English
The cited simp lemmas characterize the components of the finite product construction, ensuring the product behaves well with respect to source/target openness and continuity in the multivariate setting.
Русский
Указанные упрощённые леммы описывают компоненты конечного произведения, обеспечивая корректное поведение источника/м_directs цели и непрерывности в многомерном случае.
LaTeX
$$$\text{IsOpen}(\text{Set.univ.pi}...)$$$
Lean4
/-- Continuity at a point can be read under left composition with an open partial homeomorphism if a
neighborhood of the initial point is sent to the source of the partial homeomorphism -/
theorem continuousAt_iff_continuousAt_comp_left {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x) :
ContinuousAt f x ↔ ContinuousAt (e ∘ f) x :=
by
have hx : f x ∈ e.source := (mem_of_mem_nhds h :)
have h' : f ⁻¹' e.source ∈ 𝓝[univ] x := by rwa [nhdsWithin_univ]
rw [← continuousWithinAt_univ, ← continuousWithinAt_univ, e.continuousWithinAt_iff_continuousWithinAt_comp_left hx h']
