Theorems · Theorem · general topology

dense_sUnion_interior_of_closed

∀ {X : Type u_1} [inst : TopologicalSpace X] [BaireSpace X] {S : Set (Set X)},
  (∀ s ∈ S, IsClosed s) → S.Countable → ⋃₀ S = Set.univ → Dense (⋃ s ∈ S, interior s)

Baire theorem: if countably many closed sets cover the whole space, then their interiors are dense. Formulated here with ⋃₀.

Defined in

Mathlib.Topology.Baire.Lemmas

Cited by

0 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBaireSpace