extend_symm_continuousWithinAt_comp_right_iff

English

The preimage-intersection of the extended map with its inverse preserves source∩image structure.

Русский

Сохранение структуры прообраза и образа при расширенном отображении и его обратном отображении.

LaTeX

$$$ \\text{Extend image preimage property: } (f^{\\mathrm{extend}}_I)^{-1}[s] \\cap (f^{\\mathrm{extend}}_I).\\mathrm{source} \\text{ behaves well under extension.}$$$

Lean4

theorem extend_symm_continuousWithinAt_comp_right_iff {X} [TopologicalSpace X] {g : M → X} {s : Set M} {x : M} :
    ContinuousWithinAt (g ∘ (f.extend I).symm) ((f.extend I).symm ⁻¹' s ∩ range I) (f.extend I x) ↔
      ContinuousWithinAt (g ∘ f.symm) (f.symm ⁻¹' s) (f x) :=
  by rw [← I.symm_continuousWithinAt_comp_right_iff]; rfl

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