Theorems · Theorem · general topology
preconnectedSpace_iff_clopen
∀ {α : Type u} [inst : TopologicalSpace α], PreconnectedSpace α ↔ ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = Set.univ- Defined in
- Mathlib.Topology.Connected.Clopen
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 79 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.
Cites15
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.preimageproof · cited by 4,946
- Set.univstatement and proof · cited by 3,945
- Compl.complproof · cited by 2,925
- Continuousproof · cited by 2,592
- IsClopenstatement and proof · cited by 189
- PreconnectedSpacestatement and proof · cited by 64
- Set.preimage_complproof · cited by 43
- Set.compl_emptyproof · cited by 33
- IsClopen.preimageproof · cited by 10
- isClopen_discreteproof · cited by 8
Cited by3
Results whose statement or proof uses this declaration.
- connectedSpace_iff_clopenproof · cited by 0
- ExtremallyDisconnected.toPreirreducibleSpaceproof · cited by 0
- TopologicalSpace.NonemptyCompacts.preconnectedSpace_iffproof · cited by 0