statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_open_iff_open_ball_subset {s : set α} :
is_open s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, is_open V ∧ ball x V ⊆ s | begin
rw is_open_iff_ball_subset,
split; intros h x hx,
{ obtain ⟨V, hV, hV'⟩ := h x hx,
exact ⟨interior V, interior_mem_uniformity hV, is_open_interior,
(ball_mono interior_subset x).trans hV'⟩, },
{ obtain ⟨V, hV, -, hV'⟩ := h x hx,
exact ⟨V, hV, hV'⟩, },
end | lemma | is_open_iff_open_ball_subset | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono",
"interior_mem_uniformity",
"interior_subset",
"is_open",
"is_open_iff_ball_subset",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.bUnion_uniformity_ball {s : set α} {U : set (α × α)} (hs : dense s) (hU : U ∈ 𝓤 α) :
(⋃ x ∈ s, ball x U) = univ | begin
refine Union₂_eq_univ_iff.2 (λ y, _),
rcases hs.inter_nhds_nonempty (mem_nhds_right y hU) with ⟨x, hxs, hxy : (x, y) ∈ U⟩,
exact ⟨x, hxs, hxy⟩
end | lemma | dense.bUnion_uniformity_ball | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"dense",
"mem_nhds_right"
] | The uniform neighborhoods of all points of a dense set cover the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_has_basis_open : has_basis (𝓤 α) (λ V : set (α × α), V ∈ 𝓤 α ∧ is_open V) id | has_basis_self.2 $ λ s hs,
⟨interior s, interior_mem_uniformity hs, is_open_interior, interior_subset⟩ | lemma | uniformity_has_basis_open | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"interior_mem_uniformity",
"is_open",
"is_open_interior"
] | Open elements of `𝓤 α` form a basis of `𝓤 α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.mem_uniformity_iff {p : β → Prop} {s : β → set (α×α)}
(h : (𝓤 α).has_basis p s) {t : set (α × α)} :
t ∈ 𝓤 α ↔ ∃ i (hi : p i), ∀ a b, (a, b) ∈ s i → (a, b) ∈ t | h.mem_iff.trans $ by simp only [prod.forall, subset_def] | lemma | filter.has_basis.mem_uniformity_iff | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_has_basis_open_symmetric :
has_basis (𝓤 α) (λ V : set (α × α), V ∈ 𝓤 α ∧ is_open V ∧ symmetric_rel V) id | begin
simp only [← and_assoc],
refine uniformity_has_basis_open.restrict (λ s hs, ⟨symmetrize_rel s, _⟩),
exact ⟨⟨symmetrize_mem_uniformity hs.1, is_open.inter hs.2 (hs.2.preimage continuous_swap)⟩,
symmetric_symmetrize_rel s, symmetrize_rel_subset_self s⟩
end | lemma | uniformity_has_basis_open_symmetric | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_swap",
"is_open",
"is_open.inter",
"symmetric_rel",
"symmetric_symmetrize_rel",
"symmetrize_rel_subset_self"
] | Open elements `s : set (α × α)` of `𝓤 α` such that `(x, y) ∈ s ↔ (y, x) ∈ s` form a basis
of `𝓤 α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_open_symm_mem_uniformity_sets {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, is_open t ∧ symmetric_rel t ∧ t ○ t ⊆ s | begin
obtain ⟨t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs,
obtain ⟨u, ⟨hu₁, hu₂, hu₃⟩, hu₄ : u ⊆ t⟩ := uniformity_has_basis_open_symmetric.mem_iff.mp ht₁,
exact ⟨u, hu₁, hu₂, hu₃, (comp_rel_mono hu₄ hu₄).trans ht₂⟩,
end | lemma | comp_open_symm_mem_uniformity_sets | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_mem_uniformity_sets",
"comp_rel_mono",
"is_open",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.has_seq_basis [is_countably_generated $ 𝓤 α] :
∃ V : ℕ → set (α × α), has_antitone_basis (𝓤 α) V ∧ ∀ n, symmetric_rel (V n) | let ⟨U, hsym, hbasis⟩ := uniform_space.has_basis_symmetric.exists_antitone_subbasis
in ⟨U, hbasis, λ n, (hsym n).2⟩ | lemma | uniform_space.has_seq_basis | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.bInter_bUnion_ball {p : ι → Prop} {U : ι → set (α × α)}
(h : has_basis (𝓤 α) p U) (s : set α) :
(⋂ i (hi : p i), ⋃ x ∈ s, ball x (U i)) = closure s | begin
ext x,
simp [mem_closure_iff_nhds_basis (nhds_basis_uniformity h), ball]
end | lemma | filter.has_basis.bInter_bUnion_ball | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"closure",
"mem_closure_iff_nhds_basis",
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous [uniform_space β] (f : α → β) | tendsto (λx:α×α, (f x.1, f x.2)) (𝓤 α) (𝓤 β) | def | uniform_continuous | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | A function `f : α → β` is *uniformly continuous* if `(f x, f y)` tends to the diagonal
as `(x, y)` tends to the diagonal. In other words, if `x` is sufficiently close to `y`, then
`f x` is close to `f y` no matter where `x` and `y` are located in `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_on [uniform_space β] (f : α → β) (s : set α) : Prop | tendsto (λ x : α × α, (f x.1, f x.2)) (𝓤 α ⊓ principal (s ×ˢ s)) (𝓤 β) | def | uniform_continuous_on | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | A function `f : α → β` is *uniformly continuous* on `s : set α` if `(f x, f y)` tends to
the diagonal as `(x, y)` tends to the diagonal while remaining in `s ×ˢ s`.
In other words, if `x` is sufficiently close to `y`, then `f x` is close to
`f y` no matter where `x` and `y` are located in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_def [uniform_space β] {f : α → β} :
uniform_continuous f ↔ ∀ r ∈ 𝓤 β, { x : α × α | (f x.1, f x.2) ∈ r} ∈ 𝓤 α | iff.rfl | theorem | uniform_continuous_def | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_iff_eventually [uniform_space β] {f : α → β} :
uniform_continuous f ↔ ∀ r ∈ 𝓤 β, ∀ᶠ (x : α × α) in 𝓤 α, (f x.1, f x.2) ∈ r | iff.rfl | theorem | uniform_continuous_iff_eventually | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on_univ [uniform_space β] {f : α → β} :
uniform_continuous_on f univ ↔ uniform_continuous f | by rw [uniform_continuous_on, uniform_continuous, univ_prod_univ, principal_univ, inf_top_eq] | theorem | uniform_continuous_on_univ | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"inf_top_eq",
"uniform_continuous",
"uniform_continuous_on",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_of_const [uniform_space β] {c : α → β} (h : ∀a b, c a = c b) :
uniform_continuous c | have (λ (x : α × α), (c (x.fst), c (x.snd))) ⁻¹' id_rel = univ, from
eq_univ_iff_forall.2 $ assume ⟨a, b⟩, h a b,
le_trans (map_le_iff_le_comap.2 $ by simp [comap_principal, this, univ_mem]) refl_le_uniformity | lemma | uniform_continuous_of_const | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel",
"refl_le_uniformity",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_id : uniform_continuous (@id α) | by simp [uniform_continuous]; exact tendsto_id | lemma | uniform_continuous_id | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_const [uniform_space β] {b : β} : uniform_continuous (λa:α, b) | uniform_continuous_of_const $ λ _ _, rfl | lemma | uniform_continuous_const | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_of_const",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.comp [uniform_space β] [uniform_space γ] {g : β → γ} {f : α → β}
(hg : uniform_continuous g) (hf : uniform_continuous f) : uniform_continuous (g ∘ f) | hg.comp hf | lemma | uniform_continuous.comp | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_continuous_iff {ι'} [uniform_space β] {p : ι → Prop}
{s : ι → set (α×α)} (ha : (𝓤 α).has_basis p s) {q : ι' → Prop} {t : ι' → set (β×β)}
(hb : (𝓤 β).has_basis q t) {f : α → β} :
uniform_continuous f ↔ ∀ i (hi : q i), ∃ j (hj : p j), ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ t i | (ha.tendsto_iff hb).trans $ by simp only [prod.forall] | lemma | filter.has_basis.uniform_continuous_iff | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_continuous_on_iff {ι'} [uniform_space β] {p : ι → Prop}
{s : ι → set (α×α)} (ha : (𝓤 α).has_basis p s) {q : ι' → Prop} {t : ι' → set (β×β)}
(hb : (𝓤 β).has_basis q t) {f : α → β} {S : set α} :
uniform_continuous_on f S ↔
∀ i (hi : q i), ∃ j (hj : p j), ∀ x y ∈ S, (x, y) ∈ s j → (f x... | ((ha.inf_principal (S ×ˢ S)).tendsto_iff hb).trans $
by simp_rw [prod.forall, set.inter_comm (s _), ball_mem_comm, mem_inter_iff, mem_prod, and_imp] | lemma | filter.has_basis.uniform_continuous_on_iff | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"and_imp",
"ball_mem_comm",
"set.inter_comm",
"uniform_continuous_on",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_le {tt : set (uniform_space α)} {t : uniform_space α} (h : t ∈ tt) :
Inf tt ≤ t | show (⨅ u ∈ tt, 𝓤[u]) ≤ 𝓤[t], from infi₂_le t h | lemma | Inf_le | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"infi₂_le",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_Inf {tt : set (uniform_space α)} {t : uniform_space α} (h : ∀t'∈tt, t ≤ t') :
t ≤ Inf tt | show 𝓤[t] ≤ (⨅ u ∈ tt, 𝓤[u]), from le_infi₂ h | lemma | le_Inf | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"le_infi₂",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_uniformity {ι : Sort*} {u : ι → uniform_space α} : 𝓤[infi u] = (⨅i, 𝓤[u i]) | infi_range | lemma | infi_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"infi",
"infi_range",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_uniformity {u v : uniform_space α} : 𝓤[u ⊓ v] = 𝓤[u] ⊓ 𝓤[v] | rfl | lemma | inf_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited_uniform_space : inhabited (uniform_space α) | ⟨⊥⟩ | instance | inhabited_uniform_space | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited_uniform_space_core : inhabited (uniform_space.core α) | ⟨@uniform_space.to_core _ default⟩ | instance | inhabited_uniform_space_core | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space.core"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.comap (f : α → β) (u : uniform_space β) : uniform_space α | { uniformity := 𝓤[u].comap (λp:α×α, (f p.1, f p.2)),
