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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
symmetric_rel.inter {U V : set (α × α)} (hU : symmetric_rel U) (hV : symmetric_rel V) :
symmetric_rel (U ∩ V) | by rw [symmetric_rel, preimage_inter, hU.eq, hV.eq] | lemma | symmetric_rel.inter | 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 | |
uniform_space.core (α : Type u) | (uniformity : filter (α × α))
(refl : 𝓟 id_rel ≤ uniformity)
(symm : tendsto prod.swap uniformity uniformity)
(comp : uniformity.lift' (λs, s ○ s) ≤ uniformity) | structure | uniform_space.core | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"id_rel",
"prod.swap",
"uniformity"
] | This core description of a uniform space is outside of the type class hierarchy. It is useful
for constructions of uniform spaces, when the topology is derived from the uniform space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.core.mk' {α : Type u} (U : filter (α × α))
(refl : ∀ (r ∈ U) x, (x, x) ∈ r)
(symm : ∀ r ∈ U, prod.swap ⁻¹' r ∈ U)
(comp : ∀ r ∈ U, ∃ t ∈ U, t ○ t ⊆ r) : uniform_space.core α | ⟨U, λ r ru, id_rel_subset.2 (refl _ ru), symm,
λ r ru, let ⟨s, hs, hsr⟩ := comp _ ru in mem_of_superset (mem_lift' hs) hsr⟩ | def | uniform_space.core.mk' | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"prod.swap",
"uniform_space.core"
] | An alternative constructor for `uniform_space.core`. This version unfolds various
`filter`-related definitions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.core.mk_of_basis {α : Type u} (B : filter_basis (α × α))
(refl : ∀ (r ∈ B) x, (x, x) ∈ r)
(symm : ∀ r ∈ B, ∃ t ∈ B, t ⊆ prod.swap ⁻¹' r)
(comp : ∀ r ∈ B, ∃ t ∈ B, t ○ t ⊆ r) : uniform_space.core α | { uniformity := B.filter,
refl := B.has_basis.ge_iff.mpr (λ r ru, id_rel_subset.2 $ refl _ ru),
symm := (B.has_basis.tendsto_iff B.has_basis).mpr symm,
comp := (has_basis.le_basis_iff (B.has_basis.lift' (monotone_id.comp_rel monotone_id))
B.has_basis).mpr comp } | def | uniform_space.core.mk_of_basis | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter_basis",
"monotone_id",
"prod.swap",
"uniform_space.core",
"uniformity"
] | Defining an `uniform_space.core` from a filter basis satisfying some uniformity-like axioms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.core.to_topological_space {α : Type u} (u : uniform_space.core α) :
topological_space α | { is_open := λs, ∀x∈s, { p : α × α | p.1 = x → p.2 ∈ s } ∈ u.uniformity,
is_open_univ := by simp; intro; exact univ_mem,
is_open_inter :=
assume s t hs ht x ⟨xs, xt⟩, by filter_upwards [hs x xs, ht x xt]; simp {contextual := tt},
is_open_sUnion :=
assume s hs x ⟨t, ts, xt⟩, by filter_upwards [hs... | def | uniform_space.core.to_topological_space | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open",
"is_open_sUnion",
"is_open_univ",
"topological_space",
"uniform_space.core"
] | A uniform space generates a topological space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.core_eq :
∀{u₁ u₂ : uniform_space.core α}, u₁.uniformity = u₂.uniformity → u₁ = u₂ | | ⟨u₁, _, _, _⟩ ⟨u₂, _, _, _⟩ rfl := by congr | lemma | uniform_space.core_eq | 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 (α : Type u) extends topological_space α, uniform_space.core α | (is_open_uniformity : ∀s, @_root_.is_open _ to_topological_space s ↔
(∀x∈s, { p : α × α | p.1 = x → p.2 ∈ s } ∈ uniformity)) | class | uniform_space | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_uniformity",
"topological_space",
"uniform_space.core",
"uniformity"
] | A uniform space is a generalization of the "uniform" topological aspects of a
metric space. It consists of a filter on `α × α` called the "uniformity", which
satisfies properties analogous to the reflexivity, symmetry, and triangle properties
of a metric.
