem_iff.mp (h... | theorem | IsCompact.nhds_hausdorff_eq_nhds_vietoris | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.le_principal_iff",
"Filter.mem_lift'",
"IsCompact",
"Set",
"Set.mem_biUnion",
"SetRel",
"TopologicalSpace.nhds_generateFrom",
"UniformSpace.ball_mem_nhds",
"UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpen",
"le_antisymm"
] | A compact set has the same neighborhoods in the Hausdorff uniformity and the Vietoris topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformSpace : UniformSpace (Closeds α) | .comap (↑) (.hausdorff α) | instance | TopologicalSpace.Closeds.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformSpace",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_def :
𝓤 (Closeds α) = .comap (Prod.map (↑) (↑)) ((𝓤 α).lift' hausdorffEntourage) | rfl | theorem | TopologicalSpace.Closeds.uniformity_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"comap",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Filter.HasBasis.uniformity_closeds
{ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
(𝓤 (Closeds α)).HasBasis p (fun i => Prod.map (↑) (↑) ⁻¹' (hausdorffEntourage (s i))) | h.uniformity_hausdorff.comap _ | theorem | Filter.HasBasis.uniformity_closeds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_coe : IsUniformEmbedding ((↑) : Closeds α → Set α) | where
injective := SetLike.coe_injective
comap_uniformity := rfl | theorem | TopologicalSpace.Closeds.isUniformEmbedding_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding",
"Set",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_coe : UniformContinuous ((↑) : Closeds α → Set α) | isUniformEmbedding_coe.uniformContinuous | theorem | TopologicalSpace.Closeds.uniformContinuous_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_inter_nonempty_of_isOpen {s : Set α} (hs : IsOpen s) :
IsOpen {t : Closeds α | ((t : Set α) ∩ s).Nonempty} | isOpen_induced (UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpen hs) | theorem | TopologicalSpace.Closeds.isOpen_inter_nonempty_of_isOpen | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsOpen",
"Set",
"UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpen",
"isOpen_induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_subsets_of_isClosed {s : Set α} (hs : IsClosed s) :
IsClosed {t : Closeds α | (t : Set α) ⊆ s} | isClosed_induced hs.powerset_hausdorff | theorem | TopologicalSpace.Closeds.isClosed_subsets_of_isClosed | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsClosed",
"Set",
"isClosed_induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClopen_singleton_bot : IsClopen {(⊥ : Closeds α)} | by
convert! UniformSpace.hausdorff.isClopen_singleton_empty.preimage uniformContinuous_coe.continuous
ext; simp | theorem | TopologicalSpace.Closeds.isClopen_singleton_bot | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsClopen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_subsets_of_totallyBounded {t : Set α} (ht : TotallyBounded t) :
TotallyBounded {F : Closeds α | ↑F ⊆ t} | totallyBounded_preimage isUniformEmbedding_coe.isUniformInducing ht.powerset_hausdorff | theorem | TopologicalSpace.Closeds.totallyBounded_subsets_of_totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"TotallyBounded",
"totallyBounded_preimage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_setOf_totallyBounded : IsClosed {s : Closeds α | TotallyBounded (s : Set α)} | UniformSpace.hausdorff.isClosed_setOf_totallyBounded.preimage uniformContinuous_coe.continuous | theorem | TopologicalSpace.Closeds.isClosed_setOf_totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsClosed",
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_singleton : IsUniformEmbedding ({·} : α → Closeds α) | isUniformEmbedding_coe.of_comp_iff.mp UniformSpace.hausdorff.isUniformEmbedding_singleton | theorem | TopologicalSpace.Closeds.isUniformEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding",
"UniformSpace.hausdorff.isUniformEmbedding_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_singleton : UniformContinuous ({·} : α → Closeds α) | isUniformEmbedding_singleton.uniformContinuous | theorem | TopologicalSpace.Closeds.uniformContinuous_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmbedding_singleton : IsEmbedding ({·} : α → Closeds α) | isUniformEmbedding_singleton.isEmbedding | theorem | TopologicalSpace.Closeds.isEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_singleton : Continuous ({·} : α → Closeds α) | isEmbedding_singleton.continuous | theorem | TopologicalSpace.Closeds.continuous_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedEmbedding_singleton : Topology.IsClosedEmbedding ({·} : α → Closeds α) | where
__ := isUniformEmbedding_singleton.isEmbedding
isClosed_range := by
rw [← SetLike.coe_injective.preimage_image (s := Set.range ({·})), ← Set.range_comp]
exact UniformSpace.hausdorff.isClosedEmbedding_singleton.isClosed_range.preimage
uniformContinuous_coe.continuous | theorem | TopologicalSpace.Closeds.isClosedEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set.range",
"Set.range_comp",
"Topology.IsClosedEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discreteUniformity_iff : DiscreteUniformity (Closeds α) ↔ DiscreteUniformity α | ⟨fun _ => isUniformEmbedding_singleton.discreteUniformity, fun _ => inferInstance⟩ | theorem | TopologicalSpace.Closeds.discreteUniformity_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"DiscreteUniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_sup : UniformContinuous (fun x : Closeds α × Closeds α => x.1 ⊔ x.2) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_union.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe | theorem | TopologicalSpace.Closeds.uniformContinuous_sup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.sup_closeds
{f g : α → Closeds β} (hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous (fun x => f x ⊔ g x) | uniformContinuous_sup.comp <| hf.prodMk hg | theorem | UniformContinuous.sup_closeds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_prod : UniformContinuous (fun x : Closeds α × Closeds β => x.1 ×ˢ x.2) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_prod.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe | theorem | TopologicalSpace.Closeds.uniformContinuous_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.prod_closeds {f : α → Closeds β} {g : α → Closeds γ}
(hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous (fun x => f x ×ˢ g x) | uniformContinuous_prod.comp (hf.prodMk hg) | theorem | UniformContinuous.prod_closeds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_prod : Continuous (fun x : Closeds α × Closeds β => x.1 ×ˢ x.2) | uniformContinuous_prod.continuous | theorem | TopologicalSpace.Closeds.continuous_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing_closure : IsUniformInducing (Closeds.closure (α := α)) | isUniformEmbedding_coe.isUniformInducing.of_comp_iff.mp
UniformSpace.hausdorff.isUniformInducing_closure | theorem | TopologicalSpace.Closeds.isUniformInducing_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformInducing",
"UniformSpace.hausdorff.isUniformInducing_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_closure : UniformContinuous (Closeds.closure (α := α)) | isUniformInducing_closure.uniformContinuous | theorem | TopologicalSpace.Closeds.uniformContinuous_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_closure : Continuous (Closeds.closure (α := α)) | uniformContinuous_closure.continuous | theorem | TopologicalSpace.Closeds.continuous_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compactSpace_iff : CompactSpace (Closeds α) ↔ CompactSpace α | by
refine ⟨fun _ => compactSpace_of_finite_subfamily_closed fun {ι} F hF₁ hF₂ => ?_,
fun _ => inferInstance⟩
have := isClopen_singleton_bot.compl.isClosed.isCompact.elim_finite_subfamily_closed
(fun i => {C : Closeds α | ↑C ⊆ F i})
(fun i => isClosed_subsets_of_isClosed (hF₁ i))
simp_rw [← Set.disjoin... | theorem | TopologicalSpace.Closeds.compactSpace_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"CompactSpace",
"Set.disjoint_compl_left_iff_subset",
"Set.disjoint_iff_inter_eq_empty",
"Set.setOf_eq_eq_singleton",
"Set.setOf_forall",
"Set.subset_empty_iff",
"Set.subset_iInter_iff",
"compactSpace_of_finite_subfamily_closed",
"isClosed_biInter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
noncompactSpace_iff : NoncompactSpace (Closeds α) ↔ NoncompactSpace α | by
