Theorems · Theorem · general topology

exists_continuous_zero_one_of_isCompact

∀ {X : Type u_1} [inst : TopologicalSpace X] [RegularSpace X] [LocallyCompactSpace X] {s t : Set X},
  IsCompact s → IsClosed t → Disjoint s t → ∃ f, Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1

Urysohn's lemma: if s and t are two disjoint sets in a regular locally compact topological space X, with s compact and t closed, then there exists a continuous function f : X → ℝ such that * f equals zero on s; * f equals one on t; * 0 ≤ f x ≤ 1 for all x.

Defined in

Mathlib.Topology.UrysohnsLemma

Cited by

2 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceRegularSpaceLocallyCompactSpace