English
If U is open in X × X and T is continuous, then for every n the dynamical entourage dynEntourage(T,U,n) is open.
Русский
Если U открыто в X×X и T непрерывно, то для каждого n динамическое окрестность dynEntourage(T,U,n) открыта.
LaTeX
$$$\text{IsOpen}(\operatorname{dynEntourage}(T,U,n))$ for all n, given U открыто и T непрерывно.$$
Lean4
theorem _root_.isOpen.dynEntourage [TopologicalSpace X] {T : X → X} (T_cont : Continuous T) {U : Set (X × X)}
(U_open : IsOpen U) (n : ℕ) : IsOpen (dynEntourage T U n) :=
by
rw [dynEntourage_eq_inter_Ico T U n]
refine isOpen_iInter_of_finite fun k ↦ ?_
exact U_open.preimage ((T_cont.prodMap T_cont).iterate k)