to_topological_space := u.to_topological_space.induced f,
refl := le_trans (by simp; exact assume ⟨a, b⟩ (h : a = b), h ▸ rfl) (comap_mono u.refl),
symm := by simp [tendsto_comap_iff, prod.swap, (∘)];
exact tendsto_swap_uniformity.comp tendsto_coma... | def | uniform_space.comap | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_fold",
"is_open_iff_mem_nhds",
"is_open_induced",
"is_open_uniformity",
"monotone_id",
"nhds_eq_comap_uniformity",
"nhds_induced",
"prod.swap",
"uniform_space",
"uniformity"
] | Given `f : α → β` and a uniformity `u` on `β`, the inverse image of `u` under `f`
is the inverse image in the filter sense of the induced function `α × α → β × β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_comap [uniform_space β] (f : α → β) :
𝓤[uniform_space.comap f ‹_›] = comap (prod.map f f) (𝓤 β) | rfl | lemma | uniformity_comap | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space",
"uniform_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space_comap_id {α : Type*} : uniform_space.comap (id : α → α) = id | by { ext : 2, rw [uniformity_comap, prod.map_id, comap_id] } | lemma | uniform_space_comap_id | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.map_id",
"uniform_space.comap",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.comap_comap {α β γ} [uγ : uniform_space γ] {f : α → β} {g : β → γ} :
uniform_space.comap (g ∘ f) uγ = uniform_space.comap f (uniform_space.comap g uγ) | by { ext1, simp only [uniformity_comap, comap_comap, prod.map_comp_map] } | lemma | uniform_space.comap_comap | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.map_comp_map",
"uniform_space",
"uniform_space.comap",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.comap_inf {α γ} {u₁ u₂ : uniform_space γ} {f : α → γ} :
(u₁ ⊓ u₂).comap f = u₁.comap f ⊓ u₂.comap f | uniform_space_eq comap_inf | lemma | uniform_space.comap_inf | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space",
"uniform_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.comap_infi {ι α γ} {u : ι → uniform_space γ} {f : α → γ} :
(⨅ i, u i).comap f = ⨅ i, (u i).comap f | begin
ext : 1,
simp [uniformity_comap, infi_uniformity]
end | lemma | uniform_space.comap_infi | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"infi_uniformity",
"uniform_space",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.comap_mono {α γ} {f : α → γ} :
monotone (λ u : uniform_space γ, u.comap f) | begin
intros u₁ u₂ hu,
change (𝓤 _) ≤ (𝓤 _),
rw uniformity_comap,
exact comap_mono hu
end | lemma | uniform_space.comap_mono | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"monotone",
"uniform_space",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_iff {α β} {uα : uniform_space α} {uβ : uniform_space β} {f : α → β} :
uniform_continuous f ↔ uα ≤ uβ.comap f | filter.map_le_iff_le_comap | lemma | uniform_continuous_iff | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter.map_le_iff_le_comap",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_uniform_continuous_id {u v : uniform_space α} :
u ≤ v ↔ @uniform_continuous _ _ u v id | by rw [uniform_continuous_iff, uniform_space_comap_id, id] | lemma | le_iff_uniform_continuous_id | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_iff",
"uniform_space",
"uniform_space_comap_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_comap {f : α → β} [u : uniform_space β] :
@uniform_continuous α β (uniform_space.comap f u) u f | tendsto_comap | lemma | uniform_continuous_comap | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space",
"uniform_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_comap {f : α → β} {u : uniform_space β} :
@uniform_space.to_topological_space _ (uniform_space.comap f u) =
topological_space.induced f (@uniform_space.to_topological_space β u) | rfl | theorem | to_topological_space_comap | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"topological_space.induced",
"uniform_space",
"uniform_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_comap' {f : γ → β} {g : α → γ} [v : uniform_space β] [u : uniform_space α]
(h : uniform_continuous (f ∘ g)) : @uniform_continuous α γ u (uniform_space.comap f v) g | tendsto_comap_iff.2 h | lemma | uniform_continuous_comap' | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space",
"uniform_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nhds_mono {u₁ u₂ : uniform_space α} (h : u₁ ≤ u₂) (a : α) :
@nhds _ (@uniform_space.to_topological_space _ u₁) a ≤
@nhds _ (@uniform_space.to_topological_space _ u₂) a | by rw [@nhds_eq_uniformity α u₁ a, @nhds_eq_uniformity α u₂ a]; exact (lift'_mono h le_rfl) | lemma | to_nhds_mono | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"le_rfl",
"nhds",
"nhds_eq_uniformity",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_mono {u₁ u₂ : uniform_space α} (h : u₁ ≤ u₂) :
