Theorems · Theorem · general topology
IsPreconnected.infinite_of_nontrivial
∀ {X : Type u_1} [inst : TopologicalSpace X] [T1Space X] {s : Set X}, IsPreconnected s → s.Nontrivial → s.Infinite- Defined in
- Mathlib.Topology.Separation.Connected
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT1Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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.Elemproof · cited by 7,166
- Set.Finiteproof · cited by 1,814
- DiscreteTopologyproof · cited by 373
- T1Spacestatement and proof · cited by 275
- Set.Infinitestatement · cited by 263
- IsPreconnectedstatement and proof · cited by 205
- Set.Nontrivialstatement and proof · cited by 145
- Set.subsingleton_coeproof · cited by 12
- Set.not_subsingleton_iffproof · cited by 7
- PreconnectedSpace.trivial_of_discreteproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- PreconnectedSpace.infiniteproof · cited by 0