Theorems · Definition · general topology
IsPreirreducible
{X : Type u_1} → [TopologicalSpace X] → Set X → PropA preirreducible set s is one where there is no non-trivial pair of disjoint opens on s.
- Defined in
- Mathlib.Topology.Irreducible
- Cited by
- 43 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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.Nonemptyproof · cited by 2,627
- IsOpenproof · cited by 2,400
Cited by47
Results whose statement or proof uses this declaration.
- IsIrreducibleproof · cited by 59
- IsIrreducible.isPreirreduciblestatement · cited by 7
- PreirreducibleSpace.isPreirreducible_univstatement · cited by 7
- subset_closure_inter_of_isPreirreducible_of_isOpenstatement and proof · cited by 6
- Set.Subsingleton.isPreirreduciblestatement · cited by 4
- IsPreirreducible.imagestatement and proof · cited by 3
- IsPreirreducible.of_subtypestatement and proof · cited by 3
- isPreirreducible_emptystatement · cited by 3
- IsPreirreducible.preimage_of_isPreirreducible_fiberstatement and proof · cited by 3
- IsIrreducible.preimage_of_isPreirreducible_fiberstatement and proof · cited by 3
- irreducibleComponent_propertystatement · cited by 3
- TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducibleproof · cited by 2