@uniform_space.to_topological_space _ u₁ ≤ @uniform_space.to_topological_space _ u₂ | le_of_nhds_le_nhds $ to_nhds_mono h | lemma | to_topological_space_mono | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"le_of_nhds_le_nhds",
"to_nhds_mono",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.continuous [uniform_space α] [uniform_space β] {f : α → β}
(hf : uniform_continuous f) : continuous f | continuous_iff_le_induced.mpr $ to_topological_space_mono $ uniform_continuous_iff.1 hf | lemma | uniform_continuous.continuous | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous",
"to_topological_space_mono",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_bot : @uniform_space.to_topological_space α ⊥ = ⊥ | rfl | lemma | to_topological_space_bot | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_top : @uniform_space.to_topological_space α ⊤ = ⊤ | top_unique $ assume s hs, s.eq_empty_or_nonempty.elim
(assume : s = ∅, this.symm ▸ @is_open_empty _ ⊤)
(assume ⟨x, hx⟩,
have s = univ, from top_unique $ assume y hy, hs x hx (x, y) rfl,
this.symm ▸ @is_open_univ _ ⊤) | lemma | to_topological_space_top | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_empty",
"is_open_univ",
"top_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_infi {ι : Sort*} {u : ι → uniform_space α} :
(infi u).to_topological_space = ⨅i, (u i).to_topological_space | begin
refine (eq_of_nhds_eq_nhds $ assume a, _),
simp only [nhds_infi, nhds_eq_uniformity, infi_uniformity],
exact lift'_infi_of_map_univ (ball_inter _) preimage_univ
end | lemma | to_topological_space_infi | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_inter",
"eq_of_nhds_eq_nhds",
"infi",
"infi_uniformity",
"nhds_eq_uniformity",
"nhds_infi",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_Inf {s : set (uniform_space α)} :
(Inf s).to_topological_space = (⨅i∈s, @uniform_space.to_topological_space α i) | begin
rw [Inf_eq_infi],
simp only [← to_topological_space_infi],
end | lemma | to_topological_space_Inf | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"Inf_eq_infi",
"to_topological_space_infi",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_inf {u v : uniform_space α} :
(u ⊓ v).to_topological_space = u.to_topological_space ⊓ v.to_topological_space | rfl | lemma | to_topological_space_inf | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.uniform_space [uniform_space α] : uniform_space (ulift α) | uniform_space.comap ulift.down ‹_› | instance | ulift.uniform_space | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space",
"uniform_space.comap"
] | Uniform space structure on `ulift α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_inf_rng {f : α → β} {u₁ : uniform_space α} {u₂ u₃ : uniform_space β}
(h₁ : @@uniform_continuous u₁ u₂ f) (h₂ : @@uniform_continuous u₁ u₃ f) :
@@uniform_continuous u₁ (u₂ ⊓ u₃) f | tendsto_inf.mpr ⟨h₁, h₂⟩ | lemma | uniform_continuous_inf_rng | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_inf_dom_left {f : α → β} {u₁ u₂ : uniform_space α} {u₃ : uniform_space β}
(hf : @@uniform_continuous u₁ u₃ f) : @@uniform_continuous (u₁ ⊓ u₂) u₃ f | tendsto_inf_left hf | lemma | uniform_continuous_inf_dom_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_inf_dom_right {f : α → β} {u₁ u₂ : uniform_space α} {u₃ : uniform_space β}
(hf : @@uniform_continuous u₂ u₃ f) : @@uniform_continuous (u₁ ⊓ u₂) u₃ f | tendsto_inf_right hf | lemma | uniform_continuous_inf_dom_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_Inf_dom {f : α → β} {u₁ : set (uniform_space α)} {u₂ : uniform_space β}
{u : uniform_space α} (h₁ : u ∈ u₁) (hf : @@uniform_continuous u u₂ f) :
@@uniform_continuous (Inf u₁) u₂ f | begin
rw [uniform_continuous, Inf_eq_infi', infi_uniformity],
exact tendsto_infi' ⟨u, h₁⟩ hf
end | lemma | uniform_continuous_Inf_dom | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"Inf_eq_infi'",
"infi_uniformity",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_Inf_rng {f : α → β} {u₁ : uniform_space α} {u₂ : set (uniform_space β)}
(h : ∀u∈u₂, @@uniform_continuous u₁ u f) : @@uniform_continuous u₁ (Inf u₂) f | begin
rw [uniform_continuous, Inf_eq_infi', infi_uniformity],
exact tendsto_infi.mpr (λ ⟨u, hu⟩, h u hu)
end | lemma | uniform_continuous_Inf_rng | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"Inf_eq_infi'",
"infi_uniformity",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_infi_dom {f : α → β} {u₁ : ι → uniform_space α} {u₂ : uniform_space β}
{i : ι} (hf : @@uniform_continuous (u₁ i) u₂ f) : @@uniform_continuous (infi u₁) u₂ f | begin
rw [uniform_continuous, infi_uniformity],
exact tendsto_infi' i hf
