Theorems · Theorem · general topology
isClopen_discrete
∀ {X : Type u} [inst : TopologicalSpace X] [DiscreteTopology X] (s : Set X), IsClopen s- Defined in
- Mathlib.Topology.Clopen
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- DiscreteTopologystatement and proof · cited by 373
- IsClopenstatement · cited by 189
- isOpen_discreteproof · cited by 36
- isClosed_discreteproof · cited by 9
Cited by8
Results whose statement or proof uses this declaration.
- preconnectedSpace_iff_clopenproof · cited by 3
- DiscreteQuotient.isClopen_preimageproof · cited by 3
- PreconnectedSpace.trivial_of_discreteproof · cited by 2
- Profinite.exists_lift_of_finite_of_injective_of_surjectiveproof · cited by 1
- IsOpenMap.enatCard_connectedComponents_le_encard_preimage_singletonproof · cited by 1
- IsOpenMap.finite_connectedComponents_of_finite_preimage_singletonproof · cited by 0