A metric space has a natural uniformity, and a uniform ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.mk' {α} (t : topological_space α)
(c : uniform_space.core α)
(is_open_uniformity : ∀s:set α, is_open s ↔
(∀x∈s, { p : α × α | p.1 = x → p.2 ∈ s } ∈ c.uniformity)) :
uniform_space α | ⟨c, is_open_uniformity⟩ | def | uniform_space.mk' | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open",
"is_open_uniformity",
"topological_space",
"uniform_space",
"uniform_space.core"
] | Alternative constructor for `uniform_space α` when a topology is already given. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.of_core {α : Type u} (u : uniform_space.core α) : uniform_space α | { to_core := u,
to_topological_space := u.to_topological_space,
is_open_uniformity := assume a, iff.rfl } | def | uniform_space.of_core | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_uniformity",
"uniform_space",
"uniform_space.core"
] | Construct a `uniform_space` from a `uniform_space.core`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.of_core_eq {α : Type u} (u : uniform_space.core α) (t : topological_space α)
(h : t = u.to_topological_space) : uniform_space α | { to_core := u,
to_topological_space := t,
is_open_uniformity := assume a, h.symm ▸ iff.rfl } | def | uniform_space.of_core_eq | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_uniformity",
"topological_space",
"uniform_space",
"uniform_space.core"
] | Construct a `uniform_space` from a `u : uniform_space.core` and a `topological_space` structure
that is equal to `u.to_topological_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.to_core_to_topological_space (u : uniform_space α) :
u.to_core.to_topological_space = u.to_topological_space | topological_space_eq $ funext $ λ s, by rw [uniform_space.is_open_uniformity, is_open_mk] | lemma | uniform_space.to_core_to_topological_space | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_mk",
"topological_space_eq",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity (α : Type u) [uniform_space α] : filter (α × α) | (@uniform_space.to_core α _).uniformity | def | uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"uniform_space"
] | The uniformity is a filter on α × α (inferred from an ambient uniform space
structure on α). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space_eq : ∀ {u₁ u₂ : uniform_space α}, 𝓤[u₁] = 𝓤[u₂] → u₁ = u₂ | | (uniform_space.mk' t₁ u₁ o₁) (uniform_space.mk' t₂ u₂ o₂) h :=
have u₁ = u₂, from uniform_space.core_eq h,
have t₁ = t₂, from topological_space_eq $ funext $ assume s, by rw [o₁, o₂]; simp [this],
by simp [*] | lemma | uniform_space_eq | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"topological_space_eq",
"uniform_space",
"uniform_space.core_eq",
"uniform_space.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.of_core_eq_to_core
(u : uniform_space α) (t : topological_space α) (h : t = u.to_core.to_topological_space) :
uniform_space.of_core_eq u.to_core t h = u | uniform_space_eq rfl | lemma | uniform_space.of_core_eq_to_core | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"topological_space",
"uniform_space",
"uniform_space.of_core_eq",
"uniform_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.replace_topology {α : Type*} [i : topological_space α]
(u : uniform_space α) (h : i = u.to_topological_space) : uniform_space α | uniform_space.of_core_eq u.to_core i $ h.trans u.to_core_to_topological_space.symm | def | uniform_space.replace_topology | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"topological_space",
"uniform_space",
"uniform_space.of_core_eq"
] | Replace topology in a `uniform_space` instance with a propositionally (but possibly not
definitionally) equal one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.replace_topology_eq {α : Type*} [i : topological_space α] (u : uniform_space α)
(h : i = u.to_topological_space) : u.replace_topology h = u | u.of_core_eq_to_core _ _ | lemma | uniform_space.replace_topology_eq | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"topological_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.of_fun {α β : Type*} [ordered_add_comm_monoid β]
(d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
(triangle : ∀ x y z, d x z ≤ d x y + d y z)
(half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) :
uniform_space α | uniform_space.of_core
{ uniformity := ⨅ r > 0, 𝓟 { x | d x.1 x.2 < r },
refl := le_infi₂ $ λ r hr, principal_mono.2 $ id_rel_subset.2 $ λ x, by simpa [refl],
symm := tendsto_infi_infi $ λ r, tendsto_infi_infi $ λ _, tendsto_principal_principal.2 $
λ x hx, by rwa [mem_set_of, symm],
comp := le_infi₂... | def | uniform_space.of_fun | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"le_infi₂",
"ordered_add_comm_monoid",
"uniform_space",
"uniform_space.of_core",
"uniformity"
] | Define a `uniform_space` using a "distance" function. The function can be, e.g., the distance in
a (usual or extended) metric space or an absolute value on a ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.has_basis_of_fun {α β : Type*} [linear_ordered_add_comm_monoid β]
(h₀ : ∃ x : β, 0 < x) (d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
(triangle : ∀ x y z, d x z ≤ d x y + d y z)
(half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) :