end | lemma | uniform_continuous_infi_dom | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"infi",
"infi_uniformity",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_infi_rng {f : α → β} {u₁ : uniform_space α} {u₂ : ι → uniform_space β}
(h : ∀i, @@uniform_continuous u₁ (u₂ i) f) : @@uniform_continuous u₁ (infi u₂) f | by rwa [uniform_continuous, infi_uniformity, tendsto_infi] | lemma | uniform_continuous_infi_rng | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"infi",
"infi_uniformity",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_of_discrete_uniformity [hα : uniform_space α]
(h : uniformity α = 𝓟 id_rel) :
discrete_topology α | ⟨(uniform_space_eq h.symm : ⊥ = hα) ▸ rfl⟩ | lemma | discrete_topology_of_discrete_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"discrete_topology",
"id_rel",
"uniform_space",
"uniform_space_eq",
"uniformity"
] | A uniform space with the discrete uniformity has the discrete topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_of_mul : uniform_continuous (of_mul : α → additive α) | uniform_continuous_id | lemma | uniform_continuous_of_mul | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"additive",
"uniform_continuous",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_to_mul : uniform_continuous (to_mul : additive α → α) | uniform_continuous_id | lemma | uniform_continuous_to_mul | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"additive",
"uniform_continuous",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_of_add : uniform_continuous (of_add : α → multiplicative α) | uniform_continuous_id | lemma | uniform_continuous_of_add | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"multiplicative",
"uniform_continuous",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_to_add : uniform_continuous (to_add : multiplicative α → α) | uniform_continuous_id | lemma | uniform_continuous_to_add | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"multiplicative",
"uniform_continuous",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_additive : 𝓤 (additive α) = (𝓤 α).map (prod.map of_mul of_mul) | by { convert map_id.symm, exact prod.map_id } | lemma | uniformity_additive | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"additive",
"prod.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_multiplicative : 𝓤 (multiplicative α) = (𝓤 α).map (prod.map of_add of_add) | by { convert map_id.symm, exact prod.map_id } | lemma | uniformity_multiplicative | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"multiplicative",
"prod.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_subtype {p : α → Prop} [t : uniform_space α] :
𝓤 (subtype p) = comap (λq:subtype p × subtype p, (q.1.1, q.2.1)) (𝓤 α) | rfl | lemma | uniformity_subtype | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_set_coe {s : set α} [t : uniform_space α] :
𝓤 s = comap (prod.map (coe : s → α) (coe : s → α)) (𝓤 α) | rfl | lemma | uniformity_set_coe | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_subtype_val {p : α → Prop} [uniform_space α] :
uniform_continuous (subtype.val : {a : α // p a} → α) | uniform_continuous_comap | lemma | uniform_continuous_subtype_val | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_comap",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_subtype_coe {p : α → Prop} [uniform_space α] :
uniform_continuous (coe : {a : α // p a} → α) | uniform_continuous_subtype_val | lemma | uniform_continuous_subtype_coe | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_subtype_val",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.subtype_mk {p : α → Prop} [uniform_space α] [uniform_space β]
{f : β → α} (hf : uniform_continuous f) (h : ∀x, p (f x)) :
uniform_continuous (λx, ⟨f x, h x⟩ : β → subtype p) | uniform_continuous_comap' hf | lemma | uniform_continuous.subtype_mk | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_comap'",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on_iff_restrict [uniform_space α] [uniform_space β] {f : α → β}
{s : set α} :
uniform_continuous_on f s ↔ uniform_continuous (s.restrict f) | begin
unfold uniform_continuous_on set.restrict uniform_continuous tendsto,
conv_rhs { rw [show (λ x : s × s, (f x.1, f x.2)) = prod.map f f ∘ prod.map coe coe, from rfl,
uniformity_set_coe, ← map_map, map_comap, range_prod_map, subtype.range_coe] },
refl
end | lemma | uniform_continuous_on_iff_restrict | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"set.restrict",
"subtype.range_coe",
"uniform_continuous",
"uniform_continuous_on",
"uniform_space",
"uniformity_set_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_uniform_continuous_subtype
[uniform_space α] [uniform_space β] {f : α → β} {s : set α} {a : α}
(hf : uniform_continuous (λx:s, f x.val)) (ha : s ∈ 𝓝 a) :
