fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
totally_bounded_iff_subset {s : set α} : totally_bounded s ↔
∀d ∈ 𝓤 α, ∃t ⊆ s, set.finite t ∧ s ⊆ (⋃y∈t, {x | (x,y) ∈ d}) :=
⟨λ H d hd, H.exists_subset hd, λ H d hd, let ⟨t, _, ht⟩ := H d hd in ⟨t, ht⟩⟩ | totally_bounded_iff_subset {s : set α} : totally_bounded s ↔
∀d ∈ 𝓤 α, ∃t ⊆ s, set.finite t ∧ s ⊆ (⋃y∈t, {x | (x,y) ∈ d}) | ⟨λ H d hd, H.exists_subset hd, λ H d hd, let ⟨t, _, ht⟩ := H d hd in ⟨t, ht⟩⟩ | theorem | totally_bounded_iff_subset | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"set.finite",
"totally_bounded"
] | null | 450 | 452 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.totally_bounded_iff {ι} {p : ι → Prop} {U : ι → set (α × α)}
(H : (𝓤 α).has_basis p U) {s : set α} :
totally_bounded s ↔ ∀ i, p i → ∃ t : set α, set.finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U i} :=
H.forall_iff $ λ U V hUV h, h.imp $ λ t ht, ⟨ht.1, ht.2.trans $ Union₂_mono $ λ x hx y hy, hUV hy⟩ | filter.has_basis.totally_bounded_iff {ι} {p : ι → Prop} {U : ι → set (α × α)}
(H : (𝓤 α).has_basis p U) {s : set α} :
totally_bounded s ↔ ∀ i, p i → ∃ t : set α, set.finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U i} | H.forall_iff $ λ U V hUV h, h.imp $ λ t ht, ⟨ht.1, ht.2.trans $ Union₂_mono $ λ x hx y hy, hUV hy⟩ | lemma | filter.has_basis.totally_bounded_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"set.finite",
"totally_bounded"
] | null | 454 | 457 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_of_forall_symm {s : set α}
(h : ∀ V ∈ 𝓤 α, symmetric_rel V → ∃ t : set α, set.finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
totally_bounded s :=
uniform_space.has_basis_symmetric.totally_bounded_iff.2 $ λ V hV,
by simpa only [ball_eq_of_symmetry hV.2] using h V hV.1 hV.2 | totally_bounded_of_forall_symm {s : set α}
(h : ∀ V ∈ 𝓤 α, symmetric_rel V → ∃ t : set α, set.finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
totally_bounded s | uniform_space.has_basis_symmetric.totally_bounded_iff.2 $ λ V hV,
by simpa only [ball_eq_of_symmetry hV.2] using h V hV.1 hV.2 | lemma | totally_bounded_of_forall_symm | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"ball_eq_of_symmetry",
"set.finite",
"symmetric_rel",
"totally_bounded"
] | null | 459 | 463 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_subset {s₁ s₂ : set α} (hs : s₁ ⊆ s₂)
(h : totally_bounded s₂) : totally_bounded s₁ :=
assume d hd, let ⟨t, ht₁, ht₂⟩ := h d hd in ⟨t, ht₁, subset.trans hs ht₂⟩ | totally_bounded_subset {s₁ s₂ : set α} (hs : s₁ ⊆ s₂)
(h : totally_bounded s₂) : totally_bounded s₁ | assume d hd, let ⟨t, ht₁, ht₂⟩ := h d hd in ⟨t, ht₁, subset.trans hs ht₂⟩ | lemma | totally_bounded_subset | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"totally_bounded"
] | null | 465 | 467 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_empty : totally_bounded (∅ : set α) :=
λ d hd, ⟨∅, finite_empty, empty_subset _⟩ | totally_bounded_empty : totally_bounded (∅ : set α) | λ d hd, ⟨∅, finite_empty, empty_subset _⟩ | lemma | totally_bounded_empty | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"totally_bounded"
] | null | 469 | 470 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded.closure {s : set α} (h : totally_bounded s) :
totally_bounded (closure s) :=
uniformity_has_basis_closed.totally_bounded_iff.2 $ λ V hV, let ⟨t, htf, hst⟩ := h V hV.1
in ⟨t, htf, closure_minimal hst $ is_closed_bUnion htf $
λ y hy, hV.2.preimage (continuous_id.prod_mk continuous_const)⟩ | totally_bounded.closure {s : set α} (h : totally_bounded s) :
totally_bounded (closure s) | uniformity_has_basis_closed.totally_bounded_iff.2 $ λ V hV, let ⟨t, htf, hst⟩ := h V hV.1
in ⟨t, htf, closure_minimal hst $ is_closed_bUnion htf $
λ y hy, hV.2.preimage (continuous_id.prod_mk continuous_const)⟩ | lemma | totally_bounded.closure | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"closure",
"closure_minimal",
"continuous_const",
"is_closed_bUnion",
"totally_bounded"
] | The closure of a totally bounded set is totally bounded. | 473 | 477 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded.image [uniform_space β] {f : α → β} {s : set α}
(hs : totally_bounded s) (hf : uniform_continuous f) : totally_bounded (f '' s) :=
assume t ht,
have {p:α×α | (f p.1, f p.2) ∈ t} ∈ 𝓤 α,
from hf ht,
let ⟨c, hfc, hct⟩ := hs _ this in
⟨f '' c, hfc.image f,
begin
simp [image_subset_iff],
simp ... | totally_bounded.image [uniform_space β] {f : α → β} {s : set α}
(hs : totally_bounded s) (hf : uniform_continuous f) : totally_bounded (f '' s) | assume t ht,
have {p:α×α | (f p.1, f p.2) ∈ t} ∈ 𝓤 α,
from hf ht,
let ⟨c, hfc, hct⟩ := hs _ this in
⟨f '' c, hfc.image f,
begin
simp [image_subset_iff],
simp [subset_def] at hct,
intros x hx, simp,
exact hct x hx
end⟩ | lemma | totally_bounded.image | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"totally_bounded",
"uniform_continuous",
"uniform_space"
] | The image of a totally bounded set under a uniformly continuous map is totally bounded. | 480 | 492 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter.cauchy_of_totally_bounded {s : set α} (f : ultrafilter α)
(hs : totally_bounded s) (h : ↑f ≤ 𝓟 s) : cauchy (f : filter α) :=
⟨f.ne_bot', assume t ht,
let ⟨t', ht'₁, ht'_symm, ht'_t⟩ := comp_symm_of_uniformity ht in
let ⟨i, hi, hs_union⟩ := hs t' ht'₁ in
have (⋃y∈i, {x | (x,y) ∈ t'}) ∈ f,
from m... | ultrafilter.cauchy_of_totally_bounded {s : set α} (f : ultrafilter α)
(hs : totally_bounded s) (h : ↑f ≤ 𝓟 s) : cauchy (f : filter α) | ⟨f.ne_bot', assume t ht,
let ⟨t', ht'₁, ht'_symm, ht'_t⟩ := comp_symm_of_uniformity ht in
let ⟨i, hi, hs_union⟩ := hs t' ht'₁ in
have (⋃y∈i, {x | (x,y) ∈ t'}) ∈ f,
from mem_of_superset (le_principal_iff.mp h) hs_union,
have ∃y∈i, {x | (x,y) ∈ t'} ∈ f,
from (ultrafilter.finite_bUnion_mem_iff hi).1 this,
... | lemma | ultrafilter.cauchy_of_totally_bounded | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"comp_rel",
"comp_symm_of_uniformity",
"filter",
"totally_bounded",
"ultrafilter",
"ultrafilter.finite_bUnion_mem_iff"
] | null | 494 | 507 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_iff_filter {s : set α} :
totally_bounded s ↔ (∀f, ne_bot f → f ≤ 𝓟 s → ∃c ≤ f, cauchy c) :=
begin
split,
{ introsI H f hf hfs,
exact ⟨ultrafilter.of f, ultrafilter.of_le f,
(ultrafilter.of f).cauchy_of_totally_bounded H ((ultrafilter.of_le f).trans hfs)⟩ },
{ intros H d hd,
contra... | totally_bounded_iff_filter {s : set α} :
totally_bounded s ↔ (∀f, ne_bot f → f ≤ 𝓟 s → ∃c ≤ f, cauchy c) | begin
split,
{ introsI H f hf hfs,
exact ⟨ultrafilter.of f, ultrafilter.of_le f,
(ultrafilter.of f).cauchy_of_totally_bounded H ((ultrafilter.of_le f).trans hfs)⟩ },
{ intros H d hd,
contrapose! H with hd_cover,
set f := ⨅ t : finset α, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d}),
have : ne_bot f,
... | lemma | totally_bounded_iff_filter | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"directed_of_sup",
"finset",
"infi_le_of_le",
"le_rfl",
"totally_bounded",
"ultrafilter.of",
"ultrafilter.of_le"
] | null | 509 | 537 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_iff_ultrafilter {s : set α} :
totally_bounded s ↔ (∀f : ultrafilter α, ↑f ≤ 𝓟 s → cauchy (f : filter α)) :=
begin
refine ⟨λ hs f, f.cauchy_of_totally_bounded hs, λ H, totally_bounded_iff_filter.2 _⟩,
introsI f hf hfs,
exact ⟨ultrafilter.of f, ultrafilter.of_le f, H _ ((ultrafilter.of_le f).tran... | totally_bounded_iff_ultrafilter {s : set α} :
totally_bounded s ↔ (∀f : ultrafilter α, ↑f ≤ 𝓟 s → cauchy (f : filter α)) | begin
refine ⟨λ hs f, f.cauchy_of_totally_bounded hs, λ H, totally_bounded_iff_filter.2 _⟩,
introsI f hf hfs,
exact ⟨ultrafilter.of f, ultrafilter.of_le f, H _ ((ultrafilter.of_le f).trans hfs)⟩
end | lemma | totally_bounded_iff_ultrafilter | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"totally_bounded",
"ultrafilter",
"ultrafilter.of_le"
] | null | 539 | 545 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_iff_totally_bounded_is_complete {s : set α} :
is_compact s ↔ totally_bounded s ∧ is_complete s :=
⟨λ hs, ⟨totally_bounded_iff_ultrafilter.2 (λ f hf,
let ⟨x, xs, fx⟩ := is_compact_iff_ultrafilter_le_nhds.1 hs f hf in cauchy_nhds.mono fx),
λ f fc fs,
let ⟨a, as, fa⟩ := @hs f fc.1 fs in
⟨a, as, ... | is_compact_iff_totally_bounded_is_complete {s : set α} :
is_compact s ↔ totally_bounded s ∧ is_complete s | ⟨λ hs, ⟨totally_bounded_iff_ultrafilter.2 (λ f hf,
let ⟨x, xs, fx⟩ := is_compact_iff_ultrafilter_le_nhds.1 hs f hf in cauchy_nhds.mono fx),
λ f fc fs,
let ⟨a, as, fa⟩ := @hs f fc.1 fs in
⟨a, as, le_nhds_of_cauchy_adhp fc fa⟩⟩,
λ ⟨ht, hc⟩, is_compact_iff_ultrafilter_le_nhds.2
(λf hf, hc _ (totally_boun... | lemma | is_compact_iff_totally_bounded_is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_compact",
"is_complete",
"le_nhds_of_cauchy_adhp",
"totally_bounded"
] | null | 547 | 555 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.totally_bounded {s : set α} (h : is_compact s) : totally_bounded s :=
(is_compact_iff_totally_bounded_is_complete.1 h).1 | is_compact.totally_bounded {s : set α} (h : is_compact s) : totally_bounded s | (is_compact_iff_totally_bounded_is_complete.1 h).1 | lemma | is_compact.totally_bounded | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_compact",
"totally_bounded"
] | null | 557 | 558 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.is_complete {s : set α} (h : is_compact s) : is_complete s :=
(is_compact_iff_totally_bounded_is_complete.1 h).2 | is_compact.is_complete {s : set α} (h : is_compact s) : is_complete s | (is_compact_iff_totally_bounded_is_complete.1 h).2 | lemma | is_compact.is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_compact",
"is_complete"
] | null | 560 | 561 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_of_compact {α : Type u} [uniform_space α] [compact_space α] : complete_space α :=
⟨λf hf, by simpa using (is_compact_iff_totally_bounded_is_complete.1 is_compact_univ).2 f hf⟩ | complete_of_compact {α : Type u} [uniform_space α] [compact_space α] : complete_space α | ⟨λf hf, by simpa using (is_compact_iff_totally_bounded_is_complete.1 is_compact_univ).2 f hf⟩ | instance | complete_of_compact | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"compact_space",
"complete_space",
"is_compact_univ",
"uniform_space"
] | null | 563 | 565 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_of_totally_bounded_is_closed [complete_space α] {s : set α}
(ht : totally_bounded s) (hc : is_closed s) : is_compact s :=
(@is_compact_iff_totally_bounded_is_complete α _ s).2 ⟨ht, hc.is_complete⟩ | is_compact_of_totally_bounded_is_closed [complete_space α] {s : set α}
(ht : totally_bounded s) (hc : is_closed s) : is_compact s | (@is_compact_iff_totally_bounded_is_complete α _ s).2 ⟨ht, hc.is_complete⟩ | lemma | is_compact_of_totally_bounded_is_closed | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"is_closed",
"is_compact",
"is_compact_iff_totally_bounded_is_complete",
"totally_bounded"
] | null | 567 | 569 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq.totally_bounded_range {s : ℕ → α} (hs : cauchy_seq s) :
totally_bounded (range s) :=
begin
refine totally_bounded_iff_subset.2 (λ a ha, _),
cases cauchy_seq_iff.1 hs a ha with n hn,
refine ⟨s '' {k | k ≤ n}, image_subset_range _ _, (finite_le_nat _).image _, _⟩,
rw [range_subset_iff, bUnion_image],... | cauchy_seq.totally_bounded_range {s : ℕ → α} (hs : cauchy_seq s) :
totally_bounded (range s) | begin
refine totally_bounded_iff_subset.2 (λ a ha, _),
cases cauchy_seq_iff.1 hs a ha with n hn,
refine ⟨s '' {k | k ≤ n}, image_subset_range _ _, (finite_le_nat _).image _, _⟩,
rw [range_subset_iff, bUnion_image],
intro m,
rw [mem_Union₂],
cases le_total m n with hm hm,
exacts [⟨m, hm, refl_mem_uniform... | lemma | cauchy_seq.totally_bounded_range | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"refl_mem_uniformity",
"totally_bounded"
] | Every Cauchy sequence over `ℕ` is totally bounded. | 572 | 583 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq_aux (n : ℕ) : {s : set α // ∃ (_ : s ∈ f), s ×ˢ s ⊆ U n } :=
indefinite_description _ $ (cauchy_iff.1 hf).2 (U n) (U_mem n) | set_seq_aux (n : ℕ) : {s : set α // ∃ (_ : s ∈ f), s ×ˢ s ⊆ U n } | indefinite_description _ $ (cauchy_iff.1 hf).2 (U n) (U_mem n) | def | sequentially_complete.set_seq_aux | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | An auxiliary sequence of sets approximating a Cauchy filter. | 608 | 609 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq (n : ℕ) : set α := ⋂ m ∈ set.Iic n, (set_seq_aux hf U_mem m).val | set_seq (n : ℕ) : set α | ⋂ m ∈ set.Iic n, (set_seq_aux hf U_mem m).val | def | sequentially_complete.set_seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"set.Iic"
] | Given a Cauchy filter `f` and a sequence `U` of entourages, `set_seq` provides
an antitone sequence of sets `s n ∈ f` such that `s n ×ˢ s n ⊆ U`. | 613 | 613 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq_mem (n : ℕ) : set_seq hf U_mem n ∈ f :=
(bInter_mem (finite_le_nat n)).2 (λ m _, (set_seq_aux hf U_mem m).2.fst) | set_seq_mem (n : ℕ) : set_seq hf U_mem n ∈ f | (bInter_mem (finite_le_nat n)).2 (λ m _, (set_seq_aux hf U_mem m).2.fst) | lemma | sequentially_complete.set_seq_mem | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | null | 615 | 616 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq_mono ⦃m n : ℕ⦄ (h : m ≤ n) : set_seq hf U_mem n ⊆ set_seq hf U_mem m :=
bInter_subset_bInter_left (λ k hk, le_trans hk h) | set_seq_mono ⦃m n : ℕ⦄ (h : m ≤ n) : set_seq hf U_mem n ⊆ set_seq hf U_mem m | bInter_subset_bInter_left (λ k hk, le_trans hk h) | lemma | sequentially_complete.set_seq_mono | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | null | 618 | 619 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq_sub_aux (n : ℕ) : set_seq hf U_mem n ⊆ set_seq_aux hf U_mem n :=
bInter_subset_of_mem right_mem_Iic | set_seq_sub_aux (n : ℕ) : set_seq hf U_mem n ⊆ set_seq_aux hf U_mem n | bInter_subset_of_mem right_mem_Iic | lemma | sequentially_complete.set_seq_sub_aux | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | null | 621 | 622 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq_prod_subset {N m n} (hm : N ≤ m) (hn : N ≤ n) :
set_seq hf U_mem m ×ˢ set_seq hf U_mem n ⊆ U N :=
begin
assume p hp,
refine (set_seq_aux hf U_mem N).2.snd ⟨_, _⟩;
apply set_seq_sub_aux,
exact set_seq_mono hf U_mem hm hp.1,
exact set_seq_mono hf U_mem hn hp.2
end | set_seq_prod_subset {N m n} (hm : N ≤ m) (hn : N ≤ n) :
set_seq hf U_mem m ×ˢ set_seq hf U_mem n ⊆ U N | begin
assume p hp,
refine (set_seq_aux hf U_mem N).2.snd ⟨_, _⟩;
apply set_seq_sub_aux,
exact set_seq_mono hf U_mem hm hp.1,
exact set_seq_mono hf U_mem hn hp.2
end | lemma | sequentially_complete.set_seq_prod_subset | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | null | 624 | 632 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq (n : ℕ) : α := some $ hf.1.nonempty_of_mem (set_seq_mem hf U_mem n) | seq (n : ℕ) : α | some $ hf.1.nonempty_of_mem (set_seq_mem hf U_mem n) | def | sequentially_complete.seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | A sequence of points such that `seq n ∈ set_seq n`. Here `set_seq` is an antitone
sequence of sets `set_seq n ∈ f` with diameters controlled by a given sequence
of entourages. | 637 | 637 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq_mem (n : ℕ) : seq hf U_mem n ∈ set_seq hf U_mem n :=
some_spec $ hf.1.nonempty_of_mem (set_seq_mem hf U_mem n) | seq_mem (n : ℕ) : seq hf U_mem n ∈ set_seq hf U_mem n | some_spec $ hf.1.nonempty_of_mem (set_seq_mem hf U_mem n) | lemma | sequentially_complete.seq_mem | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | null | 639 | 640 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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 :=
set_seq_prod_subset hf U_mem hm hn ⟨seq_mem hf U_mem m, seq_mem hf U_mem n⟩ | 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 | set_seq_prod_subset hf U_mem hm hn ⟨seq_mem hf U_mem m, seq_mem hf U_mem n⟩ | lemma | sequentially_complete.seq_pair_mem | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | null | 642 | 644 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq_is_cauchy_seq : cauchy_seq $ seq hf U_mem :=
cauchy_seq_of_controlled U U_le $ seq_pair_mem hf U_mem | seq_is_cauchy_seq : cauchy_seq $ seq hf U_mem | cauchy_seq_of_controlled U U_le $ seq_pair_mem hf U_mem | theorem | sequentially_complete.seq_is_cauchy_seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"cauchy_seq_of_controlled"
] | null | 648 | 649 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_nhds_of_seq_tendsto_nhds ⦃a : α⦄ (ha : tendsto (seq hf U_mem) at_top (𝓝 a)) :
f ≤ 𝓝 a :=
le_nhds_of_cauchy_adhp_aux
begin
assume s hs,
rcases U_le s hs with ⟨m, hm⟩,
rcases tendsto_at_top'.1 ha _ (mem_nhds_left a (U_mem m)) with ⟨n, hn⟩,
refine ⟨set_seq hf U_mem (max m n), set_seq_mem hf U_mem _, _,
... | le_nhds_of_seq_tendsto_nhds ⦃a : α⦄ (ha : tendsto (seq hf U_mem) at_top (𝓝 a)) :
f ≤ 𝓝 a | le_nhds_of_cauchy_adhp_aux
begin
assume s hs,
rcases U_le s hs with ⟨m, hm⟩,
rcases tendsto_at_top'.1 ha _ (mem_nhds_left a (U_mem m)) with ⟨n, hn⟩,
refine ⟨set_seq hf U_mem (max m n), set_seq_mem hf U_mem _, _,
seq hf U_mem (max m n), _, seq_mem hf U_mem _⟩,
{ have := le_max_left m n,
exact set... | theorem | sequentially_complete.le_nhds_of_seq_tendsto_nhds | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"le_nhds_of_cauchy_adhp_aux",
"mem_nhds_left",
"set.subset.trans"
] | If the sequence `sequentially_complete.seq` converges to `a`, then `f ≤ 𝓝 a`. | 652 | 664 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_of_convergent_controlled_sequences (U : ℕ → set (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 at_top (𝓝 a)) :
complete_space α :=
begin
obtain ⟨U', U'_mono, hU'⟩ := (𝓤 α).exists_antitone_seq,
have Hmem : ∀ n, U n ∩ U' n ∈ 𝓤 α,
... | complete_of_convergent_controlled_sequences (U : ℕ → set (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 at_top (𝓝 a)) :
complete_space α | begin
obtain ⟨U', U'_mono, hU'⟩ := (𝓤 α).exists_antitone_seq,
have Hmem : ∀ n, U n ∩ U' n ∈ 𝓤 α,
from λ n, inter_mem (U_mem n) (hU'.2 ⟨n, subset.refl _⟩),
refine ⟨λ f hf, (HU (seq hf Hmem) (λ N m n hm hn, _)).imp $
le_nhds_of_seq_tendsto_nhds _ _ (λ s hs, _)⟩,
{ rcases (hU'.1 hs) with ⟨N, hN⟩,
exa... | theorem | uniform_space.complete_of_convergent_controlled_sequences | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space"
] | 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. | 676 | 688 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_of_cauchy_seq_tendsto
(H' : ∀ u : ℕ → α, cauchy_seq u → ∃a, tendsto u at_top (𝓝 a)) :
complete_space α :=
let ⟨U', U'_mono, hU'⟩ := (𝓤 α).exists_antitone_seq in
complete_of_convergent_controlled_sequences U' (λ n, hU'.2 ⟨n, subset.refl _⟩)
(λ u hu, H' u $ cauchy_seq_of_controlled U' (λ s hs, hU'.1 hs) ... | complete_of_cauchy_seq_tendsto
(H' : ∀ u : ℕ → α, cauchy_seq u → ∃a, tendsto u at_top (𝓝 a)) :
complete_space α | let ⟨U', U'_mono, hU'⟩ := (𝓤 α).exists_antitone_seq in
complete_of_convergent_controlled_sequences U' (λ n, hU'.2 ⟨n, subset.refl _⟩)
(λ u hu, H' u $ cauchy_seq_of_controlled U' (λ s hs, hU'.1 hs) hu) | theorem | uniform_space.complete_of_cauchy_seq_tendsto | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"cauchy_seq_of_controlled",
"complete_space"
] | A sequentially complete uniform space with a countable basis of the uniformity filter is
complete. | 692 | 697 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
first_countable_topology : first_countable_topology α :=
⟨λ a, by { rw nhds_eq_comap_uniformity, apply_instance }⟩ | first_countable_topology : first_countable_topology α | ⟨λ a, by { rw nhds_eq_comap_uniformity, apply_instance }⟩ | instance | uniform_space.first_countable_topology | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"nhds_eq_comap_uniformity"
] | null | 701 | 703 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_of_separable [separable_space α] : second_countable_topology α :=
begin
rcases exists_countable_dense α with ⟨s, hsc, hsd⟩,
obtain ⟨t : ℕ → set (α × α),
hto : ∀ (i : ℕ), t i ∈ (𝓤 α).sets ∧ is_open (t i) ∧ symmetric_rel (t i),
h_basis : (𝓤 α).has_antitone_basis t⟩ :=
(@uniformity_has... | second_countable_of_separable [separable_space α] : second_countable_topology α | begin
rcases exists_countable_dense α with ⟨s, hsc, hsd⟩,
obtain ⟨t : ℕ → set (α × α),
hto : ∀ (i : ℕ), t i ∈ (𝓤 α).sets ∧ is_open (t i) ∧ symmetric_rel (t i),
h_basis : (𝓤 α).has_antitone_basis t⟩ :=
(@uniformity_has_basis_open_symmetric α _).exists_antitone_subbasis,
choose ht_mem hto hts using ... | lemma | uniform_space.second_countable_of_separable | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"ball_subset_of_comp_subset",
"comp_symm_of_uniformity",
"exists_prop",
"is_open",
"is_open.mem_nhds",
"symmetric_rel",
"uniform_space.mem_ball_self",
"uniformity_has_basis_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`. We do not
register this as an instance, as there is alread... | 710 | 732 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_set_diagonal_eq_uniformity [compact_space α] : 𝓝ˢ (diagonal α) = 𝓤 α :=
begin
refine nhds_set_diagonal_le_uniformity.antisymm _,
have : (𝓤 (α × α)).has_basis (λ U, U ∈ 𝓤 α)
(λ U, (λ p : (α × α) × α × α, ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U),
{ rw [uniformity_prod_eq_comap_prod],
exact (𝓤 α... | nhds_set_diagonal_eq_uniformity [compact_space α] : 𝓝ˢ (diagonal α) = 𝓤 α | begin
refine nhds_set_diagonal_le_uniformity.antisymm _,
have : (𝓤 (α × α)).has_basis (λ U, U ∈ 𝓤 α)
(λ U, (λ p : (α × α) × α × α, ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U),
{ rw [uniformity_prod_eq_comap_prod],
exact (𝓤 α).basis_sets.prod_self.comap _ },
refine (is_compact_diagonal.nhds_set_basis_... | lemma | nhds_set_diagonal_eq_uniformity | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"refl_mem_uniformity",
"uniformity_prod_eq_comap_prod"
] | On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. | 50 | 59 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space_uniformity [compact_space α] : 𝓤 α = ⨆ x, 𝓝 (x, x) :=
nhds_set_diagonal_eq_uniformity.symm.trans (nhds_set_diagonal _) | compact_space_uniformity [compact_space α] : 𝓤 α = ⨆ x, 𝓝 (x, x) | nhds_set_diagonal_eq_uniformity.symm.trans (nhds_set_diagonal _) | lemma | compact_space_uniformity | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"nhds_set_diagonal"
] | On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. | 63 | 64 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_uniformity_of_compact [t : topological_space γ] [compact_space γ]
{u u' : uniform_space γ} (h : u.to_topological_space = t) (h' : u'.to_topological_space = t) :
u = u' :=
begin
apply uniform_space_eq,
change uniformity _ = uniformity _,
haveI : @compact_space γ u.to_topological_space, { rwa h },
have... | unique_uniformity_of_compact [t : topological_space γ] [compact_space γ]
{u u' : uniform_space γ} (h : u.to_topological_space = t) (h' : u'.to_topological_space = t) :
u = u' | begin
apply uniform_space_eq,
change uniformity _ = uniformity _,
haveI : @compact_space γ u.to_topological_space, { rwa h },
haveI : @compact_space γ u'.to_topological_space, { rwa h' },
rw [compact_space_uniformity, compact_space_uniformity, h, h']
end | lemma | unique_uniformity_of_compact | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"compact_space_uniformity",
"topological_space",
"uniform_space",
"uniform_space_eq",
"uniformity"
] | null | 66 | 75 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space_of_compact_t2 [topological_space γ] [compact_space γ] [t2_space γ] :
uniform_space γ :=
{ uniformity := 𝓝ˢ (diagonal γ),
refl := principal_le_nhds_set,
symm := continuous_swap.tendsto_nhds_set $ λ x, eq.symm,
comp := begin
/-
This is the difficult part of the proof. We need to prove that,... | uniform_space_of_compact_t2 [topological_space γ] [compact_space γ] [t2_space γ] :
uniform_space γ | { uniformity := 𝓝ˢ (diagonal γ),
refl := principal_le_nhds_set,
symm := continuous_swap.tendsto_nhds_set $ λ x, eq.symm,
comp := begin
/-
This is the difficult part of the proof. We need to prove that, for each neighborhood `W`
of the diagonal `Δ`, there exists a smaller neighborhood `V` such that `V... | def | uniform_space_of_compact_t2 | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"by_contra",
"closure",
"closure_compl",
"cluster_point_of_compact",
"cluster_pt",
"compact_space",
"compl_singleton_mem_nhds",
"disjoint_nested_nhds",
"em",
"filter.comap",
"interior",
"is_open",
"is_open.mem_nhds",
"is_open.prod",
"is_open.union",
"is_open_fold",
"is_open_iff_mem_n... | The unique uniform structure inducing a given compact topological structure. | 78 | 158 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space.uniform_continuous_of_continuous [compact_space α]
{f : α → β} (h : continuous f) : uniform_continuous f :=
have tendsto (prod.map f f) (𝓝ˢ (diagonal α)) (𝓝ˢ (diagonal β)),
from (h.prod_map h).tendsto_nhds_set maps_to_prod_map_diagonal,
(this.mono_left nhds_set_diagonal_eq_uniformity.ge).mono_right ... | compact_space.uniform_continuous_of_continuous [compact_space α]
{f : α → β} (h : continuous f) : uniform_continuous f | have tendsto (prod.map f f) (𝓝ˢ (diagonal α)) (𝓝ˢ (diagonal β)),
from (h.prod_map h).tendsto_nhds_set maps_to_prod_map_diagonal,
(this.mono_left nhds_set_diagonal_eq_uniformity.ge).mono_right nhds_set_diagonal_le_uniformity | lemma | compact_space.uniform_continuous_of_continuous | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"continuous",
"nhds_set_diagonal_le_uniformity",
"uniform_continuous"
] | Heine-Cantor: a continuous function on a compact uniform space is uniformly
continuous. | 166 | 170 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.uniform_continuous_on_of_continuous {s : set α} {f : α → β}
(hs : is_compact s) (hf : continuous_on f s) : uniform_continuous_on f s :=
begin
rw uniform_continuous_on_iff_restrict,
rw is_compact_iff_compact_space at hs,
rw continuous_on_iff_continuous_restrict at hf,
resetI,
exact compact_space.u... | is_compact.uniform_continuous_on_of_continuous {s : set α} {f : α → β}
(hs : is_compact s) (hf : continuous_on f s) : uniform_continuous_on f s | begin
rw uniform_continuous_on_iff_restrict,
rw is_compact_iff_compact_space at hs,
rw continuous_on_iff_continuous_restrict at hf,
resetI,
exact compact_space.uniform_continuous_of_continuous hf,
end | lemma | is_compact.uniform_continuous_on_of_continuous | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space.uniform_continuous_of_continuous",
"continuous_on",
"continuous_on_iff_continuous_restrict",
"is_compact",
"is_compact_iff_compact_space",
"uniform_continuous_on",
"uniform_continuous_on_iff_restrict"
] | Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly
continuous. | 174 | 182 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.uniform_continuous_at_of_continuous_at {r : set (β × β)} {s : set α}
(hs : is_compact s) (f : α → β) (hf : ∀ a ∈ s, continuous_at f a) (hr : r ∈ 𝓤 β) :
{x : α × α | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ 𝓤 α :=
begin
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr,
choose U hU T hT hb usin... | is_compact.uniform_continuous_at_of_continuous_at {r : set (β × β)} {s : set α}
(hs : is_compact s) (f : α → β) (hf : ∀ a ∈ s, continuous_at f a) (hr : r ∈ 𝓤 β) :
{x : α × α | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ 𝓤 α | begin
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr,
choose U hU T hT hb using λ a ha, exists_mem_nhds_ball_subset_of_mem_nhds
((hf a ha).preimage_mem_nhds $ mem_nhds_left _ ht),
obtain ⟨fs, hsU⟩ := hs.elim_nhds_subcover' U hU,
apply mem_of_superset ((bInter_finset_mem fs).2 $ λ a _, hT a ... | lemma | is_compact.uniform_continuous_at_of_continuous_at | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"comp_symm_mem_uniformity_sets",
"continuous_at",
"exists_mem_nhds_ball_subset_of_mem_nhds",
"is_compact",
"mem_nhds_left"
] | If `s` is compact and `f` is continuous at all points of `s`, then `f` is
"uniformly continuous at the set `s`", i.e. `f x` is close to `f y` whenever `x ∈ s` and `y` is
close to `x` (even if `y` is not itself in `s`, so this is a stronger assertion than
`uniform_continuous_on s`). | 188 | 202 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.uniform_continuous_of_tendsto_cocompact {f : α → β} {x : β}
(h_cont : continuous f) (hx : tendsto f (cocompact α) (𝓝 x)) : uniform_continuous f :=
uniform_continuous_def.2 $ λ r hr, begin
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr,
obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx $ mem_... | continuous.uniform_continuous_of_tendsto_cocompact {f : α → β} {x : β}
(h_cont : continuous f) (hx : tendsto f (cocompact α) (𝓝 x)) : uniform_continuous f | uniform_continuous_def.2 $ λ r hr, begin
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr,
obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx $ mem_nhds_left _ ht),
apply mem_of_superset (symmetrize_mem_uniformity $ hs.uniform_continuous_at_of_continuous_at
f (λ _ _, h_cont.continuous_at) $ symmetrize_... | lemma | continuous.uniform_continuous_of_tendsto_cocompact | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"comp_symm_mem_uniformity_sets",
"continuous",
"mem_nhds_left",
"symmetrize_mem_uniformity",
"uniform_continuous"
] | null | 204 | 216 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_mul_support.is_one_at_infty {f : α → γ} [topological_space γ] [has_one γ]
(h : has_compact_mul_support f) : tendsto f (cocompact α) (𝓝 1) :=
begin
-- porting note: move to src/topology/support.lean once the port is over
intros N hN,
rw [mem_map, mem_cocompact'],
refine ⟨mul_tsupport f, h.is_compa... | has_compact_mul_support.is_one_at_infty {f : α → γ} [topological_space γ] [has_one γ]
(h : has_compact_mul_support f) : tendsto f (cocompact α) (𝓝 1) | begin
-- porting note: move to src/topology/support.lean once the port is over
intros N hN,
rw [mem_map, mem_cocompact'],
refine ⟨mul_tsupport f, h.is_compact, _⟩,
rw compl_subset_comm,
intros v hv,
rw [mem_preimage, image_eq_one_of_nmem_mul_tsupport hv],
exact mem_of_mem_nhds hN,
end | lemma | has_compact_mul_support.is_one_at_infty | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"has_compact_mul_support",
"image_eq_one_of_nmem_mul_tsupport",
"mem_map",
"mem_of_mem_nhds",
"topological_space"
] | If `f` has compact multiplicative support, then `f` tends to 1 at infinity. | 219 | 231 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_mul_support.uniform_continuous_of_continuous {f : α → β} [has_one β]
(h1 : has_compact_mul_support f) (h2 : continuous f) : uniform_continuous f :=
h2.uniform_continuous_of_tendsto_cocompact h1.is_one_at_infty | has_compact_mul_support.uniform_continuous_of_continuous {f : α → β} [has_one β]
(h1 : has_compact_mul_support f) (h2 : continuous f) : uniform_continuous f | h2.uniform_continuous_of_tendsto_cocompact h1.is_one_at_infty | lemma | has_compact_mul_support.uniform_continuous_of_continuous | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"continuous",
"has_compact_mul_support",
"uniform_continuous"
] | null | 233 | 236 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.tendsto_uniformly [locally_compact_space α] [compact_space β]
[uniform_space γ] {f : α → β → γ} {x : α} {U : set α}
(hxU : U ∈ 𝓝 x) (h : continuous_on ↿f (U ×ˢ univ)) :
tendsto_uniformly f (f x) (𝓝 x) :=
begin
rcases locally_compact_space.local_compact_nhds _ _ hxU with ⟨K, hxK, hKU, hK⟩,
have... | continuous_on.tendsto_uniformly [locally_compact_space α] [compact_space β]
[uniform_space γ] {f : α → β → γ} {x : α} {U : set α}
(hxU : U ∈ 𝓝 x) (h : continuous_on ↿f (U ×ˢ univ)) :
tendsto_uniformly f (f x) (𝓝 x) | begin
rcases locally_compact_space.local_compact_nhds _ _ hxU with ⟨K, hxK, hKU, hK⟩,
have : uniform_continuous_on ↿f (K ×ˢ univ),
from is_compact.uniform_continuous_on_of_continuous (hK.prod is_compact_univ)
(h.mono $ prod_mono hKU subset.rfl),
exact this.tendsto_uniformly hxK
end | lemma | continuous_on.tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"continuous_on",
"is_compact.uniform_continuous_on_of_continuous",
"is_compact_univ",
"locally_compact_space",
"tendsto_uniformly",
"uniform_continuous_on",
"uniform_space"
] | A family of functions `α → β → γ` tends uniformly to its value at `x` if `α` is locally compact,
`β` is compact and `f` is continuous on `U × (univ : set β)` for some neighborhood `U` of `x`. | 240 | 250 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.tendsto_uniformly [locally_compact_space α] [compact_space β] [uniform_space γ]
(f : α → β → γ) (h : continuous ↿f) (x : α) : tendsto_uniformly f (f x) (𝓝 x) :=
h.continuous_on.tendsto_uniformly univ_mem | continuous.tendsto_uniformly [locally_compact_space α] [compact_space β] [uniform_space γ]
(f : α → β → γ) (h : continuous ↿f) (x : α) : tendsto_uniformly f (f x) (𝓝 x) | h.continuous_on.tendsto_uniformly univ_mem | lemma | continuous.tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"continuous",
"locally_compact_space",
"tendsto_uniformly",
"uniform_space"
] | A continuous family of functions `α → β → γ` tends uniformly to its value at `x` if `α` is
locally compact and `β` is compact. | 254 | 256 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space.uniform_equicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α}
[compact_space β] (h : equicontinuous F) :
uniform_equicontinuous F :=
begin
rw equicontinuous_iff_continuous at h,
rw uniform_equicontinuous_iff_uniform_continuous,
exact compact_space.uniform_continuous_of_continuous h
end | compact_space.uniform_equicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α}
[compact_space β] (h : equicontinuous F) :
uniform_equicontinuous F | begin
rw equicontinuous_iff_continuous at h,
rw uniform_equicontinuous_iff_uniform_continuous,
exact compact_space.uniform_continuous_of_continuous h
end | lemma | compact_space.uniform_equicontinuous_of_equicontinuous | topology.uniform_space | src/topology/uniform_space/compact.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.equicontinuity",
"topology.separation",
"topology.support"
] | [
"compact_space",
"compact_space.uniform_continuous_of_continuous",
"equicontinuous",
"equicontinuous_iff_continuous",
"uniform_equicontinuous",
"uniform_equicontinuous_iff_uniform_continuous"
] | An equicontinuous family of functions defined on a compact uniform space is automatically
uniformly equicontinuous. | 262 | 269 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd : set C(α, β) := { g | ∀ (x ∈ K), (f x, g x) ∈ V } | compact_conv_nhd : set C(α, β) | { g | ∀ (x ∈ K), (f x, g x) ∈ V } | def | continuous_map.compact_conv_nhd | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [] | Given `K ⊆ α`, `V ⊆ β × β`, and `f : C(α, β)`, we define `compact_conv_nhd K V f` to be the set
of `g : C(α, β)` that are `V`-close to `f` on `K`. | 92 | 92 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_mem_compact_conv_nhd (hV : V ∈ 𝓤 β) : f ∈ compact_conv_nhd K V f :=
λ x hx, refl_mem_uniformity hV | self_mem_compact_conv_nhd (hV : V ∈ 𝓤 β) : f ∈ compact_conv_nhd K V f | λ x hx, refl_mem_uniformity hV | lemma | continuous_map.self_mem_compact_conv_nhd | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"refl_mem_uniformity"
] | null | 96 | 97 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_mono {V' : set (β × β)} (hV' : V' ⊆ V) :
compact_conv_nhd K V' f ⊆ compact_conv_nhd K V f :=
λ x hx a ha, hV' (hx a ha) | compact_conv_nhd_mono {V' : set (β × β)} (hV' : V' ⊆ V) :
compact_conv_nhd K V' f ⊆ compact_conv_nhd K V f | λ x hx a ha, hV' (hx a ha) | lemma | continuous_map.compact_conv_nhd_mono | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [] | null | 99 | 101 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_mem_comp {g₁ g₂ : C(α, β)} {V' : set (β × β)}
(hg₁ : g₁ ∈ compact_conv_nhd K V f) (hg₂ : g₂ ∈ compact_conv_nhd K V' g₁) :
g₂ ∈ compact_conv_nhd K (V ○ V') f :=
λ x hx, ⟨g₁ x, hg₁ x hx, hg₂ x hx⟩ | compact_conv_nhd_mem_comp {g₁ g₂ : C(α, β)} {V' : set (β × β)}
(hg₁ : g₁ ∈ compact_conv_nhd K V f) (hg₂ : g₂ ∈ compact_conv_nhd K V' g₁) :
g₂ ∈ compact_conv_nhd K (V ○ V') f | λ x hx, ⟨g₁ x, hg₁ x hx, hg₂ x hx⟩ | lemma | continuous_map.compact_conv_nhd_mem_comp | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [] | null | 103 | 106 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_nhd_basis (hV : V ∈ 𝓤 β) :
∃ (V' ∈ 𝓤 β), V' ⊆ V ∧ ∀ (g ∈ compact_conv_nhd K V' f),
compact_conv_nhd K V' g ⊆ compact_conv_nhd K V f :=
begin
obtain ⟨V', h₁, h₂⟩ := comp_mem_uniformity_sets hV,
exact ⟨V', h₁, subset.trans (subset_comp_self_of_mem_uniformity h₁) h₂, λ g hg g' hg',
compact... | compact_conv_nhd_nhd_basis (hV : V ∈ 𝓤 β) :
∃ (V' ∈ 𝓤 β), V' ⊆ V ∧ ∀ (g ∈ compact_conv_nhd K V' f),
compact_conv_nhd K V' g ⊆ compact_conv_nhd K V f | begin
obtain ⟨V', h₁, h₂⟩ := comp_mem_uniformity_sets hV,
exact ⟨V', h₁, subset.trans (subset_comp_self_of_mem_uniformity h₁) h₂, λ g hg g' hg',
compact_conv_nhd_mono f h₂ (compact_conv_nhd_mem_comp f hg hg')⟩,
end | lemma | continuous_map.compact_conv_nhd_nhd_basis | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"comp_mem_uniformity_sets",
"subset_comp_self_of_mem_uniformity"
] | A key property of `compact_conv_nhd`. It allows us to apply
`topological_space.nhds_mk_of_nhds_filter_basis` below. | 110 | 117 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_subset_inter (K₁ K₂ : set α) (V₁ V₂ : set (β × β)) :
compact_conv_nhd (K₁ ∪ K₂) (V₁ ∩ V₂) f ⊆
compact_conv_nhd K₁ V₁ f ∩ compact_conv_nhd K₂ V₂ f :=
λ g hg, ⟨λ x hx, mem_of_mem_inter_left (hg x (mem_union_left K₂ hx)),
λ x hx, mem_of_mem_inter_right (hg x (mem_union_right K₁ hx))⟩ | compact_conv_nhd_subset_inter (K₁ K₂ : set α) (V₁ V₂ : set (β × β)) :
compact_conv_nhd (K₁ ∪ K₂) (V₁ ∩ V₂) f ⊆
