unique_uniformity_of_compact — Mathlib

English

If K is compact and U is open with K ⊆ U, then there exists an entourage V such that balls around K of size V lie inside U.

Русский

Если K компактно и U открыто с K ⊆ U, существует окружность V с шариками вокруг K размерности V лежат внутри U.

LaTeX

$$$$ \\exists V \\in 𝓤 α, \\; IsOpen V \\wedge \\forall x\\in K, B(x,V) \\subseteq U. $$$$

Lean4

theorem unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ] {u u' : UniformSpace γ}
    (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) : u = u' :=
  by
  refine UniformSpace.ext ?_
  have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
  have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
  rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h']

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