Theorems · Theorem · general topology
isOpen_empty
∀ {X : Type u} [inst : TopologicalSpace X], IsOpen ∅- Defined in
- Mathlib.Topology.Basic
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- IsOpenstatement and proof · cited by 2,400
- Set.sUnion_emptyproof · cited by 17
- isOpen_sUnionproof · cited by 15
Cited by24
Results whose statement or proof uses this declaration.
- MeasureTheory.Content.outerMeasureproof · cited by 24
- t1Space_TFAEproof · cited by 8
- nhdsSet_emptyproof · cited by 7
- MeasurableSet.induction_on_openproof · cited by 7
- interior_emptyproof · cited by 5
- isOpenMap_sigmaMkproof · cited by 5
- IsEvenlyCovered.of_preimage_eq_emptyproof · cited by 4
- Topology.IsConstructible.empty_union_inductionproof · cited by 3
- Profinite.exists_locallyConstantproof · cited by 2
- TopologicalSpace.vietoris.isClopen_singleton_emptyproof · cited by 2
- MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOnproof · cited by 2
- IndiscreteTopology.isOpen_iffproof · cited by 2