tendsto f (𝓝 a) (𝓝 (f a)) | by rw [(@map_nhds_subtype_coe_eq α _ s a (mem_of_mem_nhds ha) ha).symm]; exact
tendsto_map' (continuous_iff_continuous_at.mp hf.continuous _) | lemma | tendsto_of_uniform_continuous_subtype | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"map_nhds_subtype_coe_eq",
"mem_of_mem_nhds",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on.continuous_on [uniform_space α] [uniform_space β] {f : α → β}
{s : set α} (h : uniform_continuous_on f s) : continuous_on f s | begin
rw uniform_continuous_on_iff_restrict at h,
rw continuous_on_iff_continuous_restrict,
exact h.continuous
end | lemma | uniform_continuous_on.continuous_on | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_on",
"continuous_on_iff_continuous_restrict",
"uniform_continuous_on",
"uniform_continuous_on_iff_restrict",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_mul_opposite [uniform_space α] :
𝓤 (αᵐᵒᵖ) = comap (λ q : αᵐᵒᵖ × αᵐᵒᵖ, (q.1.unop, q.2.unop)) (𝓤 α) | rfl | lemma | uniformity_mul_opposite | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_uniformity_mul_opposite [uniform_space α] :
comap (λ p : α × α, (mul_opposite.op p.1, mul_opposite.op p.2)) (𝓤 αᵐᵒᵖ) = 𝓤 α | by simpa [uniformity_mul_opposite, comap_comap, (∘)] using comap_id | lemma | comap_uniformity_mul_opposite | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mul_opposite.op",
"uniform_space",
"uniformity_mul_opposite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_unop [uniform_space α] : uniform_continuous (unop : αᵐᵒᵖ → α) | uniform_continuous_comap | lemma | mul_opposite.uniform_continuous_unop | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_comap",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_op [uniform_space α] : uniform_continuous (op : α → αᵐᵒᵖ) | uniform_continuous_comap' uniform_continuous_id | lemma | mul_opposite.uniform_continuous_op | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_comap'",
"uniform_continuous_id",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_prod [uniform_space α] [uniform_space β] : 𝓤 (α × β) =
(𝓤 α).comap (λp:(α × β) × α × β, (p.1.1, p.2.1)) ⊓
(𝓤 β).comap (λp:(α × β) × α × β, (p.1.2, p.2.2)) | rfl | theorem | uniformity_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_prod_eq_comap_prod [uniform_space α] [uniform_space β] :
𝓤 (α × β) = comap (λ p : (α × β) × (α × β), ((p.1.1, p.2.1), (p.1.2, p.2.2))) (𝓤 α ×ᶠ 𝓤 β) | by rw [uniformity_prod, filter.prod, comap_inf, comap_comap, comap_comap] | lemma | uniformity_prod_eq_comap_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter.prod",
"uniform_space",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_prod_eq_prod [uniform_space α] [uniform_space β] :
𝓤 (α × β) = map (λ p : (α × α) × (β × β), ((p.1.1, p.2.1), (p.1.2, p.2.2))) (𝓤 α ×ᶠ 𝓤 β) | by rw [map_swap4_eq_comap, uniformity_prod_eq_comap_prod] | lemma | uniformity_prod_eq_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space",
"uniformity_prod_eq_comap_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity_of_uniform_continuous_invariant [uniform_space α] [uniform_space β]
{s : set (β × β)} {f : α → α → β} (hf : uniform_continuous (λ p : α × α, f p.1 p.2))
(hs : s ∈ 𝓤 β) :
∃ u ∈ 𝓤 α, ∀ a b c, (a, b) ∈ u → (f a c, f b c) ∈ s | begin
rw [uniform_continuous, uniformity_prod_eq_prod, tendsto_map'_iff, (∘)] at hf,
rcases mem_prod_iff.1 (mem_map.1 $ hf hs) with ⟨u, hu, v, hv, huvt⟩,
exact ⟨u, hu, λ a b c hab, @huvt ((_, _), (_, _)) ⟨hab, refl_mem_uniformity hv⟩⟩
end | lemma | mem_uniformity_of_uniform_continuous_invariant | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"refl_mem_uniformity",
"uniform_continuous",
"uniform_space",
"uniformity_prod_eq_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniform_prod [t₁ : uniform_space α] [t₂ : uniform_space β] {a : set (α × α)}
{b : set (β × β)} (ha : a ∈ 𝓤 α) (hb : b ∈ 𝓤 β) :
{p:(α×β)×(α×β) | (p.1.1, p.2.1) ∈ a ∧ (p.1.2, p.2.2) ∈ b } ∈ 𝓤 (α × β) | by rw [uniformity_prod]; exact inter_mem_inf (preimage_mem_comap ha) (preimage_mem_comap hb) | lemma | mem_uniform_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_uniformity_fst [uniform_space α] [uniform_space β] :
tendsto (λp:(α×β)×(α×β), (p.1.1, p.2.1)) (𝓤 (α × β)) (𝓤 α) | le_trans (map_mono inf_le_left) map_comap_le | lemma | tendsto_prod_uniformity_fst | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"inf_le_left",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_uniformity_snd [uniform_space α] [uniform_space β] :
tendsto (λp:(α×β)×(α×β), (p.1.2, p.2.2)) (𝓤 (α × β)) (𝓤 β) | le_trans (map_mono inf_le_right) map_comap_le | lemma | tendsto_prod_uniformity_snd | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"inf_le_right",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_fst [uniform_space α] [uniform_space β] :