𝓤[uniform_space.of_fun d r... | has_basis_binfi_principal'
(λ ε₁ h₁ ε₂ h₂, ⟨min ε₁ ε₂, lt_min h₁ h₂, λ _x hx, lt_of_lt_of_le hx (min_le_left _ _),
λ _x hx, lt_of_lt_of_le hx (min_le_right _ _)⟩) h₀ | lemma | uniform_space.has_basis_of_fun | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"linear_ordered_add_comm_monoid",
"uniform_space.of_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_uniformity {s : set α} :
is_open s ↔ (∀x∈s, { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α) | uniform_space.is_open_uniformity s | lemma | is_open_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_le_uniformity : 𝓟 id_rel ≤ 𝓤 α | (@uniform_space.to_core α _).refl | lemma | refl_le_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity.ne_bot [nonempty α] : ne_bot (𝓤 α) | diagonal_nonempty.principal_ne_bot.mono refl_le_uniformity | instance | uniformity.ne_bot | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"refl_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_mem_uniformity {x : α} {s : set (α × α)} (h : s ∈ 𝓤 α) :
(x, x) ∈ s | refl_le_uniformity h rfl | lemma | refl_mem_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"refl_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity_of_eq {x y : α} {s : set (α × α)} (h : s ∈ 𝓤 α) (hx : x = y) :
(x, y) ∈ s | refl_le_uniformity h hx | lemma | mem_uniformity_of_eq | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"refl_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_le_uniformity : map (@prod.swap α α) (𝓤 _) ≤ (𝓤 _) | (@uniform_space.to_core α _).symm | lemma | symm_le_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_le_uniformity : (𝓤 α).lift' (λs:set (α×α), s ○ s) ≤ 𝓤 α | (@uniform_space.to_core α _).comp | lemma | comp_le_uniformity | 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 | |
tendsto_swap_uniformity : tendsto (@prod.swap α α) (𝓤 α) (𝓤 α) | symm_le_uniformity | lemma | tendsto_swap_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mem_uniformity_sets {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, t ○ t ⊆ s | have s ∈ (𝓤 α).lift' (λt:set (α×α), t ○ t),
from comp_le_uniformity hs,
(mem_lift'_sets $ monotone_id.comp_rel monotone_id).mp this | lemma | comp_mem_uniformity_sets | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_le_uniformity",
"monotone_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_uniformity_iterate_comp_subset {s : set (α × α)} (hs : s ∈ 𝓤 α) (n : ℕ) :
∀ᶠ t in (𝓤 α).small_sets, ((○) t) ^[n] t ⊆ s | begin
suffices : ∀ᶠ t in (𝓤 α).small_sets, t ⊆ s ∧ (((○) t) ^[n] t ⊆ s),
from (eventually_and.1 this).2,
induction n with n ihn generalizing s, { simpa },
rcases comp_mem_uniformity_sets hs with ⟨t, htU, hts⟩,
refine (ihn htU).mono (λ U hU, _),
rw [function.iterate_succ_apply'],
exact ⟨hU.1.trans $ (su... | lemma | eventually_uniformity_iterate_comp_subset | 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",
"function.iterate_succ_apply'",
"refl_le_uniformity",
"subset_comp_self"
] | If `s ∈ 𝓤 α`, then for any natural `n`, for a subset `t` of a sufficiently small set in `𝓤 α`,
we have `t ○ t ○ ... ○ t ⊆ s` (`n` compositions). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_uniformity_comp_subset {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∀ᶠ t in (𝓤 α).small_sets, t ○ t ⊆ s | eventually_uniformity_iterate_comp_subset hs 1 | lemma | eventually_uniformity_comp_subset | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"eventually_uniformity_iterate_comp_subset"
] | If `s ∈ 𝓤 α`, then for any natural `n`, for a subset `t` of a sufficiently small set in `𝓤 α`,
we have `t ○ t ⊆ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.uniformity_trans {l : filter β} {f₁ f₂ f₃ : β → α}
(h₁₂ : tendsto (λ x, (f₁ x, f₂ x)) l (𝓤 α)) (h₂₃ : tendsto (λ x, (f₂ x, f₃ x)) l (𝓤 α)) :
tendsto (λ x, (f₁ x, f₃ x)) l (𝓤 α) | begin
refine le_trans (le_lift'.2 $ λ s hs, mem_map.2 _) comp_le_uniformity,
filter_upwards [h₁₂ hs, h₂₃ hs] with x hx₁₂ hx₂₃ using ⟨_, hx₁₂, hx₂₃⟩,
end | lemma | filter.tendsto.uniformity_trans | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_le_uniformity",
"filter"
] | Relation `λ f g, tendsto (λ x, (f x, g x)) l (𝓤 α)` is transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.uniformity_symm {l : filter β} {f : β → α × α}
(h : tendsto f l (𝓤 α)) :
tendsto (λ x, ((f x).2, (f x).1)) l (𝓤 α) | tendsto_swap_uniformity.comp h | lemma | filter.tendsto.uniformity_symm | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter"
] | Relation `λ f g, tendsto (λ x, (f x, g x)) l (𝓤 α)` is symmetric | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_diag_uniformity (f : β → α) (l : filter β) :
tendsto (λ x, (f x, f x)) l (𝓤 α) | assume s hs, mem_map.2 $ univ_mem' $ λ x, refl_mem_uniformity hs | lemma | tendsto_diag_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"refl_mem_uniformity"
] | Relation `λ f g, tendsto (λ x, (f x, g x)) l (𝓤 α)` is reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_const_uniformity {a : α} {f : filter β} : tendsto (λ _, (a, a)) f (𝓤 α) | tendsto_diag_uniformity (λ _, a) f | lemma | tendsto_const_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"tendsto_diag_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_of_uniformity {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, (∀a b, (a, b) ∈ t → (b, a) ∈ t) ∧ t ⊆ s | have preimage prod.swap s ∈ 𝓤 α, from symm_le_uniformity hs,
