English
A map is proper iff it is continuous and the induced map on X × Filter X → Y × Filter X is closed.
Русский
Отображение пропорно тогда и только тогда, когда оно непрерывно и отображение на \(X×Filter X→Y×Filter X\) замкнуто.
LaTeX
$$IsProperMap f ↔ Continuous f ∧ IsClosedMap (Prod.map f id : X × Filter X → Y × Filter X)$$
Lean4
@[to_additive]
instance lipschitzMul : LipschitzMul βᵐᵒᵖ where
lipschitz_mul :=
⟨LipschitzMul.C β, fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ =>
(lipschitzWith_lipschitz_const_mul_edist ⟨x₂.unop, x₁.unop⟩ ⟨y₂.unop, y₁.unop⟩).trans_eq
(congr_arg _ <| max_comm _ _)⟩
-- this instance could be deduced from `NormedAddCommGroup.lipschitzAdd`, but we prove it
-- separately here so that it is available earlier in the hierarchy
