eventually_nhds_iff — Mathlib

English

A T3 space with second-countable topology can be embedded into ℓ∞(ℕ, ℝ).

Русский

Т3-пространство с топологией второй счетности можно встроить в ℓ∞(ℕ, ℝ).

LaTeX

$$$\\exists f: X \\to \\ell^{\\infty}(\\mathbb{N},\\mathbb{R}),\\; \\text{IsEmbedding}(f)$$$

Lean4

/-- A predicate is true in a neighborhood of `x` iff it is true for all the points in an open set
containing `x`. -/
theorem eventually_nhds_iff {p : X → Prop} : (∀ᶠ y in 𝓝 x, p y) ↔ ∃ t : Set X, (∀ y ∈ t, p y) ∧ IsOpen t ∧ x ∈ t :=
  mem_nhds_iff.trans <| by simp only [subset_def, mem_setOf_eq]

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