⟨s ∩ preimage prod.swap s, inter_mem hs this, λ a b ⟨h₁, h₂⟩, ⟨h₂, h₁⟩, inter_subset_left _ _⟩ | lemma | symm_of_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm_of_uniformity {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, (∀{a b}, (a, b) ∈ t → (b, a) ∈ t) ∧ t ○ t ⊆ s | let ⟨t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs in
let ⟨t', ht', ht'₁, ht'₂⟩ := symm_of_uniformity ht₁ in
⟨t', ht', ht'₁, subset.trans (monotone_id.comp_rel monotone_id ht'₂) ht₂⟩ | lemma | comp_symm_of_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_mem_uniformity_sets",
"monotone_id",
"symm_of_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_le_symm : 𝓤 α ≤ (@prod.swap α α) <$> 𝓤 α | by rw [map_swap_eq_comap_swap];
from map_le_iff_le_comap.1 tendsto_swap_uniformity | lemma | uniformity_le_symm | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"tendsto_swap_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_symm : 𝓤 α = (@prod.swap α α) <$> 𝓤 α | le_antisymm uniformity_le_symm symm_le_uniformity | lemma | uniformity_eq_symm | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"symm_le_uniformity",
"uniformity_le_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_swap_uniformity : comap (@prod.swap α α) (𝓤 α) = 𝓤 α | (congr_arg _ uniformity_eq_symm).trans $ comap_map prod.swap_injective | lemma | comap_swap_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"prod.swap_injective",
"uniformity_eq_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetrize_mem_uniformity {V : set (α × α)} (h : V ∈ 𝓤 α) : symmetrize_rel V ∈ 𝓤 α | begin
apply (𝓤 α).inter_sets h,
rw [← image_swap_eq_preimage_swap, uniformity_eq_symm],
exact image_mem_map h,
end | lemma | symmetrize_mem_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetrize_rel",
"uniformity_eq_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.has_basis_symmetric :
(𝓤 α).has_basis (λ s : set (α × α), s ∈ 𝓤 α ∧ symmetric_rel s) id | has_basis_self.2 $ λ t t_in, ⟨symmetrize_rel t, symmetrize_mem_uniformity t_in,
symmetric_symmetrize_rel t, symmetrize_rel_subset_self t⟩ | lemma | uniform_space.has_basis_symmetric | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetric_rel",
"symmetric_symmetrize_rel",
"symmetrize_mem_uniformity",
"symmetrize_rel_subset_self"
] | Symmetric entourages form a basis of `𝓤 α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_lift_le_swap {g : set (α×α) → filter β} {f : filter β} (hg : monotone g)
(h : (𝓤 α).lift (λs, g (preimage prod.swap s)) ≤ f) : (𝓤 α).lift g ≤ f | calc (𝓤 α).lift g ≤ (filter.map (@prod.swap α α) $ 𝓤 α).lift g :
lift_mono uniformity_le_symm le_rfl
... ≤ _ :
by rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]; exact h | theorem | uniformity_lift_le_swap | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"filter.map",
"le_rfl",
"lift",
"monotone",
"prod.swap",
"uniformity_le_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_lift_le_comp {f : set (α×α) → filter β} (h : monotone f) :
(𝓤 α).lift (λs, f (s ○ s)) ≤ (𝓤 α).lift f | calc (𝓤 α).lift (λs, f (s ○ s)) =
((𝓤 α).lift' (λs:set (α×α), s ○ s)).lift f :
begin
rw [lift_lift'_assoc],
exact monotone_id.comp_rel monotone_id,
exact h
end
... ≤ (𝓤 α).lift f : lift_mono comp_le_uniformity le_rfl | lemma | uniformity_lift_le_comp | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_le_uniformity",
"filter",
"le_rfl",
"lift",
"monotone",
"monotone_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_le_uniformity3 :
(𝓤 α).lift' (λs:set (α×α), s ○ (s ○ s)) ≤ (𝓤 α) | calc (𝓤 α).lift' (λd, d ○ (d ○ d)) =
(𝓤 α).lift (λs, (𝓤 α).lift' (λt:set(α×α), s ○ (t ○ t))) :
begin
rw [lift_lift'_same_eq_lift'],
exact (assume x, monotone_const.comp_rel $ monotone_id.comp_rel monotone_id),
exact (assume x, monotone_id.comp_rel monotone_const),
end
... ≤ (𝓤 α).lift (λs, (𝓤 α... | lemma | comp_le_uniformity3 | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_le_uniformity",
"lift",
"monotone_const",
"monotone_id",
"uniformity_lift_le_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm_mem_uniformity_sets {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, symmetric_rel t ∧ t ○ t ⊆ s | begin
obtain ⟨w, w_in, w_sub⟩ : ∃ w ∈ 𝓤 α, w ○ w ⊆ s := comp_mem_uniformity_sets hs,
use [symmetrize_rel w, symmetrize_mem_uniformity w_in, symmetric_symmetrize_rel w],
have : symmetrize_rel w ⊆ w := symmetrize_rel_subset_self w,
calc symmetrize_rel w ○ symmetrize_rel w ⊆ w ○ w : by mono
... | lemma | comp_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",
"symmetric_rel",
"symmetric_symmetrize_rel",
"symmetrize_mem_uniformity",
"symmetrize_rel",
"symmetrize_rel_subset_self"
] | See also `comp_open_symm_mem_uniformity_sets`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_comp_self_of_mem_uniformity {s : set (α × α)} (h : s ∈ 𝓤 α) : s ⊆ s ○ s | subset_comp_self (refl_le_uniformity h) | lemma | subset_comp_self_of_mem_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"refl_le_uniformity",
"subset_comp_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_comp_symm_mem_uniformity_sets {s : set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, symmetric_rel t ∧ t ○ t ○ t ⊆ s | begin
