Theorems · Theorem · general topology
IsClopen.eq_univ
∀ {α : Type u} [inst : TopologicalSpace α] [PreconnectedSpace α] {s : Set α}, IsClopen s → s.Nonempty → s = Set.univ- Defined in
- Mathlib.Topology.Connected.Clopen
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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 · cited by 3,945
- Set.Nonemptystatement and proof · cited by 2,627
- IsClopenstatement and proof · cited by 189
- Set.Nonempty.ne_emptyproof · cited by 65
- PreconnectedSpacestatement and proof · cited by 64
- isClopen_iffproof · cited by 6
Cited by5
Results whose statement or proof uses this declaration.
- IsSeparatedMap.eq_of_comp_eqproof · cited by 4
- PreconnectedSpace.induction₂'proof · cited by 2
- PreconnectedSpace.trivial_of_discreteproof · cited by 2
- PathConnectedSpace.of_locallyPathConnectedSpaceproof · cited by 2
- IsOpenMap.enatCard_connectedComponents_le_encard_preimage_singletonproof · cited by 1