compact_conv_nhd K₁ V₁ f ∩ compact_conv_nhd K₂ V₂ f | λ g hg, ⟨λ x hx, mem_of_mem_inter_left (hg x (mem_union_left K₂ hx)),
λ x hx, mem_of_mem_inter_right (hg x (mem_union_right K₁ hx))⟩ | lemma | continuous_map.compact_conv_nhd_subset_inter | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [] | null | 119 | 123 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_compact_entourage_nonempty :
{ KV : set α × set (β × β) | is_compact KV.1 ∧ KV.2 ∈ 𝓤 β }.nonempty :=
⟨⟨∅, univ⟩, is_compact_empty, filter.univ_mem⟩ | compact_conv_nhd_compact_entourage_nonempty :
{ KV : set α × set (β × β) | is_compact KV.1 ∧ KV.2 ∈ 𝓤 β }.nonempty | ⟨⟨∅, univ⟩, is_compact_empty, filter.univ_mem⟩ | lemma | continuous_map.compact_conv_nhd_compact_entourage_nonempty | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact",
"is_compact_empty"
] | null | 125 | 127 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_filter_is_basis : filter.is_basis
(λ (KV : set α × set (β × β)), is_compact KV.1 ∧ KV.2 ∈ 𝓤 β)
(λ KV, compact_conv_nhd KV.1 KV.2 f) :=
{ nonempty := compact_conv_nhd_compact_entourage_nonempty,
inter :=
begin
rintros ⟨K₁, V₁⟩ ⟨K₂, V₂⟩ ⟨hK₁, hV₁⟩ ⟨hK₂, hV₂⟩,
exact ⟨⟨K₁ ∪ K₂, V₁... | compact_conv_nhd_filter_is_basis : filter.is_basis
(λ (KV : set α × set (β × β)), is_compact KV.1 ∧ KV.2 ∈ 𝓤 β)
(λ KV, compact_conv_nhd KV.1 KV.2 f) | { nonempty := compact_conv_nhd_compact_entourage_nonempty,
inter :=
begin
rintros ⟨K₁, V₁⟩ ⟨K₂, V₂⟩ ⟨hK₁, hV₁⟩ ⟨hK₂, hV₂⟩,
exact ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, filter.inter_mem hV₁ hV₂⟩,
compact_conv_nhd_subset_inter f K₁ K₂ V₁ V₂⟩,
end, } | lemma | continuous_map.compact_conv_nhd_filter_is_basis | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter.inter_mem",
"filter.is_basis",
"is_compact"
] | null | 129 | 138 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_convergence_filter_basis (f : C(α, β)) : filter_basis C(α, β) :=
(compact_conv_nhd_filter_is_basis f).filter_basis | compact_convergence_filter_basis (f : C(α, β)) : filter_basis C(α, β) | (compact_conv_nhd_filter_is_basis f).filter_basis | def | continuous_map.compact_convergence_filter_basis | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter_basis"
] | A filter basis for the neighbourhood filter of a point in the compact-convergence topology. | 141 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_compact_convergence_nhd_filter (Y : set C(α, β)) :
Y ∈ (compact_convergence_filter_basis f).filter ↔
∃ (K : set α) (V : set (β × β)) (hK : is_compact K) (hV : V ∈ 𝓤 β), compact_conv_nhd K V f ⊆ Y :=
begin
split,
{ rintros ⟨X, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩,
exact ⟨K, V, hK, hV, hY⟩, },
{ rintros ⟨K, V,... | mem_compact_convergence_nhd_filter (Y : set C(α, β)) :
Y ∈ (compact_convergence_filter_basis f).filter ↔
∃ (K : set α) (V : set (β × β)) (hK : is_compact K) (hV : V ∈ 𝓤 β), compact_conv_nhd K V f ⊆ Y | begin
split,
{ rintros ⟨X, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩,
exact ⟨K, V, hK, hV, hY⟩, },
{ rintros ⟨K, V, hK, hV, hY⟩,
exact ⟨compact_conv_nhd K V f, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩, },
end | lemma | continuous_map.mem_compact_convergence_nhd_filter | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter",
"is_compact"
] | null | 144 | 153 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_convergence_topology : topological_space C(α, β) :=
topological_space.mk_of_nhds $ λ f, (compact_convergence_filter_basis f).filter | compact_convergence_topology : topological_space C(α, β) | topological_space.mk_of_nhds $ λ f, (compact_convergence_filter_basis f).filter | def | continuous_map.compact_convergence_topology | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter",
"topological_space",
"topological_space.mk_of_nhds"
] | The compact-convergence topology. In fact, see `compact_open_eq_compact_convergence` this is
the same as the compact-open topology. This definition is thus an auxiliary convenience definition
and is unlikely to be of direct use. | 158 | 159 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_compact_convergence :
@nhds _ compact_convergence_topology f = (compact_convergence_filter_basis f).filter :=
begin
rw topological_space.nhds_mk_of_nhds_filter_basis;
rintros g - ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩,
{ exact self_mem_compact_conv_nhd g hV, },
{ obtain ⟨V', hV', h₁, h₂⟩ := compact_conv_nhd_nhd_basis g... | nhds_compact_convergence :
@nhds _ compact_convergence_topology f = (compact_convergence_filter_basis f).filter | begin
rw topological_space.nhds_mk_of_nhds_filter_basis;
rintros g - ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩,
{ exact self_mem_compact_conv_nhd g hV, },
{ obtain ⟨V', hV', h₁, h₂⟩ := compact_conv_nhd_nhd_basis g hV,
exact ⟨compact_conv_nhd K V' g, ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, compact_conv_nhd_mono g h₁,
λ g' hg', ⟨comp... | lemma | continuous_map.nhds_compact_convergence | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter",
"nhds",
"topological_space.nhds_mk_of_nhds_filter_basis"
] | null | 161 | 170 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_compact_convergence :
has_basis (@nhds _ compact_convergence_topology f)
(λ (p : set α × set (β × β)), is_compact p.1 ∧ p.2 ∈ 𝓤 β) (λ p, compact_conv_nhd p.1 p.2 f) :=
(nhds_compact_convergence f).symm ▸ (compact_conv_nhd_filter_is_basis f).has_basis | has_basis_nhds_compact_convergence :
has_basis (@nhds _ compact_convergence_topology f)
(λ (p : set α × set (β × β)), is_compact p.1 ∧ p.2 ∈ 𝓤 β) (λ p, compact_conv_nhd p.1 p.2 f) | (nhds_compact_convergence f).symm ▸ (compact_conv_nhd_filter_is_basis f).has_basis | lemma | continuous_map.has_basis_nhds_compact_convergence | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact",
"nhds"
] | null | 172 | 175 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_forall_compact_tendsto_uniformly_on'
{ι : Type u₃} {p : filter ι} {F : ι → C(α, β)} :
filter.tendsto F p (@nhds _ compact_convergence_topology f) ↔
∀ K, is_compact K → tendsto_uniformly_on (λ i a, F i a) f p K :=
begin
simp only [(has_basis_nhds_compact_convergence f).tendsto_right_iff, tendsto_unif... | tendsto_iff_forall_compact_tendsto_uniformly_on'
{ι : Type u₃} {p : filter ι} {F : ι → C(α, β)} :
filter.tendsto F p (@nhds _ compact_convergence_topology f) ↔
∀ K, is_compact K → tendsto_uniformly_on (λ i a, F i a) f p K | begin
simp only [(has_basis_nhds_compact_convergence f).tendsto_right_iff, tendsto_uniformly_on,
and_imp, prod.forall],
refine forall_congr (λ K, _),
rw forall_swap,
exact forall₃_congr (λ hK V hV, iff.rfl),
end | lemma | continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"and_imp",
"filter",
"filter.tendsto",
"forall_swap",
"forall₃_congr",
"is_compact",
"nhds",
"tendsto_uniformly_on"
] | This is an auxiliary lemma and is unlikely to be of direct use outside of this file. See
`tendsto_iff_forall_compact_tendsto_uniformly_on` below for the useful version where the topology
is picked up via typeclass inference. | 180 | 190 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_conv_nhd_subset_compact_open (hK : is_compact K) {U : set β} (hU : is_open U)
(hf : f ∈ compact_open.gen K U) :
∃ (V ∈ 𝓤 β), is_open V ∧ compact_conv_nhd K V f ⊆ compact_open.gen K U :=
begin
obtain ⟨V, hV₁, hV₂, hV₃⟩ := lebesgue_number_of_compact_open (hK.image f.continuous) hU hf,
refine ⟨V, hV₁, hV₂... | compact_conv_nhd_subset_compact_open (hK : is_compact K) {U : set β} (hU : is_open U)
(hf : f ∈ compact_open.gen K U) :
∃ (V ∈ 𝓤 β), is_open V ∧ compact_conv_nhd K V f ⊆ compact_open.gen K U | begin
obtain ⟨V, hV₁, hV₂, hV₃⟩ := lebesgue_number_of_compact_open (hK.image f.continuous) hU hf,
refine ⟨V, hV₁, hV₂, _⟩,
rintros g hg _ ⟨x, hx, rfl⟩,
exact hV₃ (f x) ⟨x, hx, rfl⟩ (hg x hx),
end | lemma | continuous_map.compact_conv_nhd_subset_compact_open | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact",
"is_open",
"lebesgue_number_of_compact_open"
] | Any point of `compact_open.gen K U` is also an interior point wrt the topology of compact
convergence.
The topology of compact convergence is thus at least as fine as the compact-open topology. | 196 | 204 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Inter_compact_open_gen_subset_compact_conv_nhd (hK : is_compact K) (hV : V ∈ 𝓤 β) :
∃ (ι : Sort (u₁ + 1)) [fintype ι]
(C : ι → set α) (hC : ∀ i, is_compact (C i))
(U : ι → set β) (hU : ∀ i, is_open (U i)),
(f ∈ ⋂ i, compact_open.gen (C i) (U i)) ∧
(⋂ i, compact_open.gen (C i) (U i)) ⊆ compact_conv_nhd K V f ... | Inter_compact_open_gen_subset_compact_conv_nhd (hK : is_compact K) (hV : V ∈ 𝓤 β) :
∃ (ι : Sort (u₁ + 1)) [fintype ι]
(C : ι → set α) (hC : ∀ i, is_compact (C i))
(U : ι → set β) (hU : ∀ i, is_open (U i)),
(f ∈ ⋂ i, compact_open.gen (C i) (U i)) ∧
(⋂ i, compact_open.gen (C i) (U i)) ⊆ compact_conv_nhd K V f | begin
obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV,
obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁,
let U : α → set α := λ x, f⁻¹' (ball (f x) Z),
have hU : ∀ x, is_open (U x) := λ x, f.continuous.is_open_preimage _ (is_open_ball _ hZ₄),
have hUK : K ⊆ ⋃ (x... | lemma | continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"ball_mono",
"closure",
"closure_mono",
"comp_open_symm_mem_uniformity_sets",
"exists_prop",
"fintype",
"is_closed_closure",
"is_compact",
"is_open",
"mem_ball_comp",
"mem_ball_symmetry",
"set_coe.forall",
"subset_closure",
"subtype.coe_mk"
] | The point `f` in `compact_conv_nhd K V f` is also an interior point wrt the compact-open
topology.
