mathlib documentation

topology.list

Topology on lists and vectors #

theorem nhds_list {α : Type u_1} [topological_space α] (as : list α) :
theorem nhds_cons {α : Type u_1} [topological_space α] (a : α) (l : list α) :
theorem list.tendsto_cons {α : Type u_1} [topological_space α] {a : α} {l : list α} :
filter.tendsto (λ (p : α × list α), p.fst :: p.snd) ((nhds a).prod (nhds l)) (nhds (a :: l))
theorem filter.tendsto.cons {β : Type u_2} [topological_space β] {α : Type u_1} {f : α β} {g : α list β} {a : filter α} {b : β} {l : list β} (hf : filter.tendsto f a (nhds b)) (hg : filter.tendsto g a (nhds l)) :
filter.tendsto (λ (a : α), f a :: g a) a (nhds (b :: l))
theorem list.tendsto_cons_iff {α : Type u_1} [topological_space α] {β : Type u_2} {f : list α β} {b : filter β} {a : α} {l : list α} :
filter.tendsto f (nhds (a :: l)) b filter.tendsto (λ (p : α × list α), f (p.fst :: p.snd)) ((nhds a).prod (nhds l)) b
theorem list.continuous_cons {α : Type u_1} [topological_space α] :
continuous (λ (x : α × list α), x.fst :: x.snd)
theorem list.tendsto_nhds {α : Type u_1} [topological_space α] {β : Type u_2} {f : list α β} {r : list α filter β} (h_nil : filter.tendsto f (has_pure.pure list.nil) (r list.nil)) (h_cons : (l : list α) (a : α), filter.tendsto f (nhds l) (r l) filter.tendsto (λ (p : α × list α), f (p.fst :: p.snd)) ((nhds a).prod (nhds l)) (r (a :: l))) (l : list α) :
filter.tendsto f (nhds l) (r l)
theorem list.tendsto_insert_nth' {α : Type u_1} [topological_space α] {a : α} {n : } {l : list α} :
filter.tendsto (λ (p : α × list α), list.insert_nth n p.fst p.snd) ((nhds a).prod (nhds l)) (nhds (list.insert_nth n a l))
theorem list.tendsto_insert_nth {α : Type u_1} [topological_space α] {β : Type u_2} {n : } {a : α} {l : list α} {f : β α} {g : β list α} {b : filter β} (hf : filter.tendsto f b (nhds a)) (hg : filter.tendsto g b (nhds l)) :
filter.tendsto (λ (b : β), list.insert_nth n (f b) (g b)) b (nhds (list.insert_nth n a l))
theorem list.continuous_insert_nth {α : Type u_1} [topological_space α] {n : } :
continuous (λ (p : α × list α), list.insert_nth n p.fst p.snd)
theorem list.tendsto_remove_nth {α : Type u_1} [topological_space α] {n : } {l : list α} :
filter.tendsto (λ (l : list α), l.remove_nth n) (nhds l) (nhds (l.remove_nth n))
theorem list.continuous_remove_nth {α : Type u_1} [topological_space α] {n : } :
continuous (λ (l : list α), l.remove_nth n)
theorem vector.tendsto_cons {α : Type u_1} [topological_space α] {n : } {a : α} {l : vector α n} :
filter.tendsto (λ (p : α × vector α n), p.fst ::ᵥ p.snd) ((nhds a).prod (nhds l)) (nhds (a ::ᵥ l))
theorem vector.tendsto_insert_nth {α : Type u_1} [topological_space α] {n : } {i : fin (n + 1)} {a : α} {l : vector α n} :
filter.tendsto (λ (p : α × vector α n), vector.insert_nth p.fst i p.snd) ((nhds a).prod (nhds l)) (nhds (vector.insert_nth a i l))
theorem vector.continuous_insert_nth' {α : Type u_1} [topological_space α] {n : } {i : fin (n + 1)} :
continuous (λ (p : α × vector α n), vector.insert_nth p.fst i p.snd)
theorem vector.continuous_insert_nth {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {n : } {i : fin (n + 1)} {f : β α} {g : β vector α n} (hf : continuous f) (hg : continuous g) :
continuous (λ (b : β), vector.insert_nth (f b) i (g b))
theorem vector.continuous_at_remove_nth {α : Type u_1} [topological_space α] {n : } {i : fin (n + 1)} {l : vector α (n + 1)} :