English
If β is a topological space with continuous multiplication and α has the induced topology via f: α → β, then α also has a continuous multiplication, i.e., a topological monoid structure induced from β.
Русский
Если β — топологическое пространство с непрерывным умножением, а α снабжено индуицированной топологией через f: α → β, то α тоже имеет непрерывное умножение (индуцированное сверху).
LaTeX
$$$\text{ContinuousMul}$ (induced $\alpha$ via $f$) ⇒ $\text{ContinuousMul}(\alpha)$$$
Lean4
theorem const_mul [PosMulStrictMono 𝕜] (h : Tendsto f l (𝓝[<] c)) : Tendsto (fun a => b * f a) l (𝓝[<] (b * c)) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ ((tendsto_nhds_of_tendsto_nhdsWithin h).const_mul b) <|
(tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Iio] at *; gcongr; assumption
