ofIsClopen_fiber_one — Mathlib

English

If U is clopen, the fiber at 1 is the complement Uᶜ.

Русский

Если U клинопеременный, то волокно при 1 равно композиции Uᶜ.

LaTeX

$$$\\mathrm{Set.preimage}(\\mathrm{LocallyConstant.instFunLike.coe}(\\mathrm{LocallyConstant.ofIsClopen}~hU), \\{1\\}) = U^{c}$$$

Lean4

@[simp]
theorem ofIsClopen_fiber_one {X : Type*} [TopologicalSpace X] {U : Set X} [∀ x, Decidable (x ∈ U)] (hU : IsClopen U) :
    ofIsClopen hU ⁻¹' ({ 1 } : Set (Fin 2)) = Uᶜ := by
  ext
  simp only [ofIsClopen, mem_singleton_iff, coe_mk, Fin.zero_eq_one_iff, mem_preimage, ite_eq_right_iff, mem_compl_iff,
    Nat.succ_succ_ne_one]

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