rcases comp_symm_mem_uniformity_sets hs with ⟨w, w_in, w_symm, w_sub⟩,
rcases comp_symm_mem_uniformity_sets w_in with ⟨t, t_in, t_symm, t_sub⟩,
use [t, t_in, t_symm],
have : t ⊆ t ○ t := subset_comp_self_of_mem_uniformity t_in,
calc
t ○ t ○ t ⊆ w ○ t : by mono
... ⊆ w ○ (t ○ t) : by mon... | lemma | comp_comp_symm_mem_uniformity_sets | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_symm_mem_uniformity_sets",
"subset_comp_self_of_mem_uniformity",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.ball (x : β) (V : set (β × β)) : set β | (prod.mk x) ⁻¹' V | def | uniform_space.ball | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | The ball around `(x : β)` with respect to `(V : set (β × β))`. Intended to be
used for `V ∈ 𝓤 β`, but this is not needed for the definition. Recovers the
notions of metric space ball when `V = {p | dist p.1 p.2 < r }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.mem_ball_self (x : α) {V : set (α × α)} (hV : V ∈ 𝓤 α) :
x ∈ ball x V | refl_mem_uniformity hV | lemma | uniform_space.mem_ball_self | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"refl_mem_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_comp {V W : set (β × β)} {x y z} (h : y ∈ ball x V) (h' : z ∈ ball y W) :
z ∈ ball x (V ○ W) | prod_mk_mem_comp_rel h h' | lemma | mem_ball_comp | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod_mk_mem_comp_rel"
] | The triangle inequality for `uniform_space.ball` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_subset_of_comp_subset {V W : set (β × β)} {x y} (h : x ∈ ball y W) (h' : W ○ W ⊆ V) :
ball x W ⊆ ball y V | λ z z_in, h' (mem_ball_comp h z_in) | lemma | ball_subset_of_comp_subset | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_ball_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mono {V W : set (β × β)} (h : V ⊆ W) (x : β) : ball x V ⊆ ball x W | preimage_mono h | lemma | ball_mono | 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 | |
ball_inter (x : β) (V W : set (β × β)) : ball x (V ∩ W) = ball x V ∩ ball x W | preimage_inter | lemma | ball_inter | 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 | |
ball_inter_left (x : β) (V W : set (β × β)) : ball x (V ∩ W) ⊆ ball x V | ball_mono (inter_subset_left V W) x | lemma | ball_inter_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_inter_right (x : β) (V W : set (β × β)) : ball x (V ∩ W) ⊆ ball x W | ball_mono (inter_subset_right V W) x | lemma | ball_inter_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ball_symmetry {V : set (β × β)} (hV : symmetric_rel V) {x y} :
x ∈ ball y V ↔ y ∈ ball x V | show (x, y) ∈ prod.swap ⁻¹' V ↔ (x, y) ∈ V, by { unfold symmetric_rel at hV, rw hV } | lemma | mem_ball_symmetry | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_eq_of_symmetry {V : set (β × β)} (hV : symmetric_rel V) {x} :
ball x V = {y | (y, x) ∈ V} | by { ext y, rw mem_ball_symmetry hV, exact iff.rfl } | lemma | ball_eq_of_symmetry | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_ball_symmetry",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comp_of_mem_ball {V W : set (β × β)} {x y z : β} (hV : symmetric_rel V)
(hx : x ∈ ball z V) (hy : y ∈ ball z W) : (x, y) ∈ V ○ W | begin
rw mem_ball_symmetry hV at hx,
exact ⟨z, hx, hy⟩
end | lemma | mem_comp_of_mem_ball | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_ball_symmetry",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.is_open_ball (x : α) {V : set (α × α)} (hV : is_open V) :
is_open (ball x V) | hV.preimage $ continuous_const.prod_mk continuous_id | lemma | uniform_space.is_open_ball | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_id",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comp_comp {V W M : set (β × β)} (hW' : symmetric_rel W) {p : β × β} :
p ∈ V ○ M ○ W ↔ ((ball p.1 V ×ˢ ball p.2 W) ∩ M).nonempty | begin
cases p with x y,
split,
{ rintros ⟨z, ⟨w, hpw, hwz⟩, hzy⟩,
exact ⟨(w, z), ⟨hpw, by rwa mem_ball_symmetry hW'⟩, hwz⟩, },
{ rintro ⟨⟨w, z⟩, ⟨w_in, z_in⟩, hwz⟩,
rwa mem_ball_symmetry hW' at z_in,
use [z, w] ; tauto },
end | lemma | mem_comp_comp | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_ball_symmetry",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_uniformity_iff_right {x : α} {s : set α} :
s ∈ 𝓝 x ↔ {p : α × α | p.1 = x → p.2 ∈ s} ∈ 𝓤 α | begin
refine ⟨_, λ hs, _⟩,
{ simp only [mem_nhds_iff, is_open_uniformity, and_imp, exists_imp_distrib],
intros t ts ht xt,
filter_upwards [ht x xt] using λ y h eq, ts (h eq) },
{ refine mem_nhds_iff.mpr ⟨{x | {p : α × α | p.1 = x → p.2 ∈ s} ∈ 𝓤 α}, _, _, hs⟩,
{ exact λ y hy, refl_mem_uniformity hy rf... | lemma | mem_nhds_uniformity_iff_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"and_imp",
"comp_mem_uniformity_sets",
"exists_imp_distrib",
"is_open_uniformity",
"mem_nhds_iff",
"refl_mem_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_uniformity_iff_left {x : α} {s : set α} :
s ∈ 𝓝 x ↔ {p : α × α | p.2 = x → p.1 ∈ s} ∈ 𝓤 α | by { rw [uniformity_eq_symm, mem_nhds_uniformity_iff_right], refl } | lemma | mem_nhds_uniformity_iff_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_nhds_uniformity_iff_right",
