Theorems · Theorem · general topology

exists_tsupport_one_of_isOpen_isClosed

∀ {X : Type u_1} [inst : TopologicalSpace X] [R1Space X] {s t : Set X},
  IsOpen s →
    IsCompact (closure s) → IsClosed t → t ⊆ s → ∃ f, tsupport ⇑f ⊆ s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1

A variation of Urysohn's lemma. In a R1Space X, for a closed set t and a relatively compact open set s such that t ⊆ s, there is a continuous function f supported in s, f x = 1 on t and 0 ≤ f x ≤ 1.

Defined in

Mathlib.Topology.UrysohnsLemma

Cited by

2 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceR1Space