Theorems · Theorem · general topology
IsPreconnected.isDiscrete_iff_subsingleton
∀ {α : Type u} [inst : TopologicalSpace α] {S : Set α}, IsPreconnected S → (IsDiscrete S ↔ S.Subsingleton)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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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
- DiscreteTopologyproof · cited by 373
- Set.Subsingletonstatement and proof · cited by 276
- IsPreconnectedstatement and proof · cited by 205
- IsDiscretestatement and proof · cited by 86
- PreconnectedSpaceproof · cited by 64
- isDiscrete_iff_discreteTopologyproof · cited by 6
- isPreconnected_iff_preconnectedSpaceproof · cited by 4
- subsingleton_of_preconnected_totallyDisconnectedproof · cited by 1
- Set.Subsingleton.isDiscreteproof · cited by 1
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