uniform_continuous (λp:α×β, p.1) | tendsto_prod_uniformity_fst | lemma | uniform_continuous_fst | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"tendsto_prod_uniformity_fst",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_snd [uniform_space α] [uniform_space β] :
uniform_continuous (λp:α×β, p.2) | tendsto_prod_uniformity_snd | lemma | uniform_continuous_snd | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"tendsto_prod_uniformity_snd",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.prod_mk
{f₁ : α → β} {f₂ : α → γ} (h₁ : uniform_continuous f₁) (h₂ : uniform_continuous f₂) :
uniform_continuous (λa, (f₁ a, f₂ a)) | by rw [uniform_continuous, uniformity_prod]; exact
tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩ | lemma | uniform_continuous.prod_mk | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.prod_mk_left {f : α × β → γ} (h : uniform_continuous f) (b) :
uniform_continuous (λ a, f (a,b)) | h.comp (uniform_continuous_id.prod_mk uniform_continuous_const) | lemma | uniform_continuous.prod_mk_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.prod_mk_right {f : α × β → γ} (h : uniform_continuous f) (a) :
uniform_continuous (λ b, f (a,b)) | h.comp (uniform_continuous_const.prod_mk uniform_continuous_id) | lemma | uniform_continuous.prod_mk_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.prod_map [uniform_space δ] {f : α → γ} {g : β → δ}
(hf : uniform_continuous f) (hg : uniform_continuous g) :
uniform_continuous (prod.map f g) | (hf.comp uniform_continuous_fst).prod_mk (hg.comp uniform_continuous_snd) | lemma | uniform_continuous.prod_map | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous_fst",
"uniform_continuous_snd",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_prod {α} {β} [u : uniform_space α] [v : uniform_space β] :
@uniform_space.to_topological_space (α × β) prod.uniform_space =
@prod.topological_space α β u.to_topological_space v.to_topological_space | rfl | lemma | to_topological_space_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_inf_dom_left₂ {α β γ} {f : α → β → γ}
{ua1 ua2 : uniform_space α} {ub1 ub2 : uniform_space β} {uc1 : uniform_space γ}
(h : by haveI := ua1; haveI := ub1; exact uniform_continuous (λ p : α × β, f p.1 p.2)) :
by haveI | ua1 ⊓ ua2; haveI := ub1 ⊓ ub2; exact uniform_continuous (λ p : α × β, f p.1 p.2) :=
begin
-- proof essentially copied from ``continuous_inf_dom_left₂`
have ha := @uniform_continuous_inf_dom_left _ _ id ua1 ua2 ua1 (@uniform_continuous_id _ (id _)),
have hb := @uniform_continuous_inf_dom_left _ _ id ub1 ub2 ub1 (@... | lemma | uniform_continuous_inf_dom_left₂ | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous.comp",
"uniform_continuous.prod_map",
"uniform_continuous_id",
"uniform_continuous_inf_dom_left",
"uniform_space"
] | A version of `uniform_continuous_inf_dom_left` for binary functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_inf_dom_right₂ {α β γ} {f : α → β → γ}
{ua1 ua2 : uniform_space α} {ub1 ub2 : uniform_space β} {uc1 : uniform_space γ}
(h : by haveI := ua2; haveI := ub2; exact uniform_continuous (λ p : α × β, f p.1 p.2)) :
by haveI | ua1 ⊓ ua2; haveI := ub1 ⊓ ub2; exact uniform_continuous (λ p : α × β, f p.1 p.2) :=
begin
-- proof essentially copied from ``continuous_inf_dom_right₂`
have ha := @uniform_continuous_inf_dom_right _ _ id ua1 ua2 ua2 (@uniform_continuous_id _ (id _)),
have hb := @uniform_continuous_inf_dom_right _ _ id ub1 ub2 ub2... | lemma | uniform_continuous_inf_dom_right₂ | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous.comp",
"uniform_continuous.prod_map",
"uniform_continuous_id",
"uniform_continuous_inf_dom_right",
"uniform_space"
] | A version of `uniform_continuous_inf_dom_right` for binary functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_Inf_dom₂ {α β γ} {f : α → β → γ}
{uas : set (uniform_space α)} {ubs : set (uniform_space β)}
{ua : uniform_space α} {ub : uniform_space β} {uc : uniform_space γ}
(ha : ua ∈ uas) (hb : ub ∈ ubs)
(hf : uniform_continuous (λ p : α × β, f p.1 p.2)):
by haveI | Inf uas; haveI := Inf ubs;
exact @uniform_continuous _ _ _ uc (λ p : α × β, f p.1 p.2) :=
begin
-- proof essentially copied from ``continuous_Inf_dom`
let t : uniform_space (α × β) := prod.uniform_space,
have ha := uniform_continuous_Inf_dom ha uniform_continuous_id,
have hb := uniform_continuous_Inf_dom hb... | lemma | uniform_continuous_Inf_dom₂ | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous.comp",
"uniform_continuous.prod_map",
"uniform_continuous_Inf_dom",
"uniform_continuous_id",
"uniform_space"
