English
In a topological monoid with continuous multiplication, the neighborhood of 1 multiplied by the neighborhood of a equals the neighborhood of a.
Русский
В топологическом моноиде с непрерывным умножением окружность 1 умножается на окружность a, давая окружность a.
LaTeX
$$$\mathcal{N}(1) * \mathcal{N}(a) = \mathcal{N}(a)$$$
Lean4
/-- If `f : α → M` and `g : β → M` both tend to zero on the cofinite filter, then so does
`fun i : α × β ↦ f i.1 * g i.2`. -/
theorem tendsto_mul_cofinite_nhds_zero {f : α → M} {g : β → M} (hf : Tendsto f cofinite (𝓝 0))
(hg : Tendsto g cofinite (𝓝 0)) : Tendsto (fun i : α × β ↦ f i.1 * g i.2) cofinite (𝓝 0) :=
by
letI : TopologicalSpace α := ⊥
haveI : DiscreteTopology α := discreteTopology_bot α
letI : TopologicalSpace β := ⊥
haveI : DiscreteTopology β := discreteTopology_bot β
rw [← cocompact_eq_cofinite] at *
exact tendsto_mul_cocompact_nhds_zero continuous_of_discreteTopology continuous_of_discreteTopology hf hg