Since `compact_conv_nhd K V f` are a neighbourhood basis at `f` for each `f`, it follows that
the compact-open topology is at least as fine as the topology of compact convergence. | 211 | 253 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_open_eq_compact_convergence :
continuous_map.compact_open = (compact_convergence_topology : topological_space C(α, β)) :=
begin
rw [compact_convergence_topology, continuous_map.compact_open],
refine le_antisymm _ _,
{ refine λ X hX, is_open_iff_forall_mem_open.mpr (λ f hf, _),
have hXf : X ∈ (compac... | compact_open_eq_compact_convergence :
continuous_map.compact_open = (compact_convergence_topology : topological_space C(α, β)) | begin
rw [compact_convergence_topology, continuous_map.compact_open],
refine le_antisymm _ _,
{ refine λ X hX, is_open_iff_forall_mem_open.mpr (λ f hf, _),
have hXf : X ∈ (compact_convergence_filter_basis f).filter,
{ rw ← nhds_compact_convergence,
exact @is_open.mem_nhds C(α, β) compact_convergence... | lemma | continuous_map.compact_open_eq_compact_convergence | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"and_imp",
"continuous_map.compact_open",
"continuous_map.is_open_gen",
"exists_prop",
"filter",
"filter.mem_of_superset",
"filter_basis.mem_filter_of_mem",
"forall_exists_index",
"is_open.mem_nhds",
"is_open_Inter",
"topological_space",
"topological_space.le_generate_from_iff_subset_is_open"
... | The compact-open topology is equal to the compact-convergence topology. | 256 | 275 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_convergence_uniformity : filter (C(α, β) × C(α, β)) :=
⨅ KV ∈ { KV : set α × set (β × β) | is_compact KV.1 ∧ KV.2 ∈ 𝓤 β },
𝓟 { fg : C(α, β) × C(α, β) | ∀ (x : α), x ∈ KV.1 → (fg.1 x, fg.2 x) ∈ KV.2 } | compact_convergence_uniformity : filter (C(α, β) × C(α, β)) | ⨅ KV ∈ { KV : set α × set (β × β) | is_compact KV.1 ∧ KV.2 ∈ 𝓤 β },
𝓟 { fg : C(α, β) × C(α, β) | ∀ (x : α), x ∈ KV.1 → (fg.1 x, fg.2 x) ∈ KV.2 } | def | continuous_map.compact_convergence_uniformity | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter",
"is_compact"
] | The filter on `C(α, β) × C(α, β)` which underlies the uniform space structure on `C(α, β)`. | 278 | 280 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_compact_convergence_uniformity_aux :
has_basis (@compact_convergence_uniformity α β _ _)
(λ p : set α × set (β × β), is_compact p.1 ∧ p.2 ∈ 𝓤 β)
(λ p, { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 }) :=
begin
refine filter.has_basis_binfi_principal _ compact_conv_nhd_compact_entour... | has_basis_compact_convergence_uniformity_aux :
has_basis (@compact_convergence_uniformity α β _ _)
(λ p : set α × set (β × β), is_compact p.1 ∧ p.2 ∈ 𝓤 β)
(λ p, { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 }) | begin
refine filter.has_basis_binfi_principal _ compact_conv_nhd_compact_entourage_nonempty,
rintros ⟨K₁, V₁⟩ ⟨hK₁, hV₁⟩ ⟨K₂, V₂⟩ ⟨hK₂, hV₂⟩,
refine ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, filter.inter_mem hV₁ hV₂⟩, _⟩,
simp only [le_eq_subset, prod.forall, set_of_subset_set_of, ge_iff_le, order.preimage,
← ... | lemma | continuous_map.has_basis_compact_convergence_uniformity_aux | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"filter.has_basis_binfi_principal",
"filter.inter_mem",
"forall_and_distrib",
"forall_imp",
"ge_iff_le",
"is_compact",
"order.preimage"
] | null | 282 | 293 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_compact_convergence_uniformity (X : set (C(α, β) × C(α, β))) :
X ∈ @compact_convergence_uniformity α β _ _ ↔
∃ (K : set α) (V : set (β × β)) (hK : is_compact K) (hV : V ∈ 𝓤 β),
{ fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
by simp only [has_basis_compact_convergence_uniformity_aux.mem_i... | mem_compact_convergence_uniformity (X : set (C(α, β) × C(α, β))) :
X ∈ @compact_convergence_uniformity α β _ _ ↔
∃ (K : set α) (V : set (β × β)) (hK : is_compact K) (hV : V ∈ 𝓤 β),
{ fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X | by simp only [has_basis_compact_convergence_uniformity_aux.mem_iff, exists_prop, prod.exists,
and_assoc] | lemma | continuous_map.mem_compact_convergence_uniformity | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"exists_prop",
"is_compact"
] | An intermediate lemma. Usually `mem_compact_convergence_entourage_iff` is more useful. | 296 | 301 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_convergence_uniform_space : uniform_space C(α, β) :=
{ uniformity := compact_convergence_uniformity,
refl :=
begin
simp only [compact_convergence_uniformity, and_imp, filter.le_principal_iff, prod.forall,
filter.mem_principal, mem_set_of_eq, le_infi_iff, id_rel_subset],
exact λ K V hK ... | compact_convergence_uniform_space : uniform_space C(α, β) | { uniformity := compact_convergence_uniformity,
refl :=
begin
simp only [compact_convergence_uniformity, and_imp, filter.le_principal_iff, prod.forall,
filter.mem_principal, mem_set_of_eq, le_infi_iff, id_rel_subset],
exact λ K V hK hV f x hx, refl_mem_uniformity hV,
end,
symm :=
beg... | instance | continuous_map.compact_convergence_uniform_space | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"and_imp",
"comp_mem_uniformity_sets",
"exists₄_congr",
"filter.eventually_of_mem",
"filter.le_principal_iff",
"filter.mem_lift'",
"filter.mem_of_superset",
"filter.mem_principal",
"filter.tendsto_infi",
"filter.tendsto_principal",
"forall₂_congr",
"id_rel_subset",
"is_open_uniformity",
"l... | Note that we ensure the induced topology is definitionally the compact-open topology. | 304 | 346 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_compact_convergence_entourage_iff (X : set (C(α, β) × C(α, β))) :
X ∈ 𝓤 C(α, β) ↔ ∃ (K : set α) (V : set (β × β)) (hK : is_compact K) (hV : V ∈ 𝓤 β),
{ fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
mem_compact_convergence_uniformity X | mem_compact_convergence_entourage_iff (X : set (C(α, β) × C(α, β))) :
X ∈ 𝓤 C(α, β) ↔ ∃ (K : set α) (V : set (β × β)) (hK : is_compact K) (hV : V ∈ 𝓤 β),
{ fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X | mem_compact_convergence_uniformity X | lemma | continuous_map.mem_compact_convergence_entourage_iff | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact"
] | null | 348 | 351 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_compact_convergence_uniformity :
has_basis (𝓤 C(α, β)) (λ p : set α × set (β × β), is_compact p.1 ∧ p.2 ∈ 𝓤 β)
(λ p, { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 }) :=
has_basis_compact_convergence_uniformity_aux | has_basis_compact_convergence_uniformity :
has_basis (𝓤 C(α, β)) (λ p : set α × set (β × β), is_compact p.1 ∧ p.2 ∈ 𝓤 β)
(λ p, { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 }) | has_basis_compact_convergence_uniformity_aux | lemma | continuous_map.has_basis_compact_convergence_uniformity | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact"
] | null | 353 | 356 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.filter.has_basis.compact_convergence_uniformity {ι : Type*} {pi : ι → Prop}
{s : ι → set (β × β)} (h : (𝓤 β).has_basis pi s) :
has_basis (𝓤 C(α, β)) (λ p : set α × ι, is_compact p.1 ∧ pi p.2)
(λ p, { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 }) :=
begin
refine has_basis_compact_con... | _root_.filter.has_basis.compact_convergence_uniformity {ι : Type*} {pi : ι → Prop}
{s : ι → set (β × β)} (h : (𝓤 β).has_basis pi s) :
has_basis (𝓤 C(α, β)) (λ p : set α × ι, is_compact p.1 ∧ pi p.2)
(λ p, { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 }) | begin
refine has_basis_compact_convergence_uniformity.to_has_basis _ _,
{ rintro ⟨t₁, t₂⟩ ⟨h₁, h₂⟩,
rcases h.mem_iff.1 h₂ with ⟨i, hpi, hi⟩,
exact ⟨(t₁, i), ⟨h₁, hpi⟩, λ fg hfg x hx, hi (hfg _ hx)⟩ },
{ rintro ⟨t, i⟩ ⟨ht, hi⟩,
exact ⟨(t, s i), ⟨ht, h.mem_of_mem hi⟩, subset.rfl⟩ }
end | lemma | filter.has_basis.compact_convergence_uniformity | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact"
] | null | 358 | 369 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_forall_compact_tendsto_uniformly_on :
tendsto F p (𝓝 f) ↔ ∀ K, is_compact K → tendsto_uniformly_on (λ i a, F i a) f p K :=
by rw [compact_open_eq_compact_convergence, tendsto_iff_forall_compact_tendsto_uniformly_on'] | tendsto_iff_forall_compact_tendsto_uniformly_on :
tendsto F p (𝓝 f) ↔ ∀ K, is_compact K → tendsto_uniformly_on (λ i a, F i a) f p K | by rw [compact_open_eq_compact_convergence, tendsto_iff_forall_compact_tendsto_uniformly_on'] | lemma | continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact",
"tendsto_uniformly_on"
] | null | 373 | 375 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_tendsto_locally_uniformly
(h : tendsto_locally_uniformly (λ i a, F i a) f p) : tendsto F p (𝓝 f) :=
begin
rw tendsto_iff_forall_compact_tendsto_uniformly_on,
intros K hK,
rw ← tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact hK,
exact h.tendsto_locally_uniformly_on,
end | tendsto_of_tendsto_locally_uniformly
(h : tendsto_locally_uniformly (λ i a, F i a) f p) : tendsto F p (𝓝 f) | begin
rw tendsto_iff_forall_compact_tendsto_uniformly_on,
intros K hK,
rw ← tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact hK,
exact h.tendsto_locally_uniformly_on,
end | lemma | continuous_map.tendsto_of_tendsto_locally_uniformly | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on_iff_tendsto_uniformly_on_of_compact"
] | Locally uniform convergence implies convergence in the compact-open topology. | 378 | 385 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_of_tendsto
(hα : ∀ x : α, ∃ n, is_compact n ∧ n ∈ 𝓝 x) (h : tendsto F p (𝓝 f)) :
tendsto_locally_uniformly (λ i a, F i a) f p :=
begin
rw tendsto_iff_forall_compact_tendsto_uniformly_on at h,
intros V hV x,
obtain ⟨n, hn₁, hn₂⟩ := hα x,
exact ⟨n, hn₂, h n hn₁ V hV⟩,
end | tendsto_locally_uniformly_of_tendsto
(hα : ∀ x : α, ∃ n, is_compact n ∧ n ∈ 𝓝 x) (h : tendsto F p (𝓝 f)) :
tendsto_locally_uniformly (λ i a, F i a) f p | begin
rw tendsto_iff_forall_compact_tendsto_uniformly_on at h,
intros V hV x,
obtain ⟨n, hn₁, hn₂⟩ := hα x,
exact ⟨n, hn₂, h n hn₁ V hV⟩,
end | lemma | continuous_map.tendsto_locally_uniformly_of_tendsto | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact",
"tendsto_locally_uniformly"
] | If every point has a compact neighbourhood, then convergence in the compact-open topology
implies locally uniform convergence.
