English
In a quotient setting, the lift map Quotient.lift f hs is continuous when f is continuous and hs respects the quotient.
Русский
В условиях факторного пространства подъём Quotient.lift f hs непрерывен при условии непрерывности f и соответствия hs эквивалентности.
LaTeX
$$$$ \mathrm{Continuous}(\operatorname{Quotient.lift}(f, hs)) $$$$
Lean4
theorem nhds_pi {a : ∀ i, A i} : 𝓝 a = pi fun i => 𝓝 (a i) := by simp only [nhds_iInf, nhds_induced, Filter.pi]