Theorems · Theorem · general topology
nonempty_interior_of_iUnion_of_closed
∀ {X : Type u_1} {ι : Sort u_3} [inst : TopologicalSpace X] [BaireSpace X] [Nonempty X] [Countable ι] {f : ι → Set X},
(∀ (i : ι), IsClosed (f i)) → ⋃ i, f i = Set.univ → ∃ i, (interior (f i)).NonemptyOne of the most useful consequences of Baire theorem: if a countable union of closed sets covers the space, then one of the sets has nonempty interior.
- Defined in
- Mathlib.Topology.Baire.Lemmas
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.univstatement and proof · cited by 3,945
- Set.Nonemptystatement · cited by 2,627
- Set.iUnionstatement and proof · cited by 2,483
- IsClosedstatement and proof · cited by 1,639
- interiorstatement · cited by 714
- Countablestatement and proof · cited by 633
- BaireSpacestatement and proof · cited by 36
- Dense.nonemptyproof · cited by 4
- dense_iUnion_interior_of_closedproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.exists_approx_preimage_norm_leproof · cited by 1
- smul_singleton_mem_nhds_of_sigmaCompactproof · cited by 1
- vadd_singleton_mem_nhds_of_sigmaCompactproof · cited by 1