See also `tendsto_iff_tendsto_locally_uniformly`, especially for T2 spaces. | 391 | 399 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_tendsto_locally_uniformly [locally_compact_space α] :
tendsto F p (𝓝 f) ↔ tendsto_locally_uniformly (λ i a, F i a) f p :=
⟨tendsto_locally_uniformly_of_tendsto exists_compact_mem_nhds, tendsto_of_tendsto_locally_uniformly⟩ | tendsto_iff_tendsto_locally_uniformly [locally_compact_space α] :
tendsto F p (𝓝 f) ↔ tendsto_locally_uniformly (λ i a, F i a) f p | ⟨tendsto_locally_uniformly_of_tendsto exists_compact_mem_nhds, tendsto_of_tendsto_locally_uniformly⟩ | lemma | continuous_map.tendsto_iff_tendsto_locally_uniformly | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"exists_compact_mem_nhds",
"locally_compact_space",
"tendsto_locally_uniformly"
] | Convergence in the compact-open topology is the same as locally uniform convergence on a locally
compact space.
For non-T2 spaces, the assumption `locally_compact_space α` is stronger than we need and in fact
the `←` direction is true unconditionally. See `tendsto_locally_uniformly_of_tendsto` and
`tendsto_of_tendsto_... | 407 | 409 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_compact_convergence_uniformity_of_compact :
has_basis (𝓤 C(α, β)) (λ V : set (β × β), V ∈ 𝓤 β)
(λ V, { fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V }) :=
has_basis_compact_convergence_uniformity.to_has_basis
(λ p hp, ⟨p.2, hp.2, λ fg hfg x hx, hfg x⟩)
(λ V hV, ⟨⟨univ, V⟩, ⟨is_compact... | has_basis_compact_convergence_uniformity_of_compact :
has_basis (𝓤 C(α, β)) (λ V : set (β × β), V ∈ 𝓤 β)
(λ V, { fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V }) | has_basis_compact_convergence_uniformity.to_has_basis
(λ p hp, ⟨p.2, hp.2, λ fg hfg x hx, hfg x⟩)
(λ V hV, ⟨⟨univ, V⟩, ⟨is_compact_univ, hV⟩, λ fg hfg x, hfg x (mem_univ x)⟩) | lemma | continuous_map.has_basis_compact_convergence_uniformity_of_compact | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [] | null | 415 | 420 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_tendsto_uniformly :
tendsto F p (𝓝 f) ↔ tendsto_uniformly (λ i a, F i a) f p :=
begin
rw [tendsto_iff_forall_compact_tendsto_uniformly_on, ← tendsto_uniformly_on_univ],
exact ⟨λ h, h univ is_compact_univ, λ h K hK, h.mono (subset_univ K)⟩,
end | tendsto_iff_tendsto_uniformly :
tendsto F p (𝓝 f) ↔ tendsto_uniformly (λ i a, F i a) f p | begin
rw [tendsto_iff_forall_compact_tendsto_uniformly_on, ← tendsto_uniformly_on_univ],
exact ⟨λ h, h univ is_compact_univ, λ h K hK, h.mono (subset_univ K)⟩,
end | lemma | continuous_map.tendsto_iff_tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/compact_convergence.lean | [
"topology.compact_open",
"topology.uniform_space.uniform_convergence"
] | [
"is_compact_univ",
"tendsto_uniformly",
"tendsto_uniformly_on_univ"
] | Convergence in the compact-open topology is the same as uniform convergence for sequences of
continuous functions on a compact space. | 424 | 429 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rat.uniform_space_eq :
(absolute_value.abs : absolute_value ℚ ℚ).uniform_space = pseudo_metric_space.to_uniform_space :=
begin
ext s,
rw [(absolute_value.has_basis_uniformity _).mem_iff, metric.uniformity_basis_dist_rat.mem_iff],
simp only [rat.dist_eq, absolute_value.abs_apply, ← rat.cast_sub, ← rat.cast_abs, ... | rat.uniform_space_eq :
(absolute_value.abs : absolute_value ℚ ℚ).uniform_space = pseudo_metric_space.to_uniform_space | begin
ext s,
rw [(absolute_value.has_basis_uniformity _).mem_iff, metric.uniformity_basis_dist_rat.mem_iff],
simp only [rat.dist_eq, absolute_value.abs_apply, ← rat.cast_sub, ← rat.cast_abs, rat.cast_lt,
abs_sub_comm]
end | lemma | rat.uniform_space_eq | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"abs_sub_comm",
"absolute_value",
"absolute_value.abs",
"absolute_value.has_basis_uniformity",
"rat.cast_abs",
"rat.cast_lt",
"rat.cast_sub",
"rat.dist_eq",
"uniform_space"
] | The metric space uniform structure on ℚ (which presupposes the existence
of real numbers) agrees with the one coming directly from (abs : ℚ → ℚ). | 60 | 67 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rational_cau_seq_pkg : @abstract_completion ℚ $ (@absolute_value.abs ℚ _).uniform_space :=
{ space := ℝ,
coe := (coe : ℚ → ℝ),
uniform_struct := by apply_instance,
complete := by apply_instance,
separation := by apply_instance,
uniform_inducing := by { rw rat.uniform_space_eq,
exa... | rational_cau_seq_pkg : @abstract_completion ℚ $ (@absolute_value.abs ℚ _).uniform_space | { space := ℝ,
coe := (coe : ℚ → ℝ),
uniform_struct := by apply_instance,
complete := by apply_instance,
separation := by apply_instance,
uniform_inducing := by { rw rat.uniform_space_eq,
exact rat.uniform_embedding_coe_real.to_uniform_inducing },
dense := rat.dense_embedding_coe... | def | rational_cau_seq_pkg | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"absolute_value.abs",
"abstract_completion",
"dense",
"rat.uniform_space_eq",
"uniform_inducing",
"uniform_space"
] | Cauchy reals packaged as a completion of ℚ using the absolute value route. | 70 | 78 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q := ℚ | Q | ℚ | def | compare_reals.Q | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [] | Type wrapper around ℚ to make sure the absolute value uniform space instance is picked up
instead of the metric space one. We proved in rat.uniform_space_eq that they are equal,
but they are not definitionaly equal, so it would confuse the type class system (and probably
also human readers). | 85 | 85 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: uniform_space Q := (@absolute_value.abs ℚ _).uniform_space | : uniform_space Q | (@absolute_value.abs ℚ _).uniform_space | instance | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"absolute_value.abs",
"uniform_space"
] | null | 87 | 87 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Bourbakiℝ : Type := completion Q | Bourbakiℝ : Type | completion Q | def | compare_reals.Bourbakiℝ | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [] | Real numbers constructed as in Bourbaki. | 90 | 91 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bourbaki.uniform_space: uniform_space Bourbakiℝ := completion.uniform_space Q | bourbaki.uniform_space: uniform_space Bourbakiℝ | completion.uniform_space Q | instance | compare_reals.bourbaki.uniform_space | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"uniform_space"
] | null | 93 | 93 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Bourbaki_pkg : abstract_completion Q := completion.cpkg | Bourbaki_pkg : abstract_completion Q | completion.cpkg | def | compare_reals.Bourbaki_pkg | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"abstract_completion"
] | Bourbaki reals packaged as a completion of Q using the general theory. | 96 | 96 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compare_equiv : Bourbakiℝ ≃ᵤ ℝ :=
Bourbaki_pkg.compare_equiv rational_cau_seq_pkg | compare_equiv : Bourbakiℝ ≃ᵤ ℝ | Bourbaki_pkg.compare_equiv rational_cau_seq_pkg | def | compare_reals.compare_equiv | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"rational_cau_seq_pkg"
] | The uniform bijection between Bourbaki and Cauchy reals. | 99 | 100 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compare_uc : uniform_continuous (compare_equiv) :=
Bourbaki_pkg.uniform_continuous_compare_equiv _ | compare_uc : uniform_continuous (compare_equiv) | Bourbaki_pkg.uniform_continuous_compare_equiv _ | lemma | compare_reals.compare_uc | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"uniform_continuous"
] | null | 102 | 103 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compare_uc_symm : uniform_continuous (compare_equiv).symm :=
Bourbaki_pkg.uniform_continuous_compare_equiv_symm _ | compare_uc_symm : uniform_continuous (compare_equiv).symm | Bourbaki_pkg.uniform_continuous_compare_equiv_symm _ | lemma | compare_reals.compare_uc_symm | topology.uniform_space | src/topology/uniform_space/compare_reals.lean | [
"topology.uniform_space.absolute_value",
"topology.instances.real",
"topology.instances.rat",
"topology.uniform_space.completion"
] | [
"uniform_continuous"
] | null | 105 | 106 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_complete.is_closed [uniform_space α] [separated_space α] {s : set α} (h : is_complete s) :
is_closed s :=
is_closed_iff_cluster_pt.2 $ λ a ha, begin
let f := 𝓝[s] a,
have : cauchy f := cauchy_nhds.mono' ha inf_le_left,
rcases h f this (inf_le_right) with ⟨y, ys, fy⟩,
rwa (tendsto_nhds_unique' ha inf_le_l... | is_complete.is_closed [uniform_space α] [separated_space α] {s : set α} (h : is_complete s) :
is_closed s | is_closed_iff_cluster_pt.2 $ λ a ha, begin
let f := 𝓝[s] a,
have : cauchy f := cauchy_nhds.mono' ha inf_le_left,
rcases h f this (inf_le_right) with ⟨y, ys, fy⟩,
rwa (tendsto_nhds_unique' ha inf_le_left fy : a = y)
end | lemma | is_complete.is_closed | topology.uniform_space | src/topology/uniform_space/complete_separated.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"inf_le_left",
"inf_le_right",
"is_closed",
"is_complete",
"separated_space",
"tendsto_nhds_unique'",
"uniform_space"
] | null | 25 | 32 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_extend_of_cauchy {e : α → β} {f : α → γ}
(de : dense_inducing e) (h : ∀ b : β, cauchy (map f (comap e $ 𝓝 b))) :
continuous (de.extend f) :=
de.continuous_extend $ λ b, complete_space.complete (h b) | continuous_extend_of_cauchy {e : α → β} {f : α → γ}
(de : dense_inducing e) (h : ∀ b : β, cauchy (map f (comap e $ 𝓝 b))) :
continuous (de.extend f) | de.continuous_extend $ λ b, complete_space.complete (h b) | lemma | dense_inducing.continuous_extend_of_cauchy | topology.uniform_space | src/topology/uniform_space/complete_separated.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"continuous",
"dense_inducing"
] | null | 39 | 42 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Cauchy (α : Type u) [uniform_space α] : Type u := { f : filter α // cauchy f } | Cauchy (α : Type u) [uniform_space α] : Type u | { f : filter α // cauchy f } | def | Cauchy | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"cauchy",
"filter",
"uniform_space"
] | Space of Cauchy filters
This is essentially the completion of a uniform space. The embeddings are the neighbourhood filters.
