Theorems · Theorem · general topology
IsClopen.isClosed
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClopen s → IsClosed s- Defined in
- Mathlib.Topology.Clopen
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 4 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
- IsClosedstatement · cited by 1,639
- IsClopenstatement and proof · cited by 189
Cited by12
Results whose statement or proof uses this declaration.
- TopologicalSpace.NonemptyCompacts.isClosedEmbedding_toCompactsproof · cited by 5
- MDifferentiable.isLocallyConstantproof · cited by 2
- PrimeSpectrum.isOpen_singleton_tfae_of_isNoetherian_of_isJacobsonRingproof · cited by 2
- ContinuousMap.isClopen_setOfPred_mapsToproof · cited by 2
- IsTopologicalAddGroup.exist_add_closure_nhdsproof · cited by 1
- Ideal.Fiber.lift_residueField_surjectiveproof · cited by 1
- IsTopologicalGroup.exist_mul_closure_nhdsproof · cited by 1
- TopologicalSpace.Clopens.isClosedproof · cited by 0
- TopologicalSpace.Closeds.compactSpace_iffproof · cited by 0
- TopologicalSpace.NonemptyCompacts.preconnectedSpace_iffproof · cited by 0
- Valued.isClosed_sphereproof · cited by 0
- Valuation.isClosed_sphereproof · cited by 0