mathlib documentation

topology.instances.real

Topological properties of ℝ #

@[protected, instance]
noncomputable def rat.metric_space  :
Equations
theorem rat.dist_eq (x y : ) :
dist x y = |x - y|
@[simp, norm_cast]
theorem rat.dist_cast (x y : ) :
dist x y = dist x y
@[protected, instance]
noncomputable def int.has_dist  :
Equations
theorem int.dist_eq (x y : ) :
dist x y = |x - y|
@[simp, norm_cast]
theorem int.dist_cast_real (x y : ) :
dist x y = dist x y
@[simp, norm_cast]
theorem int.dist_cast_rat (x y : ) :
dist x y = dist x y
theorem int.pairwise_one_le_dist  :
pairwise (λ (m n : ), 1 dist m n)
theorem int.ball_eq_Ioo (x : ) (r : ) :
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[protected, instance]
theorem real.mem_closure_iff {s : set } {x : } :
x closure s ∀ (ε : ), ε > 0(∃ (y : ) (H : y s), |y - x| < ε)
theorem real.uniform_continuous_inv (s : set ) {r : } (r0 : 0 < r) (H : ∀ (x : ), x sr |x|) :
theorem real.tendsto_inv {r : } (r0 : r 0) :
filter.tendsto (λ (q : ), q⁻¹) (𝓝 r) (𝓝 r⁻¹)
theorem real.continuous_inv  :
continuous (λ (a : {r // r 0}), (a.val)⁻¹)
theorem real.continuous.inv {α : Type u} [topological_space α] {f : α → } (h : ∀ (a : α), f a 0) (hf : continuous f) :
continuous (λ (a : α), (f a)⁻¹)
theorem real.uniform_continuous_mul (s : set ( × )) {r₁ r₂ : } (H : ∀ (x : × ), x s|x.fst| < r₁ |x.snd| < r₂) :
uniform_continuous (λ (p : s), (p.val.fst) * p.val.snd)
@[protected]
theorem real.continuous_mul  :
continuous (λ (p : × ), (p.fst) * p.snd)
@[protected, instance]
theorem rat.continuous_mul  :
continuous (λ (p : × ), (p.fst) * p.snd)
@[protected, instance]
@[protected, instance]
theorem closure_of_rat_image_lt {q : } :
closure (coe '' {x : | q < x}) = {r : | q r}
theorem function.periodic.compact_of_continuous' {α : Type u} [topological_space α] {f : → α} {c : } (hp : function.periodic f c) (hc : 0 < c) (hf : continuous f) :
theorem function.periodic.compact_of_continuous {α : Type u} [topological_space α] {f : → α} {c : } (hp : function.periodic f c) (hc : c 0) (hf : continuous f) :

A continuous, periodic function has compact range.

theorem function.periodic.bounded_of_continuous {α : Type u} [pseudo_metric_space α] {f : → α} {c : } (hp : function.periodic f c) (hc : c 0) (hf : continuous f) :

A continuous, periodic function is bounded.

theorem real.subgroup_dense_of_no_min {G : add_subgroup } {g₀ : } (g₀_in : g₀ G) (g₀_ne : g₀ 0) (H' : ¬∃ (a : ), is_least {g : | g G 0 < g} a) :

Given a nontrivial subgroup G ⊆ ℝ, if G ∩ ℝ_{>0} has no minimum then G is dense.

Subgroups of are either dense or cyclic. See real.subgroup_dense_of_no_min and subgroup_cyclic_of_min for more precise statements.