This space is not minimal, the separated uniform space (i.e. quotiented on the intersection of all
entourages) is necessary for this. | 58 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen (s : set (α × α)) : set (Cauchy α × Cauchy α) :=
{p | s ∈ p.1.val ×ᶠ p.2.val } | gen (s : set (α × α)) : set (Cauchy α × Cauchy α) | {p | s ∈ p.1.val ×ᶠ p.2.val } | def | Cauchy.gen | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy"
] | The pairs of Cauchy filters generated by a set. | 68 | 69 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_gen : monotone gen :=
monotone_set_of $ assume p, @filter.monotone_mem _ (p.1.val ×ᶠ p.2.val) | monotone_gen : monotone gen | monotone_set_of $ assume p, @filter.monotone_mem _ (p.1.val ×ᶠ p.2.val) | lemma | Cauchy.monotone_gen | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"filter.monotone_mem",
"monotone"
] | null | 71 | 72 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_gen : map prod.swap ((𝓤 α).lift' gen) ≤ (𝓤 α).lift' gen :=
calc map prod.swap ((𝓤 α).lift' gen) =
(𝓤 α).lift' (λs:set (α×α), {p | s ∈ p.2.val ×ᶠ p.1.val }) :
begin
delta gen,
simp [map_lift'_eq, monotone_set_of, filter.monotone_mem,
function.comp, image_swap_eq_preimage_swap, -subtype.val... | symm_gen : map prod.swap ((𝓤 α).lift' gen) ≤ (𝓤 α).lift' gen | calc map prod.swap ((𝓤 α).lift' gen) =
(𝓤 α).lift' (λs:set (α×α), {p | s ∈ p.2.val ×ᶠ p.1.val }) :
begin
delta gen,
simp [map_lift'_eq, monotone_set_of, filter.monotone_mem,
function.comp, image_swap_eq_preimage_swap, -subtype.val_eq_coe]
end
... ≤ (𝓤 α).lift' gen :
uniformity_lift_le_s... | lemma | Cauchy.symm_gen | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"filter.monotone_mem",
"filter.prod_comm",
"le_rfl",
"prod.swap",
"subtype.val_eq_coe",
"uniformity_lift_le_swap"
] | null | 74 | 90 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_rel_gen_gen_subset_gen_comp_rel {s t : set (α×α)} : comp_rel (gen s) (gen t) ⊆
(gen (comp_rel s t) : set (Cauchy α × Cauchy α)) :=
assume ⟨f, g⟩ ⟨h, h₁, h₂⟩,
let ⟨t₁, (ht₁ : t₁ ∈ f.val), t₂, (ht₂ : t₂ ∈ h.val), (h₁ : t₁ ×ˢ t₂ ⊆ s)⟩ :=
mem_prod_iff.mp h₁ in
let ⟨t₃, (ht₃ : t₃ ∈ h.val), t₄, (ht₄ : t₄ ∈ g.val), (... | comp_rel_gen_gen_subset_gen_comp_rel {s t : set (α×α)} : comp_rel (gen s) (gen t) ⊆
(gen (comp_rel s t) : set (Cauchy α × Cauchy α)) | assume ⟨f, g⟩ ⟨h, h₁, h₂⟩,
let ⟨t₁, (ht₁ : t₁ ∈ f.val), t₂, (ht₂ : t₂ ∈ h.val), (h₁ : t₁ ×ˢ t₂ ⊆ s)⟩ :=
mem_prod_iff.mp h₁ in
let ⟨t₃, (ht₃ : t₃ ∈ h.val), t₄, (ht₄ : t₄ ∈ g.val), (h₂ : t₃ ×ˢ t₄ ⊆ t)⟩ :=
mem_prod_iff.mp h₂ in
have t₂ ∩ t₃ ∈ h.val,
from inter_mem ht₂ ht₃,
let ⟨x, xt₂, xt₃⟩ :=
h.property.left.none... | lemma | Cauchy.comp_rel_gen_gen_subset_gen_comp_rel | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"comp_rel"
] | null | 92 | 108 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_gen :
((𝓤 α).lift' gen).lift' (λs, comp_rel s s) ≤ (𝓤 α).lift' gen :=
calc ((𝓤 α).lift' gen).lift' (λs, comp_rel s s) =
(𝓤 α).lift' (λs, comp_rel (gen s) (gen s)) :
begin
rw [lift'_lift'_assoc],
exact monotone_gen,
exact monotone_id.comp_rel monotone_id
end
... ≤ (𝓤 α).lift' (λs, gen $... | comp_gen :
((𝓤 α).lift' gen).lift' (λs, comp_rel s s) ≤ (𝓤 α).lift' gen | calc ((𝓤 α).lift' gen).lift' (λs, comp_rel s s) =
(𝓤 α).lift' (λs, comp_rel (gen s) (gen s)) :
begin
rw [lift'_lift'_assoc],
exact monotone_gen,
exact monotone_id.comp_rel monotone_id
end
... ≤ (𝓤 α).lift' (λs, gen $ comp_rel s s) :
lift'_mono' $ assume s hs, comp_rel_gen_gen_subset_gen_com... | lemma | Cauchy.comp_gen | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"comp_le_uniformity",
"comp_rel",
"le_rfl",
"monotone_id"
] | null | 110 | 127 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: uniform_space (Cauchy α) :=
uniform_space.of_core
{ uniformity := (𝓤 α).lift' gen,
refl := principal_le_lift'.2 $ λ s hs ⟨a, b⟩ (a_eq_b : a = b),
a_eq_b ▸ a.property.right hs,
symm := symm_gen,
comp := comp_gen } | : uniform_space (Cauchy α) | uniform_space.of_core
{ uniformity := (𝓤 α).lift' gen,
refl := principal_le_lift'.2 $ λ s hs ⟨a, b⟩ (a_eq_b : a = b),
a_eq_b ▸ a.property.right hs,
symm := symm_gen,
comp := comp_gen } | instance | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"uniform_space",
"uniform_space.of_core",
"uniformity"
] | null | 129 | 135 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity {s : set (Cauchy α × Cauchy α)} :
s ∈ 𝓤 (Cauchy α) ↔ ∃ t ∈ 𝓤 α, gen t ⊆ s :=
mem_lift'_sets monotone_gen | mem_uniformity {s : set (Cauchy α × Cauchy α)} :
s ∈ 𝓤 (Cauchy α) ↔ ∃ t ∈ 𝓤 α, gen t ⊆ s | mem_lift'_sets monotone_gen | theorem | Cauchy.mem_uniformity | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy"
] | null | 137 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_uniformity' {s : set (Cauchy α × Cauchy α)} :
s ∈ 𝓤 (Cauchy α) ↔ ∃ t ∈ 𝓤 α, ∀ f g : Cauchy α, t ∈ f.1 ×ᶠ g.1 → (f, g) ∈ s :=
mem_uniformity.trans $ bex_congr $ λ t h, prod.forall | mem_uniformity' {s : set (Cauchy α × Cauchy α)} :
s ∈ 𝓤 (Cauchy α) ↔ ∃ t ∈ 𝓤 α, ∀ f g : Cauchy α, t ∈ f.1 ×ᶠ g.1 → (f, g) ∈ s | mem_uniformity.trans $ bex_congr $ λ t h, prod.forall | theorem | Cauchy.mem_uniformity' | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"bex_congr"
] | null | 141 | 143 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pure_cauchy (a : α) : Cauchy α :=
⟨pure a, cauchy_pure⟩ | pure_cauchy (a : α) : Cauchy α | ⟨pure a, cauchy_pure⟩ | def | Cauchy.pure_cauchy | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy"
] | Embedding of `α` into its completion `Cauchy α` | 146 | 147 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing_pure_cauchy : uniform_inducing (pure_cauchy : α → Cauchy α) :=
⟨have (preimage (λ (x : α × α), (pure_cauchy (x.fst), pure_cauchy (x.snd))) ∘ gen) = id,
from funext $ assume s, set.ext $ assume ⟨a₁, a₂⟩,
by simp [preimage, gen, pure_cauchy, prod_principal_principal],
calc comap (λ (x :... | uniform_inducing_pure_cauchy : uniform_inducing (pure_cauchy : α → Cauchy α) | ⟨have (preimage (λ (x : α × α), (pure_cauchy (x.fst), pure_cauchy (x.snd))) ∘ gen) = id,
from funext $ assume s, set.ext $ assume ⟨a₁, a₂⟩,
by simp [preimage, gen, pure_cauchy, prod_principal_principal],
calc comap (λ (x : α × α), (pure_cauchy (x.fst), pure_cauchy (x.snd))) ((𝓤 α).lift' gen)
... | lemma | Cauchy.uniform_inducing_pure_cauchy | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"set.ext",
"uniform_inducing"
] | null | 149 | 157 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_pure_cauchy : uniform_embedding (pure_cauchy : α → Cauchy α) :=
{ inj := assume a₁ a₂ h, pure_injective $ subtype.ext_iff_val.1 h,
..uniform_inducing_pure_cauchy } | uniform_embedding_pure_cauchy : uniform_embedding (pure_cauchy : α → Cauchy α) | { inj := assume a₁ a₂ h, pure_injective $ subtype.ext_iff_val.1 h,
..uniform_inducing_pure_cauchy } | lemma | Cauchy.uniform_embedding_pure_cauchy | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"uniform_embedding"
] | null | 159 | 161 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range_pure_cauchy : dense_range pure_cauchy :=
assume f,
have h_ex : ∀ s ∈ 𝓤 (Cauchy α), ∃y:α, (f, pure_cauchy y) ∈ s, from
assume s hs,
let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs in
let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁ in
have t' ∈ f.val ×ᶠ f.val,
... | dense_range_pure_cauchy : dense_range pure_cauchy | assume f,
have h_ex : ∀ s ∈ 𝓤 (Cauchy α), ∃y:α, (f, pure_cauchy y) ∈ s, from
assume s hs,
let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs in
let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁ in
have t' ∈ f.val ×ᶠ f.val,
from f.property.right ht'₁,
let ⟨t, ht, (h : t ×ˢ... | lemma | Cauchy.dense_range_pure_cauchy | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"Cauchy",
"closure_eq_cluster_pts",
"cluster_pt",
"comp_mem_uniformity_sets",
"dense_range",
"nhds_eq_uniformity",
"prod_mk_mem_comp_rel",
"set.inter_comm"
] | null | 163 | 188 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_inducing_pure_cauchy : dense_inducing pure_cauchy :=
uniform_inducing_pure_cauchy.dense_inducing dense_range_pure_cauchy | dense_inducing_pure_cauchy : dense_inducing pure_cauchy | uniform_inducing_pure_cauchy.dense_inducing dense_range_pure_cauchy | lemma | Cauchy.dense_inducing_pure_cauchy | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"dense_inducing"
] | null | 190 | 191 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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