"uniformity_eq_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_comap_uniformity {x : α} : 𝓝 x = (𝓤 α).comap (prod.mk x) | by { ext s, rw [mem_nhds_uniformity_iff_right, mem_comap_prod_mk] } | lemma | nhds_eq_comap_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_nhds_uniformity_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_ball_subset {s : set α} : is_open s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, ball x V ⊆ s | begin
simp_rw [is_open_iff_mem_nhds, nhds_eq_comap_uniformity],
exact iff.rfl,
end | lemma | is_open_iff_ball_subset | 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",
"nhds_eq_comap_uniformity"
] | See also `is_open_iff_open_ball_subset`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_uniformity' {p : ι → Prop} {s : ι → set (α × α)} (h : (𝓤 α).has_basis p s)
{x : α} :
(𝓝 x).has_basis p (λ i, ball x (s i)) | by { rw [nhds_eq_comap_uniformity], exact h.comap (prod.mk x) } | lemma | nhds_basis_uniformity' | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_uniformity {p : ι → Prop} {s : ι → set (α × α)} (h : (𝓤 α).has_basis p s) {x : α} :
(𝓝 x).has_basis p (λ i, {y | (y, x) ∈ s i}) | begin
replace h := h.comap prod.swap,
rw [← map_swap_eq_comap_swap, ← uniformity_eq_symm] at h,
exact nhds_basis_uniformity' h
end | lemma | nhds_basis_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_basis_uniformity'",
"prod.swap",
"uniformity_eq_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_comap_uniformity' {x : α} : 𝓝 x = (𝓤 α).comap (λ y, (y, x)) | (nhds_basis_uniformity (𝓤 α).basis_sets).eq_of_same_basis $ (𝓤 α).basis_sets.comap _ | lemma | nhds_eq_comap_uniformity' | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.mem_nhds_iff {x : α} {s : set α} : s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, ball x V ⊆ s | begin
rw [nhds_eq_comap_uniformity, mem_comap],
exact iff.rfl,
end | lemma | uniform_space.mem_nhds_iff | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.ball_mem_nhds (x : α) ⦃V : set (α × α)⦄ (V_in : V ∈ 𝓤 α) : ball x V ∈ 𝓝 x | begin
rw uniform_space.mem_nhds_iff,
exact ⟨V, V_in, subset.refl _⟩
end | lemma | uniform_space.ball_mem_nhds | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"uniform_space.mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.mem_nhds_iff_symm {x : α} {s : set α} :
s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, symmetric_rel V ∧ ball x V ⊆ s | begin
rw uniform_space.mem_nhds_iff,
split,
{ rintros ⟨V, V_in, V_sub⟩,
use [symmetrize_rel V, symmetrize_mem_uniformity V_in, symmetric_symmetrize_rel V],
exact subset.trans (ball_mono (symmetrize_rel_subset_self V) x) V_sub },
{ rintros ⟨V, V_in, V_symm, V_sub⟩,
exact ⟨V, V_in, V_sub⟩ }
end | lemma | uniform_space.mem_nhds_iff_symm | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono",
"symmetric_rel",
"symmetric_symmetrize_rel",
"symmetrize_mem_uniformity",
"symmetrize_rel",
"symmetrize_rel_subset_self",
"uniform_space.mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.has_basis_nhds (x : α) :
has_basis (𝓝 x) (λ s : set (α × α), s ∈ 𝓤 α ∧ symmetric_rel s) (λ s, ball x s) | ⟨λ t, by simp [uniform_space.mem_nhds_iff_symm, and_assoc]⟩ | lemma | uniform_space.has_basis_nhds | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetric_rel",
"uniform_space.mem_nhds_iff_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.mem_closure_iff_symm_ball {s : set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → symmetric_rel V → (s ∩ ball x V).nonempty | by simp [mem_closure_iff_nhds_basis (has_basis_nhds x), set.nonempty] | lemma | uniform_space.mem_closure_iff_symm_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",
"set.nonempty",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space.mem_closure_iff_ball {s : set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → (ball x V ∩ s).nonempty | by simp [mem_closure_iff_nhds_basis' (nhds_basis_uniformity' (𝓤 α).basis_sets)] | lemma | uniform_space.mem_closure_iff_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_space.has_basis_nhds_prod (x y : α) :
has_basis (𝓝 (x, y)) (λ s, s ∈ 𝓤 α ∧ symmetric_rel s) $ λ s, ball x s ×ˢ ball y s | begin
rw nhds_prod_eq,
apply (has_basis_nhds x).prod_same_index (has_basis_nhds y),
rintro U V ⟨U_in, U_symm⟩ ⟨V_in, V_symm⟩,
exact ⟨U ∩ V, ⟨(𝓤 α).inter_sets U_in V_in, U_symm.inter V_symm⟩,
ball_inter_left x U V, ball_inter_right y U V⟩,
end | lemma | uniform_space.has_basis_nhds_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_inter_left",
"ball_inter_right",
"nhds_prod_eq",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_uniformity {x : α} : 𝓝 x = (𝓤 α).lift' (ball x) | (nhds_basis_uniformity' (𝓤 α).basis_sets).eq_binfi | lemma | nhds_eq_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_uniformity' {x : α} : 𝓝 x = (𝓤 α).lift' (λ s, {y | (y, x) ∈ s}) | (nhds_basis_uniformity (𝓤 α).basis_sets).eq_binfi | lemma | nhds_eq_uniformity' | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_left (x : α) {s : set (α×α)} (h : s ∈ 𝓤 α) :
{y : α | (x, y) ∈ s} ∈ 𝓝 x | ball_mem_nhds x h | lemma | mem_nhds_left | 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 | |
mem_nhds_right (y : α) {s : set (α×α)} (h : s ∈ 𝓤 α) :
{x : α | (x, y) ∈ s} ∈ 𝓝 y | mem_nhds_left _ (symm_le_uniformity h) | lemma | mem_nhds_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_nhds_left",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_nhds_ball_subset_of_mem_nhds {a : α} {U : set α} (h : U ∈ 𝓝 a) :
