instT2Space — Mathlib

English

The largest T2 quotient of a topological space is indeed T2.

Русский

Наибольший T2-квартионал пространства действительно является Хаусдорфовым.

LaTeX

$$$ T2Space(T2Quotient\\ X)$$$

Lean4

/-- The largest T2 quotient of a topological space is indeed T2. -/
instance : T2Space (T2Quotient X) := by
  rw [t2Space_iff]
  rintro ⟨x⟩ ⟨y⟩ (h : ¬T2Quotient.mk x = T2Quotient.mk y)
  obtain ⟨s, hs, hsxy⟩ : ∃ s, T2Space (Quotient s) ∧ Quotient.mk s x ≠ Quotient.mk s y := by
    simpa [T2Quotient.mk_eq] using h
  exact separated_by_continuous (continuous_map_sInf (by exact hs)) hsxy

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