simp_rw [← not_compactSpace_iff, compactSpace_iff] | theorem | TopologicalSpace.Closeds.noncompactSpace_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"NoncompactSpace",
"not_compactSpace_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace : UniformSpace (Compacts α) | .replaceTopology (.comap (↑) (.hausdorff α)) <| ext_nhds fun K ↦ by
simp_rw [nhds_induced, K.isCompact.nhds_hausdorff_eq_nhds_vietoris] | instance | TopologicalSpace.Compacts.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformSpace",
"comap",
"nhds_induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_def :
𝓤 (Compacts α) = .comap (Prod.map (↑) (↑)) ((𝓤 α).lift' hausdorffEntourage) | rfl | theorem | TopologicalSpace.Compacts.uniformity_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"comap",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Filter.HasBasis.uniformity_compacts
{ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
(𝓤 (Compacts α)).HasBasis p (fun i => Prod.map (↑) (↑) ⁻¹' (hausdorffEntourage (s i))) | h.uniformity_hausdorff.comap _ | theorem | Filter.HasBasis.uniformity_compacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_coe : IsUniformEmbedding ((↑) : Compacts α → Set α) | where
injective := SetLike.coe_injective
comap_uniformity := rfl | theorem | TopologicalSpace.Compacts.isUniformEmbedding_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding",
"Set",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_coe : UniformContinuous ((↑) : Compacts α → Set α) | isUniformEmbedding_coe.uniformContinuous | theorem | TopologicalSpace.Compacts.uniformContinuous_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_toCloseds [T2Space α] : IsUniformEmbedding (toCloseds (α := α)) | where
injective := toCloseds_injective
comap_uniformity := Filter.comap_comap | theorem | TopologicalSpace.Compacts.isUniformEmbedding_toCloseds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.comap_comap",
"IsUniformEmbedding",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_toCloseds [T2Space α] : UniformContinuous (toCloseds (α := α)) | isUniformEmbedding_toCloseds.uniformContinuous | theorem | TopologicalSpace.Compacts.uniformContinuous_toCloseds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"T2Space",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isEmbedding_toCloseds [T2Space α] : IsEmbedding (toCloseds (α := α)) | isUniformEmbedding_toCloseds.isEmbedding | theorem | TopologicalSpace.Compacts.isEmbedding_toCloseds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_toCloseds [T2Space α] : Continuous (toCloseds (α := α)) | uniformContinuous_toCloseds.continuous | theorem | TopologicalSpace.Compacts.continuous_toCloseds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Continuous",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedEmbedding_toCloseds [T2Space α] [CompleteSpace α] :
IsClosedEmbedding (toCloseds (α := α)) | where
__ := isEmbedding_toCloseds
isClosed_range := by
convert! Closeds.isClosed_setOf_totallyBounded
exact subset_antisymm
(Set.range_subset_iff.mpr fun K => K.isCompact.totallyBounded)
(fun K hK => ⟨⟨K, hK.isCompact_of_isClosed K.isClosed⟩, rfl⟩) | theorem | TopologicalSpace.Compacts.isClosedEmbedding_toCloseds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"CompleteSpace",
"T2Space",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_subsets_of_totallyBounded {t : Set α} (ht : TotallyBounded t) :
TotallyBounded {K : Compacts α | ↑K ⊆ t} | totallyBounded_preimage isUniformEmbedding_coe.isUniformInducing ht.powerset_hausdorff | theorem | TopologicalSpace.Compacts.totallyBounded_subsets_of_totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"TotallyBounded",
"totallyBounded_preimage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_singleton : IsUniformEmbedding ({·} : α → Compacts α) | isUniformEmbedding_coe.of_comp_iff.mp UniformSpace.hausdorff.isUniformEmbedding_singleton | theorem | TopologicalSpace.Compacts.isUniformEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding",
"UniformSpace.hausdorff.isUniformEmbedding_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_singleton : UniformContinuous ({·} : α → Compacts α) | isUniformEmbedding_singleton.uniformContinuous | theorem | TopologicalSpace.Compacts.uniformContinuous_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_sup :