∃ (V ∈ 𝓝 a) (t ∈ 𝓤 α), ∀ a' ∈ V, uniform_space.ball a' t ⊆ U | let ⟨t, ht, htU⟩ := comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 h) in
⟨_, mem_nhds_left a ht, t, ht, λ a₁ h₁ a₂ h₂, @htU (a, a₂) ⟨a₁, h₁, h₂⟩ rfl⟩ | lemma | exists_mem_nhds_ball_subset_of_mem_nhds | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_mem_uniformity_sets",
"mem_nhds_left",
"uniform_space.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.nhds_set_basis_uniformity {p : ι → Prop} {s : ι → set (α × α)}
(hU : (𝓤 α).has_basis p s) {K : set α} (hK : is_compact K) :
(𝓝ˢ K).has_basis p (λ i, ⋃ x ∈ K, ball x (s i)) | begin
refine ⟨λ U, _⟩,
simp only [mem_nhds_set_iff_forall, (nhds_basis_uniformity' hU).mem_iff, Union₂_subset_iff],
refine ⟨λ H, _, λ ⟨i, hpi, hi⟩ x hx, ⟨i, hpi, hi x hx⟩⟩,
replace H : ∀ x ∈ K, ∃ i : {i // p i}, ball x (s i ○ s i) ⊆ U,
{ intros x hx,
rcases H x hx with ⟨i, hpi, hi⟩,
rcases comp_mem_un... | lemma | is_compact.nhds_set_basis_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono",
"comp_mem_uniformity_sets",
"comp_rel_mono",
"is_compact",
"mem_nhds_set_iff_forall",
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.exists_uniform_thickening {A B : set α}
(hA : is_compact A) (hB : is_closed B) (h : disjoint A B) :
∃ V ∈ 𝓤 α, disjoint (⋃ x ∈ A, ball x V) (⋃ x ∈ B, ball x V) | begin
have : Bᶜ ∈ 𝓝ˢ A := hB.is_open_compl.mem_nhds_set.mpr h.le_compl_right,
rw (hA.nhds_set_basis_uniformity (filter.basis_sets _)).mem_iff at this,
rcases this with ⟨U, hU, hUAB⟩,
rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩,
refine ⟨V, hV, set.disjoint_left.mpr $ λ x, _⟩,
simp only... | lemma | disjoint.exists_uniform_thickening | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_symm_mem_uniformity_sets",
"disjoint",
"filter.basis_sets",
"is_closed",
"is_compact",
"mem_ball_symmetry",
"mem_comp_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.exists_uniform_thickening_of_basis {p : ι → Prop} {s : ι → set (α × α)}
(hU : (𝓤 α).has_basis p s) {A B : set α}
(hA : is_compact A) (hB : is_closed B) (h : disjoint A B) :
∃ i, p i ∧ disjoint (⋃ x ∈ A, ball x (s i)) (⋃ x ∈ B, ball x (s i)) | begin
rcases h.exists_uniform_thickening hA hB with ⟨V, hV, hVAB⟩,
rcases hU.mem_iff.1 hV with ⟨i, hi, hiV⟩,
exact ⟨i, hi, hVAB.mono
(Union₂_mono $ λ a _, ball_mono hiV a) (Union₂_mono $ λ b _, ball_mono hiV b)⟩,
end | lemma | disjoint.exists_uniform_thickening_of_basis | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono",
"disjoint",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_right_nhds_uniformity {a : α} : tendsto (λa', (a', a)) (𝓝 a) (𝓤 α) | assume s, mem_nhds_right a | lemma | tendsto_right_nhds_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_nhds_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_left_nhds_uniformity {a : α} : tendsto (λa', (a, a')) (𝓝 a) (𝓤 α) | assume s, mem_nhds_left a | lemma | tendsto_left_nhds_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_nhds_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nhds_left {x : α} {g : set α → filter β} (hg : monotone g) :
(𝓝 x).lift g = (𝓤 α).lift (λs:set (α×α), g (ball x s)) | by { rw [nhds_eq_comap_uniformity, comap_lift_eq2 hg], refl } | lemma | lift_nhds_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"lift",
"monotone",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nhds_right {x : α} {g : set α → filter β} (hg : monotone g) :
(𝓝 x).lift g = (𝓤 α).lift (λs:set (α×α), g {y | (y, x) ∈ s}) | by { rw [nhds_eq_comap_uniformity', comap_lift_eq2 hg], refl } | lemma | lift_nhds_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"lift",
"monotone",
"nhds_eq_comap_uniformity'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_nhds_eq_uniformity_uniformity_prod {a b : α} :
𝓝 a ×ᶠ 𝓝 b =
(𝓤 α).lift (λs:set (α×α), (𝓤 α).lift' (λt:set (α×α),
{y : α | (y, a) ∈ s} ×ˢ {y : α | (b, y) ∈ t})) | begin
rw [nhds_eq_uniformity', nhds_eq_uniformity, prod_lift'_lift'],
exacts [rfl, monotone_preimage, monotone_preimage]
end | lemma | nhds_nhds_eq_uniformity_uniformity_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"lift",
"nhds_eq_uniformity",
"nhds_eq_uniformity'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_uniformity_prod {a b : α} :
𝓝 (a, b) =
(𝓤 α).lift' (λs:set (α×α), {y : α | (y, a) ∈ s} ×ˢ {y : α | (b, y) ∈ s}) | begin
rw [nhds_prod_eq, nhds_nhds_eq_uniformity_uniformity_prod, lift_lift'_same_eq_lift'],
{ intro s, exact monotone_const.set_prod monotone_preimage },
{ intro t, exact monotone_preimage.set_prod monotone_const }
end | lemma | nhds_eq_uniformity_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"monotone_const",
"nhds_nhds_eq_uniformity_uniformity_prod",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhdset_of_mem_uniformity {d : set (α×α)} (s : set (α×α)) (hd : d ∈ 𝓤 α) :
∃(t : set (α×α)), is_open t ∧ s ⊆ t ∧ t ⊆ {p | ∃x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d} | let cl_d := {p:α×α | ∃x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d} in
have ∀p ∈ s, ∃t ⊆ cl_d, is_open t ∧ p ∈ t, from
assume ⟨x, y⟩ hp, _root_.mem_nhds_iff.mp $
show cl_d ∈ 𝓝 (x, y),
begin
rw [nhds_eq_uniformity_prod, mem_lift'_sets],
