isDetecting_iff_isSeparating — Mathlib

English

Under HasEqualizers and Balanced, IsDetecting 𝒢 holds if and only if IsSeparating 𝒢 holds.

Русский

При условии существования равноположителей и сбалансированности, IsDetecting 𝒢 эквивалентно IsSeparating 𝒢.

LaTeX

$$$IsDetecting(\mathcal{G}) \;\iff\; IsSeparating(\mathcal{G})$$$

Lean4

theorem isDetecting_iff_isSeparating [HasEqualizers C] [Balanced C] (𝒢 : Set C) : IsDetecting 𝒢 ↔ IsSeparating 𝒢 :=
  ⟨IsDetecting.isSeparating, IsSeparating.isDetecting⟩

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