English
If each C i has ContinuousStar and Star, then the function space ∀ i, C i has ContinuousStar.
Русский
Если каждый C i имеет ContinuousStar и Star, то пространство функций ∀ i, C i имеет ContinuousStar.
LaTeX
$$$\mathrm{ContinuousStar}\bigl((i:\!\iota) \to C(i)\bigr)$$$
Lean4
instance {C : ι → Type*} [∀ i, TopologicalSpace (C i)] [∀ i, Star (C i)] [∀ i, ContinuousStar (C i)] :
ContinuousStar (∀ i, C i) where continuous_star := continuous_pi fun i => Continuous.star (continuous_apply i)