exact ⟨d, hd, assume ⟨a, b⟩ ⟨ha, hb⟩, ⟨x, y, ha, hp, hb⟩⟩,
exact... | lemma | nhdset_of_mem_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open",
"is_open_Union",
"nhds_eq_uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_le_uniformity (x : α) : 𝓝 (x, x) ≤ 𝓤 α | begin
intros V V_in,
rcases comp_symm_mem_uniformity_sets V_in with ⟨w, w_in, w_symm, w_sub⟩,
have : ball x w ×ˢ ball x w ∈ 𝓝 (x, x),
{ rw nhds_prod_eq,
exact prod_mem_prod (ball_mem_nhds x w_in) (ball_mem_nhds x w_in) },
apply mem_of_superset this,
rintros ⟨u, v⟩ ⟨u_in, v_in⟩,
exact w_sub (mem_comp_... | lemma | nhds_le_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_symm_mem_uniformity_sets",
"mem_comp_of_mem_ball",
"nhds_prod_eq"
] | Entourages are neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supr_nhds_le_uniformity : (⨆ x : α, 𝓝 (x, x)) ≤ 𝓤 α | supr_le nhds_le_uniformity | lemma | supr_nhds_le_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_le_uniformity",
"supr_le"
] | Entourages are neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_set_diagonal_le_uniformity : 𝓝ˢ (diagonal α) ≤ 𝓤 α | (nhds_set_diagonal α).trans_le supr_nhds_le_uniformity | lemma | nhds_set_diagonal_le_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"nhds_set_diagonal",
"supr_nhds_le_uniformity"
] | Entourages are neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_uniformity (s : set $ α × α) :
closure s = ⋂ V ∈ {V | V ∈ 𝓤 α ∧ symmetric_rel V}, V ○ s ○ V | begin
ext ⟨x, y⟩,
simp only [mem_closure_iff_nhds_basis (uniform_space.has_basis_nhds_prod x y), mem_Inter,
mem_set_of_eq, and_imp, mem_comp_comp, exists_prop, ← mem_inter_iff, inter_comm, set.nonempty]
{ contextual := tt }
end | lemma | closure_eq_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"and_imp",
"closure",
"exists_prop",
"mem_closure_iff_nhds_basis",
"mem_comp_comp",
"set.nonempty",
"symmetric_rel",
"uniform_space.has_basis_nhds_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_has_basis_closed : has_basis (𝓤 α) (λ V : set (α × α), V ∈ 𝓤 α ∧ is_closed V) id | begin
refine filter.has_basis_self.2 (λ t h, _),
rcases comp_comp_symm_mem_uniformity_sets h with ⟨w, w_in, w_symm, r⟩,
refine ⟨closure w, mem_of_superset w_in subset_closure, is_closed_closure, _⟩,
refine subset.trans _ r,
rw closure_eq_uniformity,
apply Inter_subset_of_subset,
apply Inter_subset,
exac... | lemma | uniformity_has_basis_closed | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"closure_eq_uniformity",
"comp_comp_symm_mem_uniformity_sets",
"is_closed",
"is_closed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_uniformity_closure : 𝓤 α = (𝓤 α).lift' closure | eq.symm $ uniformity_has_basis_closed.lift'_closure_eq_self $ λ _, and.right | lemma | uniformity_eq_uniformity_closure | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniformity_closure {p : ι → Prop} {U : ι → set (α × α)}
(h : (𝓤 α).has_basis p U) : (𝓤 α).has_basis p (λ i, closure (U i)) | (@uniformity_eq_uniformity_closure α _).symm ▸ h.lift'_closure | lemma | filter.has_basis.uniformity_closure | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"closure",
"uniformity_eq_uniformity_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_has_basis_closure : has_basis (𝓤 α) (λ V : set (α × α), V ∈ 𝓤 α) closure | (𝓤 α).basis_sets.uniformity_closure | lemma | uniformity_has_basis_closure | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"closure"
] | Closed entourages form a basis of the uniformity filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_inter_uniformity {t : set (α×α)} :
closure t = (⋂ d ∈ 𝓤 α, d ○ (t ○ d)) | calc closure t = ⋂ V (hV : V ∈ 𝓤 α ∧ symmetric_rel V), V ○ t ○ V : closure_eq_uniformity t
... = ⋂ V ∈ 𝓤 α, V ○ t ○ V : eq.symm $ uniform_space.has_basis_symmetric.bInter_mem $
λ V₁ V₂ hV, comp_rel_mono (comp_rel_mono hV subset.rfl) hV
... = ⋂ V ∈ 𝓤 α, V ○ (t ○ V) : by simp only [comp_rel_assoc] | lemma | closure_eq_inter_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"closure",
"closure_eq_uniformity",
"comp_rel_assoc",
"comp_rel_mono",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_eq_uniformity_interior : 𝓤 α = (𝓤 α).lift' interior | le_antisymm
(le_infi $ assume d, le_infi $ assume hd,
let ⟨s, hs, hs_comp⟩ := (mem_lift'_sets $
monotone_id.comp_rel $ monotone_id.comp_rel monotone_id).mp
(comp_le_uniformity3 hd) in
let ⟨t, ht, hst, ht_comp⟩ := nhdset_of_mem_uniformity s hs in
have s ⊆ interior d, from
calc s ⊆ t : h... | lemma | uniformity_eq_uniformity_interior | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_le_uniformity3",
"interior",
"interior_subset",
"le_infi",
"monotone_id",
"nhdset_of_mem_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_mem_uniformity {s : set (α × α)} (hs : s ∈ 𝓤 α) :
interior s ∈ 𝓤 α | by rw [uniformity_eq_uniformity_interior]; exact mem_lift' hs | lemma | interior_mem_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"interior",
"uniformity_eq_uniformity_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity_is_closed {s : set (α×α)} (h : s ∈ 𝓤 α) :
∃t ∈ 𝓤 α, is_closed t ∧ t ⊆ s | let ⟨t, ⟨ht_mem, htc⟩, hts⟩ := uniformity_has_basis_closed.mem_iff.1 h in
⟨t, ht_mem, htc, hts⟩ | lemma | mem_uniformity_is_closed | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.