] | A version of `uniform_continuous_Inf_dom` for binary functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous₂ (f : α → β → γ) | uniform_continuous (uncurry f) | def | uniform_continuous₂ | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous"
] | Uniform continuity for functions of two variables. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous₂_def (f : α → β → γ) :
uniform_continuous₂ f ↔ uniform_continuous (uncurry f) | iff.rfl | lemma | uniform_continuous₂_def | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous₂.uniform_continuous {f : α → β → γ} (h : uniform_continuous₂ f) :
uniform_continuous (uncurry f) | h | lemma | uniform_continuous₂.uniform_continuous | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous₂_curry (f : α × β → γ) :
uniform_continuous₂ (function.curry f) ↔ uniform_continuous f | by rw [uniform_continuous₂, uncurry_curry] | lemma | uniform_continuous₂_curry | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous₂.comp {f : α → β → γ} {g : γ → δ}
(hg : uniform_continuous g) (hf : uniform_continuous₂ f) :
uniform_continuous₂ (g ∘₂ f) | hg.comp hf | lemma | uniform_continuous₂.comp | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_continuous",
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous₂.bicompl {f : α → β → γ} {ga : δ → α} {gb : δ' → β}
(hf : uniform_continuous₂ f) (hga : uniform_continuous ga) (hgb : uniform_continuous gb) :
uniform_continuous₂ (bicompl f ga gb) | hf.uniform_continuous.comp (hga.prod_map hgb) | lemma | uniform_continuous₂.bicompl | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"bicompl",
"uniform_continuous",
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_topological_space_subtype [u : uniform_space α] {p : α → Prop} :
@uniform_space.to_topological_space (subtype p) subtype.uniform_space =
@subtype.topological_space α p u.to_topological_space | rfl | lemma | to_topological_space_subtype | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.core.sum : uniform_space.core (α ⊕ β) | uniform_space.core.mk'
(map (λ p : α × α, (inl p.1, inl p.2)) (𝓤 α) ⊔ map (λ p : β × β, (inr p.1, inr p.2)) (𝓤 β))
(λ r ⟨H₁, H₂⟩ x, by cases x; [apply refl_mem_uniformity H₁, apply refl_mem_uniformity H₂])
(λ r ⟨H₁, H₂⟩, ⟨symm_le_uniformity H₁, symm_le_uniformity H₂⟩)
(λ r ⟨Hrα, Hrβ⟩, begin
rcases comp_me... | def | uniform_space.core.sum | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_mem_uniformity_sets",
"refl_mem_uniformity",
"symm_le_uniformity",
"uniform_space.core",
"uniform_space.core.mk'"
] | Uniformity on a disjoint union. Entourages of the diagonal in the union are obtained
by taking independently an entourage of the diagonal in the first part, and an entourage of
the diagonal in the second part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
union_mem_uniformity_sum
{a : set (α × α)} (ha : a ∈ 𝓤 α) {b : set (β × β)} (hb : b ∈ 𝓤 β) :
((λ p : (α × α), (inl p.1, inl p.2)) '' a ∪ (λ p : (β × β), (inr p.1, inr p.2)) '' b) ∈
(@uniform_space.core.sum α β _ _).uniformity | ⟨mem_map_iff_exists_image.2 ⟨_, ha, subset_union_left _ _⟩,
mem_map_iff_exists_image.2 ⟨_, hb, subset_union_right _ _⟩⟩ | lemma | union_mem_uniformity_sum | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space.core.sum",
"uniformity"
] | The union of an entourage of the diagonal in each set of a disjoint union is again an entourage
of the diagonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_sum_of_open_aux {s : set (α ⊕ β)} (hs : is_open s) {x : α ⊕ β} (xs : x ∈ s) :
{ p : ((α ⊕ β) × (α ⊕ β)) | p.1 = x → p.2 ∈ s } ∈ (@uniform_space.core.sum α β _ _).uniformity | begin
cases x,
{ refine mem_of_superset
(union_mem_uniformity_sum (mem_nhds_uniformity_iff_right.1 (is_open.mem_nhds hs.1 xs))
univ_mem)
(union_subset _ _);
rintro _ ⟨⟨_, b⟩, h, ⟨⟩⟩ ⟨⟩,
exact h rfl },
{ refine mem_of_superset
(union_mem_uniformity_sum univ_mem (mem_nhds_uniformit... | lemma | uniformity_sum_of_open_aux | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open",
"is_open.mem_nhds",
"uniform_space.core.sum",
"uniformity",
"union_mem_uniformity_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_of_uniformity_sum_aux {s : set (α ⊕ β)}
(hs : ∀x ∈ s, { p : ((α ⊕ β) × (α ⊕ β)) | p.1 = x → p.2 ∈ s } ∈
(@uniform_space.core.sum α β _ _).uniformity) :
is_open s | begin
split,
{ refine (@is_open_iff_mem_nhds α _ _).2 (λ a ha, mem_nhds_uniformity_iff_right.2 _),
rcases mem_map_iff_exists_image.1 (hs _ ha).1 with ⟨t, ht, st⟩,
refine mem_of_superset ht _,
rintro p pt rfl, exact st ⟨_, pt, rfl⟩ rfl },
{ refine (@is_open_iff_mem_nhds β _ _).2 (λ b hb, mem_nhds_unifo... | lemma | open_of_uniformity_sum_aux | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open",
"is_open_iff_mem_nhds",
"uniform_space.core.sum",
"uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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