Theorems · Theorem · general topology
connectedSpace_iff_connectedComponent
∀ {α : Type u} [inst : TopologicalSpace α], ConnectedSpace α ↔ ∃ x, connectedComponent x = Set.univ- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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Cites13
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 and proof · cited by 3,945
- Set.mem_univproof · cited by 416
- IsPreconnectedproof · cited by 205
- connectedComponentstatement and proof · cited by 68
- PreconnectedSpaceproof · cited by 64
- ConnectedSpacestatement and proof · cited by 37
- PreconnectedSpace.isPreconnected_univproof · cited by 26
- isPreconnected_connectedComponentproof · cited by 12
- IsPreconnected.subset_connectedComponentproof · cited by 10
- Set.eq_univ_of_univ_subsetproof · cited by 7
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