English
For a finite index set ι and families X_i, Y_i with each X_i, Y_i carrying a topology, the product of a finite family of OpenPartialHomeomorphs (ei) is defined as an OpenPartialHomeomorph between the product spaces, with the natural projections and continuity properties.
Русский
Для конечного множества индексов ι и семейств X_i, Y_i с топологиями, произведение конечной совокупности OpenPartialHomeomorphs defnируется как OpenPartialHomeomorph между произведениями пространств с естественными проекции и свойствами непрерывности.
LaTeX
$$$\pi : OpenPartialHomeomorph (\prod_{i\in ι} X_i) (\prod_{i\in ι} Y_i)$, with structure induced by ei : OpenPartialHomeomorph (X_i) (Y_i).$$
Lean4
/-- Continuity within a set at a point can be read under left composition with a local
homeomorphism if a neighborhood of the initial point is sent to the source of the local
homeomorphism -/
theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : Set Z} {x : Z} (hx : f x ∈ e.source)
(h : f ⁻¹' e.source ∈ 𝓝[s] x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt (e ∘ f) s x :=
by
refine ⟨(e.continuousAt hx).comp_continuousWithinAt, fun fe_cont => ?_⟩
rw [← continuousWithinAt_inter' h] at fe_cont ⊢
have : ContinuousWithinAt (e.symm ∘ e ∘ f) (s ∩ f ⁻¹' e.source) x :=
haveI : ContinuousWithinAt e.symm univ (e (f x)) := (e.continuousAt_symm (e.map_source hx)).continuousWithinAt
ContinuousWithinAt.comp this fe_cont (subset_univ _)
exact this.congr (fun y hy => by simp [e.left_inv hy.2]) (by simp [e.left_inv hx])
