orderClosedTopology — Mathlib

English

The topology on Γ₀ is T5 (completely normal).

Русский

Топология Γ₀ — это T5 (полностью нормальная).

LaTeX

$$$\text{T5Space}(Γ_0)$$$

Lean4

/-- The topology on a linearly ordered group with zero element adjoined is compatible with the order
structure: the set `{p : Γ₀ × Γ₀ | p.1 ≤ p.2}` is closed. -/
@[nolint defLemma]
scoped instance (priority := 100) orderClosedTopology : OrderClosedTopology Γ₀ where
  isClosed_le' := by
    simp only [← isOpen_compl_iff, compl_setOf, not_le, isOpen_iff_mem_nhds]
    rintro ⟨a, b⟩ (hab : b < a)
    rw [nhds_prod_eq, nhds_of_ne_zero (zero_le'.trans_lt hab).ne', pure_prod]
    exact Iio_mem_nhds hab

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