UniformContinuous (fun x : Compacts α × Compacts α => x.1 ⊔ x.2) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_union.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe | theorem | TopologicalSpace.Compacts.uniformContinuous_sup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.sup_compacts
{f g : α → Compacts β} (hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous (fun x => f x ⊔ g x) | uniformContinuous_sup.comp <| hf.prodMk hg | theorem | UniformContinuous.sup_compacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_prod :
UniformContinuous (fun x : Compacts α × Compacts β => x.1 ×ˢ x.2) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_prod.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe | theorem | TopologicalSpace.Compacts.uniformContinuous_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.prod_compacts {f : α → Compacts β} {g : α → Compacts γ}
(hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous (fun x => f x ×ˢ g x) | uniformContinuous_prod.comp (hf.prodMk hg) | theorem | UniformContinuous.prod_compacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.compacts_map {f : α → β} (hf : UniformContinuous f) :
UniformContinuous (Compacts.map f hf.continuous) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <| hf.image_hausdorff.comp uniformContinuous_coe | theorem | UniformContinuous.compacts_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsUniformInducing.compacts_map {f : α → β} (hf : IsUniformInducing f) :
IsUniformInducing (Compacts.map f hf.uniformContinuous.continuous) | .of_comp hf.uniformContinuous.compacts_map uniformContinuous_coe <|
hf.image_hausdorff.comp isUniformEmbedding_coe.isUniformInducing | theorem | IsUniformInducing.compacts_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsUniformEmbedding.compacts_map {f : α → β} (hf : IsUniformEmbedding f) :
IsUniformEmbedding (Compacts.map f hf.uniformContinuous.continuous) | where
__ := hf.isUniformInducing.compacts_map
injective := map_injective hf.uniformContinuous.continuous hf.injective | theorem | IsUniformEmbedding.compacts_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discreteUniformity_iff : DiscreteUniformity (Compacts α) ↔ DiscreteUniformity α | ⟨fun _ => isUniformEmbedding_singleton.discreteUniformity, fun _ => inferInstance⟩ | theorem | TopologicalSpace.Compacts.discreteUniformity_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"DiscreteUniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace : UniformSpace (NonemptyCompacts α) | .replaceTopology (.comap (↑) (.hausdorff α)) <| ext_nhds fun K ↦ by
simp_rw [nhds_induced, K.isCompact.nhds_hausdorff_eq_nhds_vietoris] | instance | TopologicalSpace.NonemptyCompacts.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformSpace",
"comap",
"nhds_induced"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_def :
𝓤 (NonemptyCompacts α) = .comap (Prod.map (↑) (↑)) ((𝓤 α).lift' hausdorffEntourage) | rfl | theorem | TopologicalSpace.NonemptyCompacts.uniformity_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"comap",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Filter.HasBasis.uniformity_nonemptyCompacts
{ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
(𝓤 (NonemptyCompacts α)).HasBasis p
(fun i => Prod.map (↑) (↑) ⁻¹' (hausdorffEntourage (s i))) | h.uniformity_hausdorff.comap _ | theorem | Filter.HasBasis.uniformity_nonemptyCompacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"hausdorffEntourage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_coe : IsUniformEmbedding ((↑) : NonemptyCompacts α → Set α) | where
injective := SetLike.coe_injective
comap_uniformity := rfl | theorem | TopologicalSpace.NonemptyCompacts.isUniformEmbedding_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding",
"Set",
"SetLike.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_coe : UniformContinuous ((↑) : NonemptyCompacts α → Set α) | isUniformEmbedding_coe.uniformContinuous | theorem | TopologicalSpace.NonemptyCompacts.uniformContinuous_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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