Theorems · Theorem · general topology

lebesgue_number_lemma

∀ {α : Type ua} [inst : UniformSpace α] {K : Set α} {ι : Sort u_2} {U : ι → Set α},
  IsCompact K → (∀ (i : ι), IsOpen (U i)) → K ⊆ ⋃ i, U i → ∃ V ∈ uniformity α, ∀ x ∈ K, ∃ i, UniformSpace.ball x V ⊆ U i

Let c : ι → Set α be an open cover of a compact set s. Then there exists an entourage n such that for each x ∈ s its n-neighborhood is contained in some c i.

Defined in

Mathlib.Topology.UniformSpace